Some observations on the Lekkerkerker-Zeckendorf decomposition of integers

In our youth, we learned of a nice arithmetic theorem of Lekkerkerker (more popularly known after Zeckendorf; hereinafter L-Z) that relates to the famous Mātrā-meru sequence M: 0, 1, 1, 2, 3, 5, 8… defined by the recurrence relationship f[n+2]=f[n+1]+f[n]. The theorem states that all positive integers can be uniquely expressed as a sum of one or more distinct non-consecutive terms of M. A proof for this theorem can be visualized through a simple geometric construction (Figure 1).

Zeckendorf_decompositionThe graphical L-Z decomposition of integers from 1..12

Pile rectangles whose sides are two successive terms of M so as to make a n \times n half-square (Figure 1). One can see that every integer can be reached by a horizontal path of such rectangles. This also specifies the algorithm for the L-Z decomposition of an integer n. Find the largest term m of M such that m \le n. If m < n then continue the same procedure on the difference n-m till n-m=0. This gives us the decompositions shown in Figure 1.

One can define sequence f that counts the length of the L-Z decomposition of each integer n. For example, we see that 12=8+3+1, i.e., it is decomposed into 3 terms. Thus, f[12]=3; similarly f[11]=2= f[10]= f[9]=2. f goes as: 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 1, 2, \cdots

One see that the value jumps by 1 for the first time at certain values of n (Figure 2): f[1]=1, f[4]=2, f[12]=3, f[33]=4. Using these n we define a new sequence f_m: 1, 4, 12, 33 \cdots We can then ask what is its convergent? We found that,

\displaystyle n \to \infty, \; \dfrac{f_m[n+1]}{f_m[n]}=\phi^2=\phi+1,

where \phi is the Golden ratio \tfrac{1+\sqrt{5}}{2}.

We can then ask if there is a closed expression for f_m. We derived this to be:

\displaystyle f_m[n] = \left\lfloor 2\sum_{k=0}^{\infty} -1^k \phi^{2n-3k-1} \right\rfloor

LZ_Fig2Figure 2

Another class of sequences we explored was f_k, the lengths of the L-Z decompositions of k^n, where k=2, 3, 4, 5 and n=0, 1, 2 \cdots, i.e., the powers of integers. For example, f_2 goes thus: 1, 1, 2, 1, 2, 3, 3, 3, 3, 6 \cdots. Plots of f_k against n show a good fit for a linear growth in the range in which we computed these values (Figure 3; it is computationally intensive), albeit with increasing dispersion as n increases. If we take their growth to be linear, we then can ask the question: what would be the slope of these lines? Interestingly, we empirically found the slopes of the lines approximating the L-Z decomposition lengths of 2^n, 3^n, 4^n, 5^n to be respectively 2^{-4/3}, 2^{-2/3}, 2^{-1/3}, 2^{-1/9}. Can this be proven or is there an alternative description of the growth of these sequences?

LZ_Fig3Figure 3

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Subjective and objective insight

The black American scientist Sylvester Gates mentioned a curious personal anecdote in a talk. To paraphrase him, when he was in college, he had to take a calculus course. He mentioned how he could cut through differentiation as it was a largely mechanical process. Then came integration, where he said he was stuck with the problems involving multiple substitutions to arrive at the final integral. The inability to crack difficult problems of that genre gave him a headache, and he fell asleep. Something happened to him in his sleep that when he awoke, he suddenly emerged with a new understanding to solve those problems, and they no longer seemed difficult. We can completely identify with that experience of his. However, it was not a night’s sleep that flipped the switch in our case. We had to wait for that testosterone burst, that elixir of masculinity, which allows males to perform great acts. We clearly remember how, a few months before it, we struggled to derive the equations of certain loci for which we had figured out mechanical constructions. But, upon the passage to manhood, suddenly we found ourselves possessed of svāyambhuva insights that allowed us to penetrate such problems with ease — it was as if the doors to a deeper realm of understanding had been opened. Other people have told us of similar experiences — we recently heard from a friend how he had a phase transition at some point in his life (overlapping with puberty), which made him suddenly grasp a mathematical entity that had previously defied him and led him to pursuing a degree in physics.

Our life-history-associated flips may be relatively easily explained in neurological terms — the gonadal hormones are known to trigger extensive neurogenesis, and these new neurons and the reorganization of neuronal connections, which they case seem to provide the firepower for apprehending mathematical and conceptual ideas that were previously difficult to process. However, the experience narrated by Gates is of a different kind. He definitely did not grow a bunch of neurons over his nap, but it seems his “subconscious” kept working and churned up the solution back to his conscious mind on awakening. Such experiences are not isolated. In fact, they might have played a big role in the history of science in the form of dream revelations. We first learnt of this from the famous story of how August von Kekulé solved the structure of benzene in a dream. Subsequently, we learnt of several other examples: 1) Ramanujan obtained formulae concerning several elliptic integrals from the goddess Śri in a dream. 2) Niels Bohr had a dream of electrons revolving like planets around the nucleus in fixed orbits. 4) Dmitri Mendeleev had a dream of the periodic table of elements. 4) AR Wallace had a dream while suffering from a tropical fever in the Far East that left him with the evolutionary theory. We have never made any of our major scientific discoveries in a dream. However, we have had a couple of mathematical problems, and the path to their solution appear in dreams — these were very rare events — we had exactly two so far in life!

While the pubertal and the dream switches might seem like different things, we hold that they have a commonality. Both are marked with the acquisition of a new insight after which the world might not appear to be the same. Before the switch, there was no way of solving the problem with a purely workmanly approach. That switch happens at a “subconscious” level, but it impinges into conscious action with a fundamentally changed framework that allows you to see a new order or a system where none seemed to exist before — everything makes sense in this framework but not outside it.

This has implications for the process of science. It has become popular to tell students that science is generated via the “scientific method”, whose realization is seen as a major development for science itself. Ideas related to the formalism of the scientific method are widespread. As we have discussed before, we encounter them in the nyāya (+vaiśeṣika) theory of knowledge production wherein from a kalpanā (tentative hypothesis) we proceed to a nirṇītā if it passes the test (vinigamaka) as opposed to the alternative hypothesis. The established hypothesis becomes the theory or siddhānta. A similar formulation emerged the Occident starting with the pioneering work of the French savant Rene Descartes (apparently, he got this framework in a dream) and culminating in the Jewish thinker Karl Popper who presented a clear “flow-chart” encompassing hypothesis generation, prediction, testing and falsification. That such a formalism sprung up convergently across cultures implies that there might be something deep to them. In general, we agree it is a good way to understand how science works. However, we should stress that this is not how it actually happens.

The actual process of scientific discovery depends heavily on the welling up of those perspective-changing insights from the subconscious to the conscious that we mentioned above (for why we term it perspective-changing, see below). However, it should not be held that the profound perspective-shift that bubbles up to you is necessarily scientifically correct even if it were mathematically beautiful, technically sound, or seemingly robust as a device. It has to be tested against actual data. Here is where the Popperian idea of making a prediction based on it and testing it comes in. We have several famous examples of how the perspective-changing explanation might be beautiful but scientifically wrong. We could mention the great German astrologer Johannes Kepler‘s original planetary model, where he fitted each of the five Platonic solids between the orbits of the six then known planets. He felt he had stumbled upon a profound insight: “The intense pleasure I have received from this discovery can never be told in words…” However, the predictions of this model did not fit the mass of astronomical observations, the great legacy of Tycho Brahe. Where Kepler emerged as a scientist was in his ultimate rejection of this hypothesis despite its beauty (he drew a diagram of it rivaling the hand of Leonard da Vinci himself) and personal appeal. Thus, the role of the Popperian process was relatively limited in this example of how science actually happened. The Popperian hypothesis rejection did not result in an automatic path ahead for Kepler. Indeed, he might have been consigned the heap of many a forgotten scientist had he stopped there. Kepler’s effort in constructing his original model and testing it provided him with many insights into the problem at hand. He also had key observations that did not fit his initial hypothesis in his head. These provided the grist for his renewed attack on the problem. Here again the subconscious churning through the paths taken by the great yavanācārya-s, Archimedes and Apollonius, going back to the Delian oracle of Apollo resulted in Kepler arriving at the correct hypothesis that was striking in its generality, even if more abstruse than the earlier one for the pre-Newtonian layman.

From a neurological perspective, this subconscious production of science is not surprising. It is well known that most of our neural processes, which might be termed thinking, are unconscious. Even in a conscious experience, like vision, there is an enormous amount of neural calculation and information processing that we are entirely unaware of. In fact, it might even be dangerous for a regular individual to be exposed to this data, its processing, and its presentation. This is strikingly illustrated by the case of the black English artist savant, Stephen Wiltshire, whose very existence might be denied by people who have not seen him in action. However, his extraordinary capture of visual detail comes at the cost of strong autistic traits that are potentially fitness-nullifying. Indeed, on very rare occasions, such capacities might get unmasked by brain injury, as in the case of Jason Padgett, suggesting that natural selection is likely working to keep them masked rather than expressed. Hence, the subconscious, which is screened from the conscious, is the most likely seat where the perspective-shifting insight arises. We hold that there is a pure Platonic realm of mathematics and “linguistic content” that contains the foundation of “all knowledge” of existence. The conscious surfing of this realm is likely not possible for most people. Many of those who are able to access it often have a cost, such as being on the autistic spectrum. Thus, this realm is in part surfed only subconsciously by most.

Anyone who has solved a difficult (to the person in doing the job) scientific or mathematical problem knows the sensation Kepler talks about — that first-person experience. The perspective-changing insight usually comes first, but it is in a sense “raw”, i.e., the details are not precise at all, but there is something in the subconscious that tells you that you have the right solution. It feels as if the “surfing” process apprehends it in the Platonic realm, but its clarity is smudged when it is dredged up to the conscious realm. After that, there is a workmanly phase wherein one implements the solution in concrete terms. In this phase, one’s intelligence and breadth of knowledge are vital determinants of how well one converts the insight into the finished product of a scientific discovery or a mathematical theorem. What emerges usually has a formalism that allows it to be communicated mechanically to the recipient. However, this communication, as well as its reception by peers, might not be easy. A common adage goes that once you announce a new insight, your peers first ridicule it — this is usually because they are not in possession of the new framework you have, and even to apply it mechanically, they need at least a limited perspective-shift. Eventually, the peers learn to apply it mechanically and see that it gives correct results. This causes them to shift towards the new framework even if they do not fully grasp it. Finally, the flip occurs in the minds of the peers and a subset of them might declare the discovery as trivial or claim that they knew it all along — this in part stems from a total conversion that makes them lose their prior framework (some of this is capture in the paradigm shift model of the Jewish thinker Thomas Kuhn).

In the end, all this still lies in the domain of what might be termed the objective because once the insight is gained, it can be formally transferred to others by a mechanical procedure. For example, most Indian “crackers” in our days who exuberantly integrated all manner of complicated functions often did not have any insight into calculus — they had merely mastered its protocol. On the other end, there might be mathematicians turgid with formalism who think all common presentations of calculus are fundamental flawed. In between are the reasonable practitioners who know that there is a certain insight, which becomes very natural at a certain point in one’s study of the field. Once one knows this, it no longer seems like a black box but as natural a procedure as 2+2=4 (unless you are possessed by the Neo-American disease). Thus, for many who have “mastered” calculus, the original perspective shifts that its discoverers might have had are no longer very important. Because of this one might also see a devolution of the field if the continuity with the original insight is lost. We believe that a good example of this is the loss of the great astronomical insights of Āryabhaṭa in Hindu astronomy until it was in a sense rediscovered by the Nambūtiri school or their (as yet unknown) predecessors. This might also be a major factor in the loss of technological insights, such as the Antikythera mechanism of the Archimedean tradition or the yantra-s of Āryabhaṭa or king Bhoja. This might even happen in our age.

In any case, the bottom line is that these perspective shifts, once realized, can be transferred to others. Hence, we see this as being in the domain of “science” or objective knowing. However, over the years, we have come to realize that there is an equivalent of this switch that intrudes into the subjective domain that might not be entirely transferable, at least by the same way we transfer the objective insights. There are several versions of this straddling domain between the purely subjective and the objective. To explore this, we start with the existence of hard biological barriers that stand in the way of the first-person experience of one group from being replicated in the other. An easily understood case of this comes from the innate differences between men and women (notwithstanding the neo-American simulacrum of West Asian diseases of the mind, which tells you that they do not exist). One domain where this is very apparent is vision — men and women literally see things differently. Women tend to see a greater richness of color than men, especially in the middle wavelengths of the visual range, and men see finer detail (especially changing light intensity) and subtler movements than women. While we do not entirely understand the biology behind this, it may proximally stem partly from the X-chromosomal linkage of opsin genes, which encode visual sensors, and partly from the massive role of testosterone in modeling the visual cortex during development. There may be good teleologies for this going back to our evolutionary past, especially given that primates are very visual animals, which recognize color on faces (among other things for mate choice), and the behavioral differences between males and females in several primate lineages. Thus, males and females have distinct subjective experiences of color and detail conditioned by their biological differences — this is analogous to the pubertal neural transformations that lead to new insights. However, in this case, the first-person experience of one group cannot be replicated by the other due to the fundamental biological distinction between them.

This leads us to the question as to whether, in some cases, this barrier to the subjective experience can be turned off by a switch such that you see things in a wholly new way — something analogous to man being able to suddenly see all the gradation of colors a woman was talking about that he never understood. For this, let us consider the effects of N,N-Dimethyltryptamine (DMT). Those who have not had a DMT experience (that includes us, to be clear), can get some picture of the self-reported objective part. For example, a survey of 561 DMT users [Footnote 1] showed significant coherence in the prominent features experienced by them. They reported an encounter with a “being, guide, spirit, alien or helper” that appeared “conscious, intelligent, and benevolent” and “continued to exist after the encounter”. The majority also stated that they received “a message or a prediction of the future”. We cannot make complete sense of what they experienced, but we can agree that the compound made them see something unusual. However, the users also show a significant trend of saying that the experience results in a profound change of world view, and they did not see things the same way after it. For example, more than half of those who identified as “atheists” no longer did so after the experience. Thus, no amount of explaining or description of the experience in an objective sense can flip the perspective switch for those who have not gone through it. Therefore, this tells us that there is a perspective switch in the subjective realm, similar to what we see in the objective sphere in the scientific process; however, that cannot be simply transmitted through a formal framework to others. Instead, one may have to subject oneself to the compound to see if such a shift might be experienced in the first person. Indeed, the commonality and distinction between these subjective and objective perspective changes is illustrated by the lysergic acid diethylamide (LSD) experiences that are said to produce both objective scientific perspective shifts, which can be formally communicated, and subjective ones which result in a “changed perspective on existence,” which seem untransmittable.

The limiting case of the subjective perspective-shift is something that educated Hindus can understand; however, others might find it incredibly difficult to grasp. At a general level, it might be something that overlaps with the flipping of the switch, which occurs with psychedelic compounds, but, typically, the Hindu praxis related to it does not go via such compounds. This may be termed, for the lack of a better word, “brahmānanda.” While the use of the term brahmānanda might indicate that we are privileging Advaita vedānta, we should clarify that it is not the case. The percipient, either due to a yogasādhanā or vicāra has a switch flip within him, which shines the light of a profound subjective experience, that might be liked to awakening from a dream. In the regular dream world, one is conscious and doing things with a unified first-person experience despite the absence of much sensory input. In that state, one takes that experience to be reality. But when one awakens, one realizes that it was not reality but some “illusion”. Similarly, in the brahmānanda experience, the percipient is said to awaken from the everyday world into that new brahmānanda state, at which point he sees the everyday world just like a dream. Some such condition and transformation into it is widely accepted in H tradition (including the vedabāhya schools). What they differ in is the ontological status they accord it and the theological framework into which they incorporate it. We will not labor on this point because educated H will get right away, and others probably will make no sense of what we are talking about.

There are more “secular” examples in the same general domain that again a subset of people can find difficult or impossible to apprehend. Below we give a couple of such anecdotes. To understand the “reality” of subjective experience, one has to be able to appreciate what is called, in modern Occidental philosophical terminology, the “hard-problem” of consciousness. It goes hand-in-hand with the “first-person-experience” available for mental reflection; it is given the technical term quale (plural: qualia). Simply put, the hard problem is then the question of how we can get to the qualia from an understanding of all the biochemistry and biophysics (the “easy problem”). This is a philosophically difficult chasm to bridge between the objective realm of science and the subjective realm of consciousness. A physicist with a prodigious head once asked us if we felt that the “human brain” and “consciousness” were the last great frontier of biology, which would draw the biggest brains in the field. We responded that it might suck in the big brains but that there were more fundamental problems in biology. This led us to talk about consciousness, and soon we realized that he thought consciousness was the same as the biochemistry and biophysics of the brain. Hard as we tried, he could neither apprehend the very existence of qualia nor the concept of the philosophical zombie — it almost seemed like he was one. We put this aside as simply an issue with our attempt at explaining the concept to him. More recently, we had a similar conversation with a set of friends. Of the two of them, one, who was formerly a physicist, again simply failed to apprehend the concept of qualia or that the hard problem could even exist. The other one, a biologist with a reasonable general knowledge of neurobiology, had considerable difficulty grasping the existence of qualia. He fumbled along, insisting, like many before him, that they must be just “illusions” not unlike optical illusions. However, midway into the conversation, a switch suddenly flipped within him. He exclaimed something like: “I get what you are saying! This is profound, a hard problem indeed! Now I see why this might be a big issue.” In this case, we could not transfer an algorithm to him for making the switch — something within him flipped while he was trying to process our words and imagery objectively.

We finally come to the specific case where there seems to be an interaction between the subjective and objective domains of knowing. We illustrate it with an example that would make the typical modern occidentally conditioned scientists (usually one with left-liberal beliefs) very uncomfortable (though the protagonists in the narrative are Occidental scientists). The narrator somehow felt we would “get it” even if we do not believe him. A senior colleague told an elderly biochemist of European ancestry of his observations on the apparent “ghostly” transmission of information from deceased individuals to those born after them in West Africa. The senior colleague had systematically gathered this information and presented what may be termed objective data with statistics to support his contention that this unbelievable thing (in the modern paradigm) happens. Unlike some who would have normally laughed it off, our biochemist heard out his colleague attentively and studied his data. He found nothing wrong in the report but could not believe that what his colleague told him could really happen. He felt there could be other mundane explanations. The said biochemist, himself a man of travel and adventure, was interested in the anthropology and genetics of certain human diseases prevalent in West Africa. Hence, he had the chance to travel there and check things out with the tribesmen himself. What he saw in “pratyakṣa” — the subjective first-person experience he had in West Africa — caused a dramatic perspective shift. After this first-hand encounter, he began believing what his senior colleague had presented to be true, even if he did not have an explanation for it. He did not publish it because he knew others would have the same disbelief as him unless perhaps they reproduced his experience for themselves — something not easily achieved when it needs serious fieldwork among the tribes of West Africa.

This is not restricted to the domain of such unusual things, though it might be enriched there. We have had at least one personal example of the same in ordinary science in our youth. A researcher had published an unusual scientific discovery whose full implication he did not grasp. When we read it, we realized how unusual it was and the major implications of it being true. However, we simply could not get ourselves to believe it, for it was not easy to reproduce it by any means at our disposal in our youth. We also found that other respected researchers in the field could not reproduce it and disregarded it. However, a few years later, we were able to reproduce it for ourselves and see it plainly with our own eyes. At that point, we managed to develop a formalism to present it quite plainly to the rest of the community. Seeing our presentation, several saw its reality and started claiming it as their profound discovery! Because it was in the realm of the objective, once the formalism was presented, people could make the flip by following it. However, to develop that, we had to have a first-person experience of it — enter a state of being a believer — before proselytizing it. However, not all such flips necessarily result in correct insights. Some of those could be false, both in the domain of science and religion.

Finally, why do we call it a “perspective shift”? Early on, we read of a mathematical construct that led us to an analogy of how these insights work. It is the famous construct of a 2D world — the flatland. For the flatlander, objects accessing and using the third dimension for motion will mysteriously appear and vanish. Moreover, a flatlander moved into the third dimension will suddenly acquire X-ray vision into other flatlanders. Thus, the insights we have discussed in this note have the feel of such a vision of a flatlander suddenly gaining access to 3D space; hence, we term them perspective shifts. Might such a thing also apply to our 3D space? Some rare people, like Henri Poincare and Alicia Stott, the daughter of the well-known mathematician Boole, had the capacity to “see” 4D space. Thus, Stott was able to construct shadows and cross-sections of 4D and higher dimensional objects in 3D space and make discoveries in this regard. This led to the great mathematician Coxeter using her extraordinary ability to assist his geometrical research even though she had never formally attended college. This was a genetic siddhi, elements of which she got from her parents and passed on to her son. For the typical individual, there might even be an inhibition against such special insights, for it could come at a fitness cost, as noted above. In this regard, we note the case of a fellow graduate student who was a virtuoso programmer. One day he had a mental quake, after which he remarked to us how he was apprehending 4D space naturally and seeing hyper-Platonic solids. Sadly, a few months later, he lapsed into dysfunctionality with a severe mental condition.

[Footnote 1] Davis et al;

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On the passing of E.O. Wilson

E.O. Wilson, one of the great biologists of the age, has fallen to the noose of the king, the black son of Vivasvān. He lived a long, productive, and eventful life, just 8 years shy of a century. He was a major influence on our scientific development. We learnt of kin and group selection and r- and K-selection from reading his classic tome, “Sociobiology: The New Synthesis” in our youth. The introduction to these concepts of the evolutionary theory kept brewing in our minds, and we kept thinking about the molecular consequences of the same. In the 13th summer of our life, we studied the immunoglobulin domain and the generation of antibody diversity in jawed vertebrates. It was then that first connections clicked into place. We realized that must be general evolutionary parallels between the immunological molecular machinery for self-non-self discrimination and the apparatus relating to kin-nonkin discrimination in social contexts. A few years later, we read John Maynard-Smith’s “Evolutionary Genetics”, which we were lucky to borrow shortly after its publication. By then, we were armed with some agility in calculus; thus, the mathematical framework provided by Maynard Smith allowed us to apprehend some key ideas of the selective process relating to the logistic growth curve and related issues. These also came together with the ideas of Pāṇini/Patañjali on linguistic systems and those of Shannon regarding the relationship between entropy in statistical mechanics and linguistic strings. Finally, one fine evening it all came together, and we realized the foundations of understanding the imprints of the selective processes we first learnt of from Wilson’s book on the information in biological macromolecules. Exploring this story has kept us occupied to this date.

We found our journey to be somewhat ironic when we learnt much later of the famous clash between J.D. Watson and Wilson when they were both at Harvard University. Old Jimmy felt that molecular biology had made Wilson’s type of biology (“stamp collector” science) unimportant. Watson is famously reputed to have said: “Smart people didn’t go into ecology … It’s not intellectually demanding.” While we also feel a degree of intellectual kinship with Watson, there is a palpable aspect of Wilson’s statement regarding Watson — “the most unpleasant human being I have ever met” — in molecular biology. Indeed, molecular biology has quite a share of the “most unpleasant” people you can meet outside of a street in some rough city of the world. We believe that some of this culture stems from the founder of that science Watson himself. While we admit this is a subjective and anecdotal impression (we do not have controls to say if scientists are more or as nasty in experimental physics or organic chemistry), it cannot be denied that the cultural defects of modern molecular biology are reflected in the mounds of fake results and credit stealing (best termed plagiarism) corrupting scientific publications from the constricted highways of the magazines and to the toxic byways of preprint servers. Even more troubling for the foundations of the science is the triumph of the Watsonian metaphor over the Wilsonian call for consilience — something that deeply resonates with the Hindu tradition of knowledge. Wilsonian consilience was put to practice by his late friend, the great entomologist, T Eisner, who brilliantly brought together the study of biological conflicts with an exploration of the chemical virtuosity of insects. Thus, we have numerous practitioners of the modern branches of biology, championed by old Jimmy, who lack an understanding of the foundational ideas of their science — imagine physicists practicing their science with only a smidgen of knowledge of the Lagrangian or the Hamiltonian. It would indeed do the science good if the practitioners were to pay more sincere attention to the Wilsonian philosophical outlook. However, this may not come to be for other reasons that intersected with Wilson’s journey through life (see below).

Kin selection was discovered by J.B.S. Haldane and elaborated in a proper theoretical framework by W. Hamilton. Wilson’s seminal contributions to hymenopteran biology were critical in establishing kin selection on a firm footing. However, ironically, Wilson tended to have a soft corner for group selection, which eventually became a full-blown attack on kin selection as the explanation for eusociality in his last years. He sought to provide this idea with a mathematical foundation with the help of Nowak and Tarnita. We feel that much of that complicated mathematics is probably more a smokescreen than real fire and does not displace kin selection, at least in the contexts that were close to him — eusociality as reported in arthropods or the mole rats. Nevertheless, unlike many other biologists, we do think Wilson had a point regarding the place of group selection in social systems. To a degree, this might have been critical in human sociality, much like the hypersocial ants that Wilson had studied. The lineage as a whole provides a way to understand this. Men unrelated to great leaders like Chingiz Khān or Shivājī sacrificed their lives for them. In return, these leaders ensured the survival of their offspring. Say they had not sacrificed themselves for the new group identity forged by the Khān, they might have been wiped out in entirety like the many bands on the steppe before them. Thus, while the Khān got to propagate his genes to leave an oversized genetic imprint that stands out even today, these men might have raised the probability of the survival of their lineage from 0 to something small but non-zero. Our investigations suggest that group selection might have a role in the stability of bacterial biofilms too.

This brings us to an important point elaborated by Wilson: the superorganism. The same genome is differentially expressed to generate a diversity of castes that dramatically diverge in appearance, size, and behavior. This provides a striking illustration of a molecular principle, namely the use of epigenetic regulatory processes to add information over and beyond that encoded in the four bases of DNA. Thus, different parts of the same code are unveiled in different individuals making them look almost as if they were different species. This led to the formulation of the evolutionary hypothesis of how epigenetic regulation in eukaryotes might provide the initial “capacitance” for changes that might then be hardwired into the genome. At the social level, it showed the remarkable success and stability of the caste system as an evolutionary strategy. It has repeatedly emerged in multiple hymenopterans and the cockroach-like clade. Thus, it should not be a surprise to see it emerge in humans though we can only be considered nearly eusocial. Nevertheless, the basic principle of a superorganism with castes can be seen as applying to our societies. That is how our ancient social theorists saw the varṇa system — the 4 varṇa-s (mirroring the numbers of castes seen in arthropods) are seen as aṅga-s of the metaphorical puruṣa who is the society. The stability of these castes for over 90 million years in hymenopterans should serve as food for thought to the left-liberals who strive to have it abolished. This should be placed against the backdrop of the many evolutionary successes of the hymenopterans and isopterans, which anticipated some of those that we pride ourselves on, like the discovery of farming or antibiotics.

This finally brings us to what brought Wilson and Watson back together. Wilson was one of the first to face the assault of the navyonmatta-s — the left-liberals with deep connections to the H-haters of American school led them — Lewontin, Gould, Kamin, and Rose, among others. They orchestrated a band of thugs, the predecessors of the kālāmukha rioters of the American gardabha-pakṣa, to attack Wilson. Watson was a member of the old mleccha guard and, like his collaborator, F. Crick, saw the reality of genetic differences between ethnicities. This made him an enemy of navyonmāda, driving him close to his old foe, Wilson. In the end, the first wave of navyonmāda orchestrated by the uparimaragata left-liberals failed to storm the scientific branches of academia completely. Instead, due to the lack of Wilsonian consilience in the Occidental academe, it festered on in the non- and less- scientific domains of the same. In the end, it has to be kept in mind that both Wilson and Watson belonged to the mleccha elite. Their fortress is still pretty strong despite being sapped by navyonmāda. Wilson was a quiet personality. He generally maintained a dignified public profile, kept writing his books, and moved to other areas of interest. Thus, the navyonmatta-s lost interest in him. In contrast, Watson has an abrasive personality who liked to focus on the most uncomfortable of human genetic differences in a public and, sometimes, crude way. This resulted in his fall from grace as an American hero. In the end, Wilson’s personality offers a better model to emulate than Watson. He was productive until late into his long life. He explored a range of ideas brought many of them to the public with elegant writing. However, this would have only been possible in the height of the mleccha academic ecosystem. Even if one had the genetic wherewithal to emulate a Wilson, it would be tough to achieve the same in the absence of that type of ecosystem, which is now under threat from navyonmāda.

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Some words on mleccha cartels

An embedded anthropological study of social substructures is vital to grasp some of their features that seem baffling to the outsider or the “uninitiated” insider. Much of what we will be talking about here has been said in some form on these pages before, and some of it will necessarily be vague for the mleccha and his agents are still in control of the world. Yet, we felt it is worth recording this as there is no guarantee that the whole story can be said in plain language. If the time does come when it can be told in such a way, then the tendency will be to focus on the distracting specifics rather than the general significance, and the focus will be through the lens of inter-personal feelings rather than objectivity. Another point to note is that in the past decade navyonmāda, whose long fuse was smoldering for decades, finally exploded on the world scene along the lines its founders wanted. This means that the old lenses used to view the mleccha are no longer a perfect fit — cognizance should be taken of the deep divergences within the mleccha world. Nevertheless, as we have often remarked before, there is an alignment of the preta, garden variety liberal and nayonmatta positions when it comes to the H. Indeed, this is a continuation of the deep anti-H sentiment in the mleccha world, especially among the āṅgla-gaṇa, which goes back to over a century and a half. For example, the popular mahāmleccha mouthpiece, the Washington Times, was as anti-H a century ago as today. Hence, despite the deep schisms within the mleccha worlds, like those between the overt Abrahamisms, there is a common pith (much like the marūnmatta scientist, Al Bīrūni, had remarked regarding the Hindu-s and the Yavana-s).

Over the past few years, two H (they not educated about H traditions but do not tilt against them either. Their knowledge of Indian politics is next to non-existent), whom we know well, brought us in contact with three influential mleccha organizations. The power of two of them, while extensive, is in no way directly related to national- or geo- politics, and their influences are relatively domain-specific. The third has a more local character but exerts considerable influence on one of the two bigger organizations. We will hereinafter refer to them as “cartels,” but it should be emphasized that they do not trade in recreational substances or weapons or the usual things the word cartel is commonly used for. However, we learnt that they operate very similar to cartels. To be clear, we are not a member of any of the three though our H contacts are. They entered those cartels through either prolonged pro bono service or playing middleman in key access networks for powerful mleccha insiders. In the process, they learnt how to play in those circles via mlecchānusāra. In short, it involves an elaborate process of delegation while claiming credit for oneself. The lower-ranked individuals actually doing the dirty work are paid primarily with “biscuits” like those tossed to the faithful dog Tom for fetching a stick. The mlecchānusāra has a subtlety to it, despite seeming simple to the naive onlooker. Indeed, a couple of such naive V3s thought they could play the boss-mleccha but ended up much like the nāpita mimicking Maṇibhadra.

While our H contacts are insiders, they still felt that there were deeper secrets that only the mleccha and mūlavātūla players could access. However, they got one key perk — immunity to random attacks from the thugs (see below). A prathamonmatta insider brought to the attention of one of our contacts the case of a non-H deśīya who entered the cartel by presenting himself as a “traditionally oppressed and excluded minority.” However, after some study, we realized that it was only half the story. He had a long track-record of serving the mleccha by playing middleman in important networks like our other H contact. There is a particular knack to the whole thing that these two individuals have mastered. It is probably how arms dealers operate. A śūlapuruṣa and a kṛśapuruṣa, who are high-level players in one of these cartels, and with deep moles in the other two, revealed to us some of their deeper dealings. The śūlapuruṣa knew well that we are an outsider with no means of harming the cartels. He had also gotten a big favor from us (to be open, a calculated action on our part that benefited us) and thought a return favor might be to reveal some secrets of his power. The kṛśapuruṣa had also benefited from us (at no particular cost or gain to us); however, he also felt some kind of “ethical discomfort” about the cartel of which he is a deep insider. Thus, they ended up revealing some of the actions that went on within.

One key feature of these cartels is the multi-leveled defensive layering like the prākāra-s of maṇḍala bristling with deities capable of deploying all manner of weapons. The outer facade is carefully painted to present a picture of being “free and fair”. However, in reality, it is anything but that. But how is that facade maintained? The main element of this is the first layer of the system — a large body of “peripherals”, who are not members of the cartel and know nothing of its inner workings. However, they are dependent for their survival on the exclusive products of the cartel and have fear and admiration for the cartel leaders, much like a low-ranked individual might have for the \alpha player. These peripherals are the ones who buy and use the cartel products and are occasionally given some small rewards for doing so. Whenever survival is hard, it is possible to easily earn the gratitude of those in the struggle by giving them a few tidbits that seem like encouragement or moral support. The second element of this involves the cartel members choosing a few individuals from among the peripherals for two kinds of things: (i) those who would do advertisements of the virtues of the “free and fair” system operated by the cartel and show how their products are chosen entirely due to merit in a competitive Turkish bazaar. (ii) The second set of individuals are chosen for doing the hard and dirty work of running the cartel’s production systems — sapping and utterly boring job if one were to have an avenue to lead a free and self-respecting life. The cartel members pay these chosen peripherals for these jobs a little more in terms of the “biscuits” they toss to them. The first mechanism is closely related to the “toolkit” approach in geopolitics that is used widely in social and legacy media and national destabilization activities of the navyonmatta-s (often backed by duṣṭa-sora and the like). Thus, at the whistle of the managers from Sora’s organization, the peripherals will start yelling “dog-whistle” and claim that the imaginary H canine is shredding them.

The next layer is that of the “thugs” who are again not cartel members but offer their services for the cartel. These thugs themselves are individuals of lower intelligence than the cartel members or the peripherals. Thus, they do not pose a significant danger to the cartel members by themselves as they cannot easily organize a rebellion against the cartel. Moreover, being good-for-nothings, they strongly depend on the cartel for their very survival. Coming from the lower rung of the social ladder, they are full of resentment and get great satisfaction from acts of vandalism and violence (even of the metaphorical kind). This feeds into their fantasies of being maverick vigilantes doing their part for the “noble cause”. Often these types are high on navyonmāda and have time on their hands to offer themselves for “policing” on behalf of the cartels. They perform two operations: (i) intimidation of the peripherals who may start discovering the cartel’s ways, fall out of line, or show independence. (ii) they attack those producers who lead lives independently of the cartels so that they cannot sell their products in the open market. These attacks are orchestrated such that they appear rather random, and the independent producers are left wondering what hit them and who is behind the attacks. The result is to force the independents into being subservient peripherals or be entirely driven out of the business. These tactics have a mirror in political navyonmāda, such as the kālāmukha thugs of the gardabha-pakṣa among the mahāmleccha. It also resembles the tactics of marūnmāda, where the kaffrs (=independents) are offered the option of either losing their foreskin or their head.

The next notable layer is made up of cartel insiders, who form the cartel’s public face. Among the mahāmleccha and their satellites, the keyword is “diversity”. Usually, the individuals in this layer are chosen so as to hide any signs of ethnic enrichment in the cartel. The members are there to create the illusion that anyone can “make it” and that the cartel does not really have any exclusive principles beyond pure merit. The cartels we are talking about includes a large number of true believers of navyonmāda, but most members would hardly give up their yachts or sprawling villas for the kṛṣṇa on the street for whom they proclaim unreserved love. Hence, they pay great attention to camouflaging this with effusive public declarations of the creed of navyonmāda. This layer also features the appointed spokespeople who direct the advertisement activities of the peripherals — e.g., updating the toolkit and setting agendas for them. A related layer is one of “managers” who interface with the thugs and peripheral workers and set goals for them.

An essential aspect of the system is an elaborate chain of scapegoating. If something bad happens and the cartel comes under fire (e.g., their egregious mistakes during the Middle Kingdom corruption or the uncovering of major sexual misconduct by a member), this chain ensures that the fuse does not burn all the way to the cartel members. Thugs and peripherals might be immolated with little remorse as sacrificial victims as long as the cartel’s interests and inner circle are preserved. Thus, it is hard to pin the blame on the cartel — it will get pinned on to someone lower down in the elaborate chain of scapegoats, and that little tentacle will be amputated like that of a Hydra leaving the rest of the animal intact to regrow it. If due to a major mishap, the blow-back happens to breach the inner rungs of the cartel, then scapegoating action follows along ethnic lines. Those ethnicities with strong internal bonds quickly close their ranks to protect and secrete away their tainted members leaving the loosely bonded ethnicities to take the blow. The H fall in that latter category as most H members of these cartels have a poor sense of religious solidarity and have a tendency to splinter along the lines of their deśa-bhāṣā. Indeed, an H player from one of the said cartels, who had achieved extraordinary power, thought himself to be immune to attack. However, his rivals trapped him using the perennial device of seduction by women. Thus exposed, he had no ethnic network to shore him up for when in power he thought he was one of the mleccha-s and shunned the H. Thus, without an ally, the mleccha-s and others closed in to finish him off. In contrast, when a mūlarogin was exposed for a major scandal affecting the core principles of the cartel, he was quickly encapsulated by his ethnic network and after a while rehabilitated with an advertisement campaign with high production values. So much so that a peripheral who spoke to us of his case was surprised that he could even have engaged in wrongdoing. A small cīna peripheral and an H thug were chosen and duly offered up as the bali-s that wiped away the enas of our mūlarogin.

Once one gets acquainted with these systems and is freed from self-censorious fear of being called a “conspiracy theorist”, one realizes that the same model is duplicated in the mleccha world across organizations diverse in size and domain of action. To our knowledge, the H have not been able to read these well and have mostly been overrun by them — the H commoner tends to believe the stated objectives of the respective cartels (we have seen H repeatedly do so with utter sincerity). The Cīna-s have realized their existence and, with their growing power, tried to penetrate them to a degree by bribery and seduction. They have definitely had a degree of success for their efforts, especially given that they have more or less captured certain other critical domains of mleccha production.

An author who experienced and described the collapse of the Soviet Rus empire said that it is not entirely bad to live a life in the margins. The basic argument is that a gale might uproot the oak at the center of the plot but do little to the pinkweed growing on its margins. Having led such a life ourselves, we agree with such an assessment for the most part. On the plus side, the margin-dweller is less-affected by catastrophic collapses and upheavals that the cartels might engineer. On the downside, this option is not easy for most as they have to make bigger gambles to sustain their families. However, we have observed that above-average but not supersmart individuals can sustain good family lives as long as they have small but reasonably talented marginal leaders. However, herein lies a potential danger. The cartels realize that such nucleations have the power to trigger marginal revolutions that can eat into their pie. Thus, the cartels try their best to quash these even threatening life in the margins. We suspect that the following factors will only make it easier for them to put down marginal revolutions: (i) the cartels gaining exclusive control of the world of the internet (e.g., culminating in the overthrow of the Nāriṅgapuruṣa among the mahāmleccha). (ii) The rise of an internet-only generation with short attention spans and a tendency to acquire knowledge from secondary sources. (iii) the belief that unreal gratifications can be achieved. (iv) the rise of “meta-software” that accesses the lives of people, which they have wholly surrendered to the cartels (the details of that might be a story for another day). The only thing that remains unclear is the timeline for this action to play out.

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Relationships between incircles of the “equilateral triangles in a square” system

This note relates to geometric relationships that may be likened to the Japanese temple-tablet problems. The inspiration for discovering and exploring it came from an origami construction presented by the pioneer in that field, Sundara Rao of Kumbhaghoṇa, in the late 1800s. Given a square piece of paper, how does one fold it into an equilateral triangle? The construction which Rao gave was that of an equilateral triangle with sides equal to that of the starting square and sharing one side with it (Figure 1). Based on Figure 1, it is easy to see how that might be achieved. When we first folded this in our youth, we realized that it is not the largest equilateral triangle that can be placed inside a square. However, examining the origami construction for the above, we realized that it also gave us an easy origami construction for the largest equilateral triangle that can be inscribed in a square (the blue triangle in Figure 1). That construction should be self-evident once the first equilateral triangle sharing a side with the square is in place). These two equilateral triangles and the square result in a configuration of 7 other triangles (Figure 1).

The below study concerns the relationship between the incircles of these 7 triangles (c_1, c_2, \cdots, c_7) and two additional incircles, namely that of the starting square c_s and the Raoian triangle constructed from it c_t. Their radii, which the relationships connect, will be respectively, r_1, r_2, \cdots, r_7 and r_s, r_t. We outline the proof rather than present all the tedious trigonometry and radical algebra. If you like to do that with paper and pencil and are good at that, you can try the same. However, we cut through the tedium of the at times complicated algebra using a combination of recognizing key patterns and the open-source computer algebra system from GeoGebra that seamlessly interfaces with its geometric constructions. Nevertheless, we will show a few obvious ones to lay the background.


Figure 1.

• First, with is straightforward trigonometry to show that


• The proportion of the radii of the incircles is equal to the proportion of the equivalent sides of their triangles.

• Let the side of the largest inscribed equilateral triangle (blue) be t_1 and the smaller one (sharing the side with the square) be t_2. We can use the half-angle formula to show that \cos\left(15^\circ\right) = \tfrac{\sqrt{2}(1+\sqrt{3})}{4}. In turn, that means \tfrac{t_1}{t_2}= \tfrac{1}{\cos\left(15^\circ\right)} = \sqrt{2}\left(\sqrt{3}-1\right)

• We can see that triangles with incircles c_2, c_4, c_5 and c_6 are similar 45^\circ-60^\circ-75^\circ triangles. Using the Sine Law we can show that the sides of such a triangle are in the ratio 1: \sqrt{\tfrac{3}{2}}: \tfrac{1+\sqrt{3}}{2}. With this and the above we can get the proportionality of these triangles. For example, we can show that the sides of the triangles with c_2 and c_4 are in the proportion: \tfrac{\sqrt{2}\left(1+\sqrt{3}\right)}{2}. :


\dfrac{r_6}{r_2}= \sqrt{2}\left(\sqrt{3}-1\right)





• Next, we use Brahmagupta’s formula for the incircle of a triangle, r=\tfrac{A(\triangle)}{s}, where s is the semiperimeter (half the perimeter) of the triangle. From the proportions of the triangle sides, we can show that the ratio of the areas and the perimeters of the triangles whose incircles are c_3 and c_4 are both 1+\tfrac{2\sqrt{3}}{3}

\therefore r_3=r_4

• Thus, from above we have: r_6=2r_4 and r_5^2 = r_4 r_6 = r_3 r_6, i.e., a geometric mean relationship.

• Similarly, we use Brahmagupta’s formula to obtain the incircle radii of c_1 and c_7. With that, we get the following relationships:


\dfrac{r_1}{r_4}= \dfrac{r_1}{r_3} = 1-\sqrt{2}+\sqrt{3}

\therefore r_1 = \dfrac{2 r_2 r_3}{r_2+r_3} = \dfrac{2 r_2 r_4}{r_2+r_4}, i.e., a harmonic mean relationship.


\dfrac{r_6}{r_2}= \sqrt{2}\left(\sqrt{3}-1\right)

\therefore r_6 = \dfrac{4 r_2 r_7}{2 r_2+r_7}, i.e., r_6 is the harmonic mean of (2 r_2, r_7).

• One can also get relationships connecting the radii of all the 7 incircles c_1 \cdots c_7 and also all 9 incircles in the configuration (Figure 1):

r_2 r_5 r_6 - r_3 r_5 r_7= 2 r_1 r_4 r_7

\dfrac{r_2}{r_3} \dfrac{r_6}{r_7}-2 \dfrac{r_1}{r_5}=\dfrac{r_s}{r_t}-\dfrac{r_5}{r_2}

Thus, one can see the “2-ness” of the square in the form of \sqrt{2} and the “3-ness” of the equilateral triangle in the form of \sqrt{3} pervades the system.

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The Rāmāyaṇa in numbers: meters, sarga- and kāṇḍa- structure

In the extant Indo-European textual corpus, only in the Hindu collection do we find two complete early epics to complement the śruti. The Iranian epics come from a much later age than the core Avestan corpus, and in the Greek and Celtic cases, the śruti-equivalents have been mostly or entirely lost. As they have come down to us, the Hindu epics postdate much of the Vedic corpus but are still in a distinct language register that largely predates the classical Sanskrit. Thus, the numerical study of the epics gives us essential information regarding the evolution of the Old Indo-Aryan language and compositional technique, with general implications for earlier branching events within Indo-European. Contrary to the deeply flawed mainstream white indological opinion (and its imitators), and in line with Hindu tradition, we hold that the original Rāmāyaṇa was composed prior to the Mahābhārata. However, it is also clear that both epics were at some point “held” by the same expositors and redactors, resulting in some convergences. We had earlier presented some key details about the structure of Rāmāyaṇa and the earliest para-Rāmāyaṇa (the Rāmopākhyāna) via numerical analysis and pointed out how the kāṇḍa-s show both a certain unity and divergence relating to the compositional history of the epic. Here, we extend that analysis further and draw some inferences regarding the above-stated issues.

The so-called Baroda “critical edition” is available in an electronic format and forms the basis of the below analysis. We have corrected several errors in that text; however, we cannot rule out that some errors remain, affecting some of the below numbers. Nevertheless, these will not affect any of the basic inferences presented below. The Rāmāyaṇa is mostly a metrical text with 17810 verses having 2 hemistiches each. There are 79 verses with 1 hemistich, i.e., standalone \tfrac{1}{2} verses; 576 verses with 3 hemistiches: 1\tfrac{1}{2} verses; 5 with 4 hemistiches which are essentially agglomerations of 2 complete verses. It is unclear if some of the 1 and 3 hemistich verses were originally complete verses that lost one hemistich. However, many of these odd-hemistich verses are genuine “capping” verses that occur at the ends of sarga-s. The 17810 “properly formed verses” fall into the below metrical classes (Table 1)

Table 1

Syllables Frequency Meter
32 16949 Anuṣṭubh (Śloka)
44 476 Triṣṭubh (Upendravajrā)
48 285 Jagati (Vaṃśastha)
50 26 Puṣpitāgrā
45 22
47 20
46 16 Aparavaktrā etc.
33 8
52 7 Rucirā

The primary meters are given in the third column with major, specific subtypes in brackets. It should be noted that the type in the bracket is just a widespread version and not the sole version found in the epic. For example, we have triṣṭubh-s of other types like jāyā, buddhi, kīrti, etc. in addition to the common upendravajra. Likewise, with the jagati-s we have versions like kumārī in addition to the prevalent Vaṃśastha. The dominant meter is, of course, the śloka anuṣṭubh. Now, some verses do not match any meter. Since we did not individually check all of them, some may be errors in the preparation of the electronic text — indeed, we corrected several of these. However, some of the 33s are genuine hypermetrical anuṣṭubh-s, like the famous ancient statement in the second hemistich that is hypermetrical:

idaṃ bhuṅkṣva mahārāja prīto yad aśanā vayam ।
yad annaḥ puruṣo bhavati tad annās tasya devatāḥ ॥
O great king, be pleased and partake this, such food as we [eat],
for the gods are offered the [same] as what food the man takes.

The 45 syllabled verses seem to be hypermetrical triṣṭubh-s, and the 47 syllabled ones seem to be primarily hypometrical jagati-s. Thus, there appears to be a total of about 50 hypo/hyper-metrical verses among the “properly formed” verses \approx .28\%. The 46 syllabled verse is a bit of a mystery. Many of these can be shown to be aparavaktrā-s; however, several do not match the aparavaktrā properly. It is not clear if these were variant aparavaktrā-s that are no longer in vogue or an error of transcription or something else.

This pattern of strong metricality is in contrast to the Veda. Looking at the most metrical of the Vedic texts, the Ṛgveda, we find below distribution (Figure 1).

RV_Syl1Figure 1. Frequency of verses of a given syllable count in the Ṛgveda.

The RV widely uses the Gāyatrī meter (2nd most common) that fell out of vogue in the later Sanskrit tradition. However, the other widely used meters Anuṣṭubh (4th most common), Triṣṭubh (most common) and Jagati (3rd most common), are shared with the epic tradition. We also have internal evidence from the śruti that their syllable count was precisely as in the later dialect, like in the epic. Hence, it is striking to note that, unlike in the epic, the meter is far more loosely maintained in the RV, with a dominance of hypometrical verses. This suggests that whereas the Rāmāyaṇa was composed more or less in the same dialect as it has come down to us, the RV was likely originally composed in an older dialect closer to the PI-Ir state, with a distinct system of saṃdhi-s than in later Sanskrit. The language in which it has come down to us has shifted in register closer to later Sanskrit, with the new saṃdhi-s resulting in losses of syllables from the old language. In a smaller number of cases, this shift in register has also resulted in the likely resolution of old saṃdhi-s and consequent hypermetricality. This shall be separately discussed in the future in the context of the Veda.

We shall next look at the distribution of the different meters in each kāṇḍa per 1000 proper verses in Table 2.

Table 2


Based on this distribution we can compute the Euclidean distance between kāṇḍa-s and construct an unrooted single linkage tree (Figure 2).

rAm_kANDa_treeFigure 2. Relationship between kāṇḍa-s based on distribution of meters.

To better understand the above groupings, we next go down to the sarga level and compute two metrics for each sarga in a kāṇḍa: (1) metrical heterogeneity, i.e., the mean syllable count per sarga and (2) length of a sarga in number of verses (as previously discussed). The metrical heterogeneity measures how “pure” a sarga is in terms of the meter. For example, a sarga composed entirely of Anuṣṭubh-s will have metrical heterogeneity of 32. We show the plots of these metrics in Figure 3.

Figure 3. Sarga heterogeneity metrics by kāṇḍa.

Here, we can see that the kāṇḍa-s 1 and 7 are dominated by sarga-s with pure Anuṣṭubh-s of similar mean length, explaining their grouping in the tree. The kāṇḍa-s 2, 3, and 4 are somewhat more heterogeneous in terms of their metrical structure and have similar mean lengths consistent with their grouping. Finally, kāṇḍa-s 5 and 6 are metrically the most heterogeneous with on an average significantly longer sarga-s. This structure and grouping throw some light on the history of the text. Kāṇḍa 1 (Bāla) states that Vālmīki composed the epic in 6 kāṇḍa-s along with an “uttara” or addendum: tathā sarga-śatān pañca ṣaṭ kāṇḍāni tathottaram ॥ (From Vulgate; absent in “Critical”). This hints that there was a memory of the uttara-kāṇḍa (7) as an addendum to the core 6. This is apparent from the nature of several parts of kāṇḍa-7, which fill in the narrative gaps in the core kāṇḍa-s or provide explanatory commentary. The same feature is evident in kāṇḍa 1 (including the above statement). Their grouping, together with an anuṣṭubh-rich structural uniformity reminiscent of the purāṇika verses, suggests that they are likely entirely (7) or partly (1) the product of a later redactional effort to fill in parts of the epic that were either lost or needed further explanation/augmentation. Even the supposed names of the sons of Rāma, Kuśa and Lava, appear to be derived from an old term for a minstrel, the kuśīlava (the twins are mentioned as such in the beginning of 1 and end of 7), suggesting the emergence of these parts within the oral tradition of such minstrels, which used the relatively-easy-to-compose anuṣṭubh-s uniformly. Kāṇḍa 1 also hints that the original epic had two subsections to it:

kāvyaṃ rāmāyaṇaṃ kṛtsnaṃ sītāyāś caritaṃ mahat ।
paulastya vadham ity evaṃ cakāra caritavrataḥ ॥
He (Vālmīki) composed the great poem, the Rāmāyaṇa, the story of Sītā.
Even so, he of firm vows composed that known as the slaying of the Paulastya.

We could interpret this as implying two larger sections of the narrative centered on the tale of Sītā (i.e., her birth and marriage to Rāma, etc.) and the killing of Rāvaṇa. Thus, we suspect the two structurally unified parts the kāṇḍa-s 2-4 (probably with parts of the ancestral 1) formed the first of these sections, and kāṇḍa-s 5-6, which are again structurally similar, and organically related to the killing of Rāvaṇa, formed the second. Kāṇḍa 5 (Sundara), which shows maximal metrical and length heterogeneity, was likely composed thus on purpose (as we noted before). This kāṇḍa foreshadows the tendency in later classical kāvya, where the kavi-s set aside specific sections of their work, to showcase their virtuosity in terms of composing in a diverse array of meters or displaying various alaṃkāra-s, including citrakāvya. We do not see the much longer and complex metrical expressions of classical kāvya nor the constraint-based composition using techniques of citrakāvya in the Sundarakāṇḍa. Yet, it is clear that Kāṇḍa 5 elegantly intersperses diverse meters on top of the basic anuṣṭubh background to bring about pleasing changes of cadence. Like later kāvya-s, it also has entire sarga-s in long meter-s.

Next, we study the structure of the sarga-s by themselves and see if we can discern: (1) specific structural classes of sarga-s; (2) whether the sarga class has a relationship to the kāṇḍa it comes from. To do this, we first construct a matrix where every row corresponds to a sarga. The first 9 columns correspond to the fractions of the sarga in verses of a particular number of syllables (32, 33, 44 etc.). Column 10 corresponds to the length of the sarga in number of verses normalized by the longest sarga (5.1). We then use this matrix for unsupervised classification of the sarga-s using the random forest predictor as implemented by Breiman and Cutler. Briefly, this is a machine-learning method that uses an ensemble of individual classification tree predictors (i.e., decision trees to classify the given data). The decision process specified by an individual tree uses each observation to vote for one “class” and the forest of such trees is used to choose the class with the plurality of votes. For the classification process, the number of randomly selected variables that are searched for deciding the best split at each node in the tree is taken to be \left\lfloor\sqrt{n}\right\rfloor, where n is the total number of variables. The unsupervised mode works by making the RF predictor discriminate the observed (i.e., the above sarga matrix in our case) from synthetically produced data. The synthetic data is made by randomly sampling from the product of marginal distributions of the variables from our input matrix. As a result, one can obtain a proximity matrix between the input observations (i.e., sarga-s in our case). This proximity matrix can be converted to a distance matrix and used as the input for multidimensional scaling (MDS), representing the observations as points in an Euclidean space of n dimensions, with the Euclidean space distances between these points approximately equal to the distances in the distance matrix. By choosing the first two dimensions in this Euclidean space and plotting them, we can reduce dimensionality and obtain visual clusters or classes of the observations. Figure 4 shows the first two dimensions of the MDS plot for our data following unsupervised random forest classification (655 trees and minimal terminal node size of 90).

rama_sarga_class_inFigure 4

We see 4 broadly delineated clusters, although their “smearing” indicates a degree of a continuum. Examining each cluster individually, we see that they provide a meaningful classification of the sarga-s: (1) The first class (ellipse to the left) is composed of sarga-s that are pure anuṣṭubh-s. (2) The second class (top right ellipse) comprises of sarga-s that have anuṣṭubh-s combined with triṣṭubh-s. The core of this class is defined by a very characteristic form of the sarga that contains a triṣṭubh as the capping (final) verse. (3) The third class (bottom right ellipse) consists of sarga-s combining anuṣṭubh-s with jagati-s. The core of this class uses a jagati, usually of the vaṃśastha type, as the capping verse. (4) Finally, the central ellipse contains a group of sarga-s typified by interspersing of different meters on an anuṣṭubh background or those with irregular (hypo/hyper-metrical) verses.

rAmAyaNa_rf_clsFigure 5

One can see from the color-coding of the sarga-s by kāṇḍa in Figure 4 that there might be distinct patterns — e.g., class 1 appears enriched in kāṇḍa-s 1 and 7, which are rare in the other classes. Hence, we next examined if each of the above classes differ significantly in terms of the kāṇḍa-s from which their sarga-s are drawn (Figure 5). This confirms that the kāṇḍa-s 1 and 7 dominate class 1 (\chi^2, p=2.87 \times 10^{-16}). The pattern is inverted for class 2 with triṣṭubh capping verses. This is in keeping with the above proposal of kāṇḍa-s 1 and 7 having a distinct compositional pattern and history. Another notable feature that emerges is the enrichment of kāṇḍa 2 in the class with jagati capping verses (\chi^2, p=5.62 \times 10^{-15}). In conclusion, this suggests that the older aitihāsika kāvya tradition had a style of capping a long run of anuṣṭubh-s with a triṣṭubh or a jagati to mark the end of a section. This practice appears to have given way to the purely anuṣṭubh composition, probably among the kuśīlava-s who subsequently preserved the itihasa-s and the emerging purāṇa-s as an oral tradition.

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Sneha, snowstorms, the sun and the moon in enigmatic ṛk-s

That the Indo-European homeland was a cold place with snow is evidenced by widespread survival of two hon-homologous words for snow. Recently, a discussion on one of these words, sparked by some linguist, landed on my timeline on Twitter. From that, it seemed that people were unaware of the attestation of its cognate in Old Indo-Aryan. This, in turn, reminded me of a discussion we had with our friend in college on the very same issue. Hence, we thought it worthwhile to put down this discussion — this apposite since the winter solstice is approaching, and as we write this note, it is a new moon with a solar eclipse in Antarctica.

The two words for snow that can be traced back to early Indo-European, probably even PIE, are attested in Sanskrit as sneha and himá/hímā. The udātta is represented by the acute sign and differs between the masculine form that is usually used for snow and the feminine for winter.

Sneha is represented by many cognates that include snow itself; Old Iranian has snaēža = to snow; Baltic: (Lithuanian) sniẽgas; Germanic: (Old English): snīwan. There are the 0-graded forms in: Greek: niphás = snowflake; Latin: nivis; Celtic (Welsh): nyf. Thus, it is a solid IE word. Interestingly, it has also been transferred to Sumerian in a form that suggests either early Indo-European or the Satem clade (Balto-Slavo-Aryan) as the possible sources. This might point to a potential trade contact between Sumerians and the Indo-Europeans in their colder lands to the north of the former. Coming to Indo-Aryan, while etymologists recognize the ancestral form as being Sanskrit sneha, they usually only cite Prakritic siṇeha or siṇhā as cognates bypassing Sanskrit as though it is not attested in it. However, as we shall see below, this is not true. In New Indo-Aryan languages that are still familiar with snow we have descendants of the Prakritic forms as the primary word for snow, e.g., Kashmiri shīna (pronounced these days with terminal schwa loss).

The second word hima, was probably paralogously polysemous right from the early stages of PIE, meaning both winter and snow. It is clearly inherited from PIE in the Indo-Hittite sense as we have Hittite gimmanza = winter. We have Baltic (Lithuanian): žiemà = winter; Greek: kheima = snowstorm/cold; khion = snow; Armenian: jiun = snow; Latin: hiems= winter; Celtic (Old Welsh) gaem = winter. In Sanskrit we see all the senses being attested. The usage śata-himā is literally a 100 winters, meaning hundred years. A similar usage is seen in Latin, e.g., bīmus from *bihimos — lasting two years. The form hemanta again denotes a season marking the end of winter. On the other hand, the form himavant means a mountain [covered] with snow. Similarly, in the Ṛgveda hima can be directly used to indicate snow, for example:

himenāgniṃ ghraṃsam avārayethām
pitumatīm ūrjam asmā adhattam ।
ṛbīse atrim aśvināvanītam
un ninyathuḥ sarvagaṇaṃ svasti ॥ RV 1.116.8

With snow, you two averted the scorching fire,
you two bestowed nourishing food for him [Atri],
You two Aśvin-s, led out Atri from the fuming crater,
into which he been led, and all the troop to weal.

This ṛk attributed to Kakṣivant Dairghatamasa [Footnote 2] is an allusion to the famous deed of the Aśvin-s that is repeatedly alluded to in the RV, where they saved Atri from a fuming crater. The simple reading of the verse would imply that Atri was led into it along with his troops. They were cooled with snow and then brought up. The phrase sarvagaṇaṃ svasti occurs only one other time in the RV, in a sūkta of the Atri-s; however, there it refers to the gaṇa-s of the god Bṛhaspati. Nevertheless, one wonders if there is a subtler, undiscovered connection furnished by this cognate phrase from these relatively late sūkta-s of the RV. Finally, one may comment that this tale hints at the possibility of Atri and his men having fallen into one of the geothermal craters in the Caspian-Black Sea region close to the IE homeland.

Returning to sneha-, we tabulate below the occurrence of this word in some old Vedic texts relative to hima- (while we count both senses of hima- we do not count the season hemanta in the below table).

Text sneha hima
Ṛgveda 2 11
Atharvaveda (vulgate) 1 16
Atharvaveda (Paippalāda) 0 35
Taittirīya Saṃhitā 0 10

Thus, it is clear that sneha, while attested in the oldest surviving Indo-Aryan text (the single AV-vulgate instance is identical to the ṛk found in the RV), it is absent in the successor texts though hima- remains as common or is more frequently used. However, we should also add that a related form snīhiti (snowstorm) occurs once in the RV and once in the Taittirīya Āraṇyaka. In the case of sneha, a semantic shift occurred in Sanskrit, where it came to denote a wide range of things, including fluidity, smoothness, oil/fat and love. We will next examine the two occurrences in the RV and suggest that they are semantically aligned with the Middle and New Indo-Aryan forms. Both occur in enigmatic sūkta-s/ṛk-s that need further discussion.

The first occurrence is in a long sūkta of Tiraścī Āṅgirasa on the many glorious acts of Indra, which needs to be described to give some context. The first three ṛk-s (1-3) describe the might of Indra and his vajra and also allude to his act of piercing the 3 \times 7 mountains with his arrow. This motif is reused by Vālmīki in the Rāmāyaṇa when Rāma pierces the seven trees to prove his might (Indra-hood displacing that of the mighty Vālin) to the ape Sugrīva. The following triad (4-6) describes how Indra, in his cosmogonic form as the generator of all beings, slew Ahi with his vajra, even as the mountains shrieked. The Marut-s seeing his valor, approached him for an alliance, like brāhmaṇa-s reciting mantra-s to praise him. The next tṛca (7-9) describes a Marut-centric version of the Vṛtra myth. When Vṛtra violently sallied forth, the other gods, who were the companions of Indra, retreated, leaving him alone in the battle. However, the 360 Marut-s (an unusual count associated with the days in an year) singing the praises of Indra joined him in battle, asking for a share of the ritual offerings. They urged Indra to scatter the anti-deva asura-s with his vajra and cakra, assisted by their battle formation fronted by their sharp spears.

In the next tṛca (10-12), the Āṅgirasa calls on his fellow ritualists to send forth their chants to Indra. In the tṛca (13-15), which includes the ṛk of specific interest to us, the narration moves to the battles fought by Indra against the enemies of the gods in alliance with Bṛhaspati. This triad is soaked in astronomical allegory, with the god Bṛhaspati himself likely being represented in the sky by the planet Jupiter. The main feature of this triad is the mention of a black (shrouding like a cloud) drop, kṛṣṇa drapsa, which is said to wander to the solstitial colure along a sinuous river Aṃsumati. We interpret this river as the lunar ecliptic. The whole myth seems to encode a solar eclipse close to the winter solstice (The old Aryan New Year), in which context the snowstorm is mentioned. The next tṛca (16-18) describes the following acts of Indra: 1. Right when he was born, he became the foe of the seven unrivaled ones (not entirely clear who they are). 2. He discovered Dyaus and Pṛthivi, which were hidden. 3. He set into motion the wide-ranging worlds. 4. He smashed the unrivaled one (Vṛtra) with his vajra. 5. He slew Śuśna. 5. He discovered the hidden cows. 6. He demolished the fortifications. 7. He released the frozen rivers and slew the demoness (an allusion to Dānu) lording over the waters. The final tṛca (19-21) praises Indra as Vṛtrahan and the lord of the Ṛbhu-s, calling on him for the soma offering.

ava drapso aṃśumatīm atiṣṭhad
iyānaḥ kṛṣṇo daśabhiḥ sahasraiḥ
āvat tam indraḥ śacyā dhamantam
apa snehitīr nṛmaṇā adhatta ॥ RV 8.96.13
The drop stood at the Aṃśumatī,
the black [drop] wanders with the ten thousand.
Indra helped it [the drop] blowing along with his skill.
The manly-minded [Indra] repulsed the snowstorm.

drapsam apaśyaṃ viṣuṇe carantam
upahvare nadyo aṃśumatyāḥ ।
nabho na kṛṣṇam avatasthivāṃsam
iṣyāmi vo vṛṣaṇo yudhyatājau ॥ RV 8.96.14
I saw the drop wandering at the solstice,
in the sinuous path of the River Aṃśumatī,
going down like a black cloud,
I impel you, bulls, to fight in the battle.

adha drapso aṃśumatyā upasthe
.adhārayat tanvaṃ titviṣāṇaḥ ।
viśo adevīr abhy ācarantīr
bṛhaspatinā yujendraḥ sasāhe ॥ RV 8.96.14
Then, the drop, in the lap of the Aṃśumatī,
bore the body sparkling with light.
As the deva-less folks moved forth [to attack],
united with Bṛhaspati, Indra conquered [them].

Commentary: The black drop (drapsa, an old IE word) is interpreted as the new Moon. The drop is associated most commonly with soma (with a lunar equivalence in several cases; on rare occasions it is used for Venus). Normally, the soma is silvery — ṛjīśin (like earlier in this sūkta) — reinforcing the lunar connection. However, here the drop is explicitly and atypically described as black, suggesting that the new Moon is alluded to. The first ṛk mentions the drop wandering with a ten thousand: we take this large number to denote the stars. The drop is seen as blowing along: we consider this an early allusion to the primitive Hindu astronomical theory (shared with the Greeks and likely of old IE provenance) of celestial bodies being blown on their paths by cosmic winds. In the second ṛk, the composer states that he sees the black drop which is likened to a cloud — suggesting its shrouding nature — wandering near a colure. He also mentions the sinuous course of the river Aṃśumatī. Given that the word aṃśu is used for soma (or the soma stalks before extraction of the juice) and metaphorically connects the soma plant and the Moon, Aṃśumatī would mean the riverine path with soma/the Moon. Hence, we take this to be an allusion to the lunar path/ecliptic [Footnote 3].

We take the colure (viṣuṇa) to be solstitial. Winter is the season that corresponds to the battle between the gods, led by Indra, and the demons. The verb, ava-sthā = going own, suggests the nether point of the ecliptic path analogized to a river. Hence, we hold that the solstice referred to is specifically the winter solstice. Thus, it is not any new moon but likely the new Moon closest to the winter solstice, corresponding to the old Aryan New Year. This, in turn, supports the idea that the snehiti which Indra averts is indeed a snowstorm. The final ṛk of this triad mentions the black drop in the lap of Aṃśumatī, where it paradoxically takes on a body sparkling with light. We take this to allude to a solar eclipse happening close to the solstice. The at the point of emergence of the sun from the total eclipse or an annular eclipse would indeed give the impression of the black drop (the Moon) taking on a glittering body. Thus, this is a variant of the famous Svarbhānu eclipse myth of the Atri-s but probably referring to a specific eclipse near the solstice. In this context, the attack adevī folks should be taken as a purposeful conflation of the earthly enemies with the asura-s causing the eclipse as in the Svarbhānu myth. Moreover, given the overall celestial setting, the specific involvement of Bṛhaspati, as a companion of Indra, in this conflict suggests the potential presence of Jupiter in the vicinity during this event (Or perhaps in the nakṣatra of Tiṣyā).

We may also point out that the deployment of snow or other “weather weapons” is a feature of the battles of Indra with the dānava-s elsewhere in the RV. For example, Hiraṇyastūpa Āṅgirasa gives an account of the battle between Indra and Ahi, when the latter had frozen the rivers and corralled the cows. Here, Ahi, first tries to pierce Indra with his spear, but Indra evades him by becoming the tail of a horse. Having evaded his strike, Indra conquered the cows and the soma and released the waters. Then he closed in for combat with Ahi:

nāsmai vidyun na tanyatuḥ siṣedha
na yām miham akirad dhrāduniṃ ca ।
indraś ca yad yuyudhāte ahiś ca
utāparībhyo maghavā vi jigye ॥ RV 1.32.13
Neither the lightning nor the thunder scared away [Indra] for him,
neither the snow (/mist) nor the hail that he [Ahi] spread out.
When Indra and Ahi fought each other,
Maghavan triumphed, [then] and also for the time that came.

Here, Ahi deploys various “weather weapons”, reminiscent of the steppe “rain-stone” magic of the Turkic and Mongolic world, but they fail to scare away Indra. The first two, lightning and thunder are unambiguous, and so is the final one, hail (hrāduni). The word miha could mean snow or mist. In either case, it supports the deployment of such weather weapons, consistent with the interpretation of sneha- as snow, i.e., in a snowstorm.

Strikingly, the second occurrence of sneha- is again in the context of the same eclipse myth. This account occurs in the monster sūkta of maṇḍala 9, which agglomerates shorter sūkta-s of various Vasiṣṭha-s and Kutsa Āṅgirasa. We shall consider the whole tṛca with this reference below:

ayā pavā pavasvainā vasūni
māṃścatva indo sarasi pra dhanva ।
bradhnaś cid atra vāto na jūtaḥ
purumedhaś cit takave naraṃ dāt ॥ RV 9.97.52
Bring (addressed to soma), by purifying yourself with this filtering, riches.
At the hiding of the Moon, O drop (moon/soma), run forth into the lake.
The yellowish (sun) is also here as if impelled by the wind.
the wise one (Soma) has indeed given us the man (Indra) for the sally.

uta na enā pavayā pavasva
adhi śrute śravāyyasya tīrthe ।
ṣaṣṭiṃ sahasrā naiguto vasūni
vṛkṣaṃ na pakvaṃ dhūnavad raṇāya ॥ RV 9.97.52
Also with this filtering purify yourself,
at the front of the famous ford of celebration.
Sixty thousand treasures the destroyer of rivals,
like a tree with ripe fruits, will shake down for triumph.

mahīme asya vṛṣanāma śūṣe
māṃścatve vā pṛśane vā vadhatre ।
asvāpayan nigutaḥ snehayac ca
apāmitrāṃ apācito acetaḥ ॥
The bull is his name [Indra], great and fierce, are his two,
deadly weapons, in the hiding of the Moon or in the touching.
He put to sleep the rivals and snowed down on them.
Repulse the enemies, repulse the senseless ones.

Commentary: Composite sūkta-s, like the one in which these ṛk-s of Kutsa Āṅgirasa occur, are typical of the final part of maṇḍala 9. Except for Kutsa, who is the author of the last 4 tṛca-s and the terminal ṛk with the classic Kutsa refrain, all the other authors are Vāsiṣṭha-s. However, throughout the long sūkta (longest in the RV) we find several allusions to the finding of the sun’s path, the holding of the sun, and soma as the Moon. Thus, it is not out of place to furnish an astronomical explanation for these ṛk-s. Key to the interpretation of these ṛk-s is a rare word māṃścatu/māṃścatva whose meaning has puzzled students over the ages. It occurs only thrice in the RV, and its meaning was already obscure to Yāska, who groups it with the words for horses in the Nighaṇṭu. Two of the occurrences are in this tṛca, and one is in RV 7.44.3 by Vasiṣṭha Maitrāvaruṇi. Thus, this word perhaps links the Vāsiṣṭha-s to Kutsa. It can be etymologized as maṃs+catu. Catu can be derived from the root cat- = to hide or vanish. Māṃs (with a pure anunāsika) is taken to mean the Moon, an older variant of mās, closely related to the form in early Indo-European. This form is supported by other Indo-European cognates like: Baltic (Latvian): mēnesis; Latin: mensis. Thus, the word is taken to mean the vanishing of the Moon. To their credit, some white indologists have correctly etymologized this word using comparisons across IE. However, they failed to understand its actual meaning. Notably, on the only occasions it occurs in RV, it is coupled with bradhna — the yellowish or reddish sun. This is also seen in the case of the verse of Vasiṣṭha:

dadhikrāvāṇam bubudhāno agnim
upa bruva uṣasaṃ sūryaṃ gām ।
bradhnam māṃścator varuṇasya babhruṃ
te viśvāsmad duritā yāvayantu ॥
Ever having awakened, to Dadhikrāvan and Agni
I speak; to Uṣas, and the sun, the cow.
The yellowish one from the hiding of the Moon, [becomes] Varuṇa’s brown one,
let them drive away all the bad things from us.

Notably, the first ṛk of Kutsa, explicitly states that the sun (bradhna) is also at at the same place as the hiding of the Moon. The sun is said to be impelled to that place by the wind — again, note an allusion to the old hypothesis of the cosmic winds moving the celestial bodies (c.f. the above ṛk of Tiraścī Āṅgirasa). This conjunction corresponds to amāvāsya or the sun and moon “dwelling together”, resulting in the new Moon (or the hiding/vanishing of the Moon). Hence, māṃścatva should be understood as the “hiding of the moon” at new Moon in all its occurrences. However, there are indications that it is a new moon with a solar eclipse. The final ṛk talks of two events, the māṃścatva and the pṛśana, i.e., the touching. We take this touching as the “contact” of the sun and Moon implying an eclipse. Moreover, the ṛk of Vasiṣṭha, states that the bradhna (usually yellow or red) is Varuṇa’s brown one from the māṃścatu. This suggests that the sun’s darkening, indicating a solar eclipse at the new moon [Footnote 1].

We also encounter the cryptic statements parallel to the ṛk-s of Tiraścī Āṅgirasa, such as the drop (Moon) running into the lake. This is also called the famous ford (tīrtha) — you can cross over to the “other side” there. We take these as allusions to the winter solstitial point on the ecliptic. Thus, we believe both Kutsa and Tiraścī are talking about the same or a similar eclipse close to the winter solstice. However, notably, in this case, it is Indra who showers snow on the enemies, putting them to “sleep” — again reminding us of the use of rain/snow stones in the Altaic warfare. Thus, the two occurrences of sneha+ and the one occurrence of snīh- in the RV indicate the use of this ancient IE word in the sense of snow. The interesting early polymorphism in the form of sneh- and snīh- suggests that the ancestral state of this word, perhaps even its form in the original dialect in which the RV was composed, was close to the ancestral Baltic version (at least the first syllable). Thus, we are probably seeing a fossil of the dialect diversity in early Indo-Aryan or Indo-Iranian itself.

To conclude, we may note, an eclipse at the solstice is a relatively rare event. However, if we give a leeway of about 5 days on either side of the winter solstice, one may get more of such events. Below is a list of such events that have happened or will happen over 3 centuries from 1801-2100 CE anywhere in the world grouped by their \approx 19 year lunar cycle from the catalog of Eclipse Predictions by Fred Espenak (provided by NASA):
21 Dec 1805; 20 Dec 1824; 21 Dec 1843; 21 Dec 1862
22 Dec 1870; 22 Dec 1889; 23 Dec 1908
24 Dec 1916; 24 Dec 1927
25 Dec 1935; 25 Dec 1954; 24 Dec 1973; 24 Dec 1992
25 Dec 2000; 26 Dec 2019; 26 Dec 2038; 26 Dec 2057
27 Dec 2065; 27 Dec 2084
17 Dec 2066

One can see that between 1900-2100 there have been/there are no events on the solstice day. However, in 1870 CE we had a very close pass within 12 hours of the solstice. In general, the 1800s saw several close events, but the following two centuries did not. Due to the 19-year clusters, we cannot be sure that the ones recorded in the RV were the same event. However, their relative rarity and clustering would mean that they might have been dramatic enough to leave a memory in the text.

Footnote 1: This occurs in the context of Dadhikrāvan, who is typically invoked in the dawn ritual. We posit that Dadhikrāvan represents a heliacally rising old Vedic constellation, although its identity still remains uncertain.

Foonote 2: While its style is similar to that of the old Gotama founders like Kakṣivant and his father, the sūkta itself mentions Kakṣivant in the third person and talks of the later Gotama-s. This suggests that it was appended later to the Kakṣivant collection by one of his successors.

Foonote 3: In later Hindu tradition, the lunar and solar paths are often depicted on temple roofs as sinuous snakes.

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The strange case of the Āpastamba sprite

This is the second of the two stories that arose from incidents during the visit of Yaśaśravas, Somakhya’s cousin. With the autumnal vacations, Somakhya was having a good time with his visiting cousin, giving him lectures on the theory and the practice of the study of variable stars. During the day, they spent their time with numerical solutions of differential equations that modeled the pulsations of stars. At night, they observed the stars in good view from Somakhya’s terrace with the smog of the fireworks having lifted. Additionally, as instructed by Somakhya’s father, he was also teaching Yashashravas sections of the hautra mantra-pāṭha. The on the second day of his cousin’s visit, the two of them were doing their evening saṃdhyā rituals together when Somakhya was disturbed by a peculiar behavior evinced by his cousin. In the midst of his prāṇāyāma or arghya, Yashashravas suddenly uttered the word Āpastamba-sūtra almost randomly a few times with a peculiar affectation and cadence. Somakhya was taken aback, but remembering his father’s words of never getting disturbed in his performance, he ignored it and continued to the end. Somakhya: “Yashashravas, what is wrong with you — have you gone crazy? Why were you randomly saying “Āpastamba-sūtra” in the course of your saṃdhyāvandana? I know you are of that school, but it makes no sense to pepper your upāsana with utterances of that word, moreover voiced in that strange manner.” Yashashravas felt as if some weight had lifted off his head and told Somakhya a strange tale. Somakhya was shocked by it, but once he had gathered himself, he said they should wait until his friend Lootika visited before taking any further action.

Somakhya’s mother had wished to invite Lootika’s mother and her daughters for the coming festival of Dattātreya that Somakhya’s family observed; she thought it might also be an opportunity for some socializing, given the visiting relatives. Somakhya was keen that his friend be around to deal with the case of his cousin Yashashravas and hear it from his own mouth. Yashashravas had already heard from their other cousin Babhru of his dramatic encounter with the four sisters; hence, he anticipated it with some excitement. However, Somakhya was a bit concerned that Yashashravas might clam up, feeling put off, if Lootika’s sisters were also around — they were only a little less formidable than his friend in terms of the impression they could produce on introduction, as Babhru had experienced first hand. Shortly after they arrived, Yashashravas quickly felt the edge of the sisters when Varoli, who was younger than him, gave them a little talk about Propynylidyne, Helium hydride, and Argon hydride their formation and energetics in interstellar space. However, as Somakhya had hoped, their mothers drew away the three younger sisters to look at jewelry, clothing and watch some video recording with them in another room. This gave Somakhya, Yashashravas and Lootika the solitude and time they wanted.

S: “Yashashravas, Lootika is our confidante and will be of great help in trying to get to the bottom of your predicament. You can reliably share all you need to with us. Let us lose no time and get started. For Lootika’s benefit, could you tell your story right from the beginning going back to your upanayana?”
Y: As you know, Somakhya, for a variety of reasons, my upanayana was performed about 6 years past the earliest recommended age at Kshayadrajanagara. It was a big affair, and a pity you could not be there due to the exams you had to give. However, our cousins and relatives like Babhru, Saumanasa, Mandara, Charuchitra and Varaha, as also our friend Indrasena and his brother, were all in attendance, and we had great fun. I recall one major untoward incident. The giant coconut tree in our grandfather’s house fell crashing the night after the upanayana and destroyed the maṇḍapa. Hence, we could not perform the Skanda-pūjā the next day, which is customary in our families. For three days after the upanayana, a snātaka from Kāśi, who, like me was an adherent of the Āpastamba school, was sent by the purohita who had performed your Atharvan upanayana. He taught me the correct performance of the saṃdhyopāsana along with some scholiastic material of Khaṇdadeva for several hours. On the afternoon of fourth day after the upanayana, Indrasena and I were seated on the parapet of our grandfather’s house yarning away about something when we saw the said snātaka come. I was a bit surprised because he was supposed to come only for 3 days, and it was not yet the time for the saṃdhyā. I thought he had come by error and wanted to tell him so. So, I jumped down from the parapet and ran out of the gate towards him. He just ignored me and kept walking ahead, all the while repeatedly saying “Āpastamba-sūtra” in the same manner you heard me, most unfortunately, utter it during the saṃdhyā.

To my surprise, he just vanished at the end of the road on which our grandfather’s house stands. It turned to shock when Indrasena said that he suddenly disappeared from his sight too when I reached him. It got worse when Indrasena’s brother, Pinakasena, who was beside the gate making a tail for a kite, said that he thought we were playing some stupid prank when he saw me speaking to the air — i.e., he did not see the snātaka at all. It was all strange and funny for them, but for me, it was just the beginning of a horror story! That evening as we sat for saṃdhyā, I began jabbering “Āpastamba-sūtra” randomly in the course of the ritual. Indrasena admonished me to be serious with the ritual and did not believe it when I said that it was just involuntarily coming to me without any effort on my part. To my horror, the same thing happened when I was doing it with my father, and he blasted me for being frivolous with the ritual. I felt too embarrassed to tell him what was happening as I feared he might think I have gone mad. With great self-effort, I acquired the ability to control it to a degree and skipped sāyam-saṃdhyā if I could, for it came upon me only in the evenings. I also got some relief, albeit incomplete, when, after vedārambha, I started studying the rakṣohā mantra-s to Viṣṇu from the Taittirīya-śruti. It returned with a frenzy on the upākarmā day and completely ruined it. I told this to Babhru, who, as you know, is quite frivolous, and he responded it was a good reason to stop saṃdhyā altogether. But thereafter, he told me the unbelievable tale of when you had visited him with Lootika and her sisters and added: “maybe you should ask them to come over to your place too”. That is why I felt some relief when you mentioned that this is something you would like to handle with Lootika around.”

S: “So, Lootika, what do you think of this?”
L: “Most remarkable and ghastly. It should be quite a problem for a V1 male not to be able to perform his saṃdhyā properly. But I’m puzzled by this utterance of “Āpastamba-sūtra”. Is that something peculiar to the Āpastamba school? As you know, my family follows the Bodhāyana-sūtra, though, as far as I know, the Āpastamba-s are not very divergent, except that they have lost the proper recitation of the Śrīsūkta and the Skandayāga in its entirety.”
Somakhya smiled saying:“I know you Bodhāyana-s are pretty proud of your Śrīsūkta and Skandayāga”, even as Yashashravas intently turned his gaze from one to the other. Suddenly, Lootika excitedly remarked: “Ah! Somakhya, I remember you telling me of this particularly malevolent sprite known as the Āpastamba-graha. My gut tells me that your cousin has been seized by that.”
S: “Yes, Gautamī! This is indeed the first time I am encountering that sprite in person. Never thought we would do so in this life.”
L: “I remember you telling me that he can be particularly nasty if we try to bind him directly. So, how do we proceed against him?”
S: “We should first hear him out. We will get some clue about how to proceed from that.”
L: “Sounds exciting. Can we get him to speak with a Ḍāmara-mantra?”
S: “We can do so, but we have to take extra precautions and prepare Yashashravas for it. I also suggest that you recall the short Vīrabhadrāstra and keep it ready in your mind.”
Somakhya went to the sacristy and brought out a special svayaṃbhu-liṅga that he and Lootika had installed, some flowers, incense, a copper plate, and a long abhicārika nail. He asked Yashashravas to perform a Śivārchana of the liṇga with the following incantations:
oṃ rudrāya namaḥ । oṃ uḍḍīśāya namaḥ । oṃ sudurlabhāya namaḥ । oṃ kapardine namaḥ । oṃ virūpākṣāya namaḥ । oṃ sarva-graha-bhayāpahāya namaḥ ॥
Somakhya: “Lootika, why don’t you go out to the veranda and ready yourself to deploy the Ḍāmara-prayoga; I’ll prep Yashashravas for it in the meantime. Yashashravas, we diagnose you as being seized by a sprite known as the Āpastambagraha. The sprite had its origin in the distant past in the Marahaṭṭa country and has been spreading like a slow virus by capturing Āpastamba V1s throughout the southern half of Bhārata and beyond. Now, recall the great Mṛtyulāṅgala incantation. It is a powerful but also an extremely dangerous incantation. It is important that you periodically keep repeating it, if not daily. The biggest danger from the sprite lies in interfering with the performance and recall of this mantra — this can prove fatal to the victim. While I have never encountered an Āpastambagraha before, I have heard of such a fatality in the case of a V1 from the Andhra country. Now perform its nyāsa:
Mṛtyulāṅgalasya Vasiṣtha ṛṣiḥ । anuṣṭubh chandaḥ । Kālāgnirudro devatā ॥
Having done that do japa of it as per the form deployed by the vipraugha with the bīja-saṃpuṭi-karaṇa that I’ll specify and not the aiśa form heard in the śruti:
ṛtaṃ satyam param-brahma-puruṣaṃ kṛṣṇa-piṅgalam ।
ūrdhvaretaṃ virūpākṣaṃ viśvarūpaṃ namāmy aham ॥
oṃ krāṃ krīṃ huṃ phaṇ namaḥ ॥

Somakhya then drew a circle and asked his cousin to sit inside it and start a japa of the said mantra. He warned him that once Lootika deployed the Ḍāmara-prayoga he could go into a trance and asked him not to resist it. Lootika then came in having fortified herself for the Ḍāmara-prayoga. She was a bit nervous from the fact she was dealing with an Āpastambagraha with a nasty reputation. Somakhya told her to calm down: “Lootika, the Āpastambagraha only possesses men, but as a male graha with affiliations to the brahmarakṣas class, it does have a tendency to grab V1 females without possessing them. I suspect he would not have interest in possessing me as Āpastamba is not my primary school but he could still lash out. That’s why I think you should deploy the Ḍāmara-prayoga solo so that I can perform a shielding prayoga for you. Yet, be warned it might break through; therefore, be ready with the Vīrabhadrāstra if it tries to attack you.” Lootika then placed a neem stave on Yaśaśravas’s head and right away deployed the Grahavādini-mantra:
oṃ namo bhagavate mahākālarudrāya tripuravināśanakāraṇāya virūpākṣāya sarvabhūta-graha-vetālādhipataye rudrasyājñayā vada vada vada vada huṃ phaṭ svāhā ॥

About 3 minutes into the prayoga, they noticed that Yashashravas was slipping into a trance. A minute later, he started prattling like Aitaśa of yore, uttering the single word — “Āpastamba-sūtra” in spurts with a strange cadence. Lootika was taken aback by the first encounter with the graha, but regaining her composure, she continued with the prayoga and sprinkled some bhūti on him. He then signaled for writing material and launched into few minutes of frenzied writing. At the end of it, he just fell flat as though exhausted from heavy exercise. Somakhya sprinkled some water on him from his kamaṇḍalu, and he returned to his senses handing over the sheets of paper to his companions. “I cannot believe I wrote all that stuff down while feeling like being in an almost catatonic state.” S: “It looks like you have covered the first page with many repeats of “Āpastamba-sūtra”. Not surprising. There are parts where you seem to have doodled away in some South Indian script with its typical twisting curves that we cannot read. Yet, there seem to be coherent, understandable blocks. Could you kindly read those parts out?”

Y: “I used to be Gaṇḍalepa and was born in the Marahaṭṭa country. As a kid, I seemed to remember elements of my past janman as a great śāstrin of the Veda and was reciting the Aṣṭādhyāyi with svara-s by the time I was an year old. I started learning the Taittirīya-śruti at age 3, even before my upanayana. By age 12, I had mastered it and moved on to study the kalpasūtra of Āpastamba, the school to which my family adhered. By 15, I had earned the reputation that if paṇḍita Haradatta Miśra were alive, he would be the one studying at my feet. When I turned 18, I realized that my scholarship would not support me, and I had a wife too. I was desperate to get employment that paid better than the paltry stipend I got from the Chatrapati’s fund. By the grace of god Viṣṇu, I found sardār Khaṭāvkar, and he appointed me as his nyāyādhīśa, a job for which I was imminently suited given my authoritative knowledge of the dharmaśāstra-s. I also helped Khaṭāvkar with the commentary he was writing on some Pāñcarātrika texts. Khaṭāvkar, in turn, trained me in arms, and I became a reasonably proficient fighter.

However, Khaṭāvkar had an evil side to him, and it is perhaps partly due to the association with his unprincipled acts that this fate has come upon me. He was a partisan of the late Padishaw Awrangzeb and a good friend of Nawab Sahib Daud Pathan. One day, Nawab Sahib hatched a plan for a raid on Khargaon and asked Khaṭāvkar to help; I joined the raiding band too. During the raid, the Nawab Sahib, to his credit, instructed the men not to attack the temple of Aṣṭhabhairava in the town. However, some Afridis and Bohras in his band disobeyed his command and burnt the temple anyway. After the arson and looting, Khaṭāvkar observed that there were gold and silver utensils of the temple and decided to loot them. He was kind enough to ask me to join him and take a share of the plunder. Losing my sense of propriety, I did so. Soon after that, Khaṭāvkar met his end with the action the new Chatrapati took against him. The gods may take multiple janman-s to punish you. Indeed, Khaṭāvkar has taken at least four and is probably still experiencing the fruits of his deeds piecemeal. I was spared and, under the patronage of the courtiers, had an opportunity to perform śrauta rituals.

I became the adhvaryu for the great yajamāna Lakṣmaṇa Śāstrin and performed several rites for him. However, in the course of that, I committed acts that I should not have done. I kept out the Mādhyaṃdina-s censuring them as false V1s, and I also kept out the Ṛgvedin-s from being elected as Hotṛ-s because my Taittirīyaka associates and I could take up their role with our hautra-pariśiṣṭa-s. An enraged Ṛgvedin put a case in the court against me and my associates. We realized that we were likely to lose it. I had exorcised a graha from an associate of mine and kept it bound by my mantra-s. I decided to dispatch that graha against the Ṛgvedin so that he would stumble in his testimony at court. I did not realize that it was an Āpastambagraha. But the Ṛgvedin, Gore by name, had some mastery of the Atharvan lore or the Kashmirian pariśiṣṭa; thus he deployed a pratīchīna-prayoga. The moment he started reciting “yāṃ kalpayanti…” the Āpastambagraha came hurtling back and seized me. It interfered with my ability to remember the Mṛtyulāṅgala incantation, and I expired six months later. As is typical of these Āpastambagraha-s, they spawn a new one each time they slay a victim, and I soon became one in search of a host.

In the meantime, Nawab Sahib had been murdered by another Nawab and had come back as a liquor-seller. I hung out at his liquor-stall, making occasional conversation with him as a friendly graha. One day, a V1 named Kuṇṭe arrived there and helped himself to a few swigs. I was thus able to seize him right away and make him jabber just like you. However, before I could finish him off, he was taken by his people to the Piśācamocana-kṣetra at Kāśi, and a gaṇa of Rudra forced me to leave him. There I hung out in a tree for centuries before I could seize the snātaka who taught you at Kāśī. He had intoned the Mṛtyulāṅgala mantra while on a commode, thus, becoming easy prey for me. He died earlier that day when I seized you spawning another Āpastambagraha. Since your family had failed to protect you with the Skanda ritual, I knew I had a new host and duly took hold of you. I have not yet been able to entirely break your defenses either because you have firmly maintained your brahmacarya. But it will not be long before I’m able to bring your chapter to a close. I also intend taking this pretty girl who had the temerity to make me speak along as a slave maid for my ghostly wanderings.”

Terrified, Yashashravas handed the script back to Somakhya: “I seem to have scrawled another page full of Āpastamba-sūtra at the end … but is there a way out of this? I don’t want to go the way of the snātaka.” Just then, Lootika’s mother called her: “Lootika, we need to be going, hurry up!” L: “Mom, we are in the midst of something important; I’ll get back by myself later.” L.M: “Remember, you don’t have your bike with you, so you have to come.” Thankfully for her, Somakhya’s mother intervened: “They seem to be engrossed in their fun — so, let them be. My husband will drop her back in the evening when he takes the offerings to the Rudra-caitya.” After some wrangling, Lootika’s mother let her stay. Just as that was settled, they all felt the ground rattle as if there was an earthquake. Lootika felt her hairband snap and someone pulling her locks: “Ouch, I fear the distraction caused by my mother to my japa has resulted in him getting me.” Suddenly Lootika saw her bag creeping on the floor towards the door: “He’s going for my siddhakāṣṭha.” However, she managed to grab her bag and retrieve her siddhakāṣṭha and deploy the mantra:
huṃ drutam muñca muñca māṃ bhadrakālī-vīrabhadrau ājñapayataḥ phaṭ ॥
With that, she managed to shake off the Āpastamba-graha. S: “That is our chance. Yashashravas return to your japa.” Somakhya got out the nail and, going over to the liṅga performed a kamaṇḍalu prayoga with the mantra:
huṃ namaḥ ṣaṇmukhāya huṃ phaṭ duṣṭaṃ graham astreṇa vitudāmi pāśena kīle badhnāmi ॥

The nail leapt out of his hand and dropped on the plate with a prolonged jingle before coming to a rest. L: “That was a close brush but I believe we have him.” S: “Indeed, did you notice how he tried to dissimulate his name thinking we may use it in place of an amum in the prayoga?” L: “Hmm… I was puzzled by that and unsure if it was written in some strange script, and didn’t know what to make of it. That’s why I made sure not to use it in any prayoga.” They sprinkled some water on Yashashravas and asked him to conclude his japa with an arghya using the Mṛtyulāṅgala mantra. S: “Yashashravas, I believe you should have a smooth saṃdhyā this evening.” L: “The tale of the late snātaka indeed reminds me of the V1 who is mentioned to have been orginally seized by a piśāca-graha in the tale of the Piśācamocana-tīrtha.” Later that evening, Somakhya’s father drove them to the caitya, where, after the initial darśana-s, they went to the sub-shrine of the Ātreya to deliver the offerings. Since Lootika and Somakhya had a fear of dogs, they let Yashashravas feed some curs while uttering vedo .asi ।. As he was doing so, Somakhya and Lootika ran up to the giant, ghostly aśvattha tree on the temple grounds and drove the nail into the ground at its base.

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Agni as the divine commander in the Veda and the Purāṇa

With the Sākamedha-parvan with the oblations to Agni Anīkavat having just passed, we present a brief note on Agni as the general of the deva army. Agni is presented as the commander of the deva-s in the brāhmaṇa literature. For example, the Śatapatha-brāhmaṇa of the Śuklayajurvedin-s makes the below statement in regard to the ratnin oblations, which are made in houses of the various functionaries of the old Indo-Aryan state. With regard to the oblation to Agnī Anīkavat we hear:

araṇyor agnī samārohya senānyo gṛhān paretyā + agnaye .anīkavate aṣṭhā-kapālam puroḍāśaṃ nirvapati । agnir vai devatānām anīkaṃ senāyā vai senānīr anīkaṃ tasmād agnaye ‘nīkavate । etad vā asyaikaṃ ratnaṃ yat senānīs tasmā evaitena sūyate । taṃ svam-anapakramiṇaṃ kurute । (in Mādhyaṃdina SB 5.3.1 \approx in Kāṇva SB 7.1.4)
Having taken up the two fires (Gārhapatya and Āhavanīya) on the two kindling-sticks, having gone to the house of the commander of the army, he prepares a cake on eight potsherds for Agni Anīkavat. Agni is indeed the leader (anīka) of the gods, and the commander is the head of the army: hence, for Agni Anīkavat. He, the commander, is verily one of his (the king’s) gems. Therefore, for him [the king], he [the commander] is thus consecrated. He [the king] makes him [the commander] his own follower.

Again, in the Gopatha-brāhmaṇa we have the following statement regarding the offering to Agni Anīkavat in the context of the Sakamedha oblations:

aindro vā eṣa yajña-kratur yat sākamedhāḥ । tad yathā mahārājaḥ purastāt senānīkāni vyuhyābhayaṃ panthānam anviyād evam evaitat purastād devatā yajati । tad yathaivādaḥ somasya mahāvratam evam evaitad iṣṭi-mahāvratam । atha yad agnim anīkavantaṃ prathamaṃ devatānāṃ yajati । agnir vai devānāṃ mukham । mukhata eva tad devān prīṇāti । (in GB 2.1.23)
These Sākamedha-s are verily that of Indra. Just as the emperor placing the commanders in the head of his army-formations advances unchecked on his path, so also, he (the ritualist) sacrifices placing the deities to the front. Just as there is the Mahāvrata of the soma sacrifices, this is the [equivalent of the] Mahāvrata for the iṣṭi-s. Now, in that, he sacrifices to Agni Anīkavat (the commander), the first of the deities. This Agni is indeed the mouth of the gods. Thus, he pleases the gods through their mouth.

As an aside, we may note that the Atharvavedic tradition sees the Sākamedha oblations as the equivalent of the Mahāvrata-s for the iṣṭi cycle. The Mahāvrata is performed at the winter solstice. The Sākamedha on the Kārttika full moon is the last full moon in autumn before the winter solstice. Hence, the two are seen as being equivalent. Now, this role of Agni as the commander of the gods is already hinted at by multiple incantations in the Ṛgveda (reproduced in the Atharvaveda) itself. For example, we have:

agnir iva manyo tviṣitaḥ sahasva senānīr naḥ sahure hūta edhi । RV 10.84.2a
Blazing like Agni, o battle fury, conquer! Our commander, the conqueror when you are invoked at the kindling.

Here Manyu (the battle fury) is implied to lead the forces, like commander Agni. We may also consider one of the Agni Anikavat incantations Atri-s:

uta svānāso divi ṣantv agnes
tigmāyudhā rakṣase hantavā u ।
made cid asya pra rujanti bhāmā
na varante paribādho adevīḥ ॥ RV 5.2.10
Also, in heaven, let there be the roars of Agni
with [his] sharp weapons for the smiting of rakṣas-es.
Indeed, in his exhilaration, his fury smashes forth,
the defense of the ungodly do not contain him.

Finally, we also have the famous Rakṣohā Agni incantations of Vāmadeva Gautama (RV 4.4.1-5), which present the most exalted account of Agni’s war-like nature in the entire śruti [Footnote 1]:

kṛṇuṣva pājaḥ prasitiṃ na pṛthvīṃ
yāhi rājevāmavāṃ ibhena ।
tṛṣvīm anu prasitiṃ drūṇāno .
astāsi vidhya rakṣasas tapiṣṭhaiḥ ॥ 1
Make your charge like a broad front.
Move forth like a mighty king with his troops,
thirsting to charge forth slaying,
You are an archer: pierce the rakṣas-es with the hottest missiles.

tava bhramāsa āśuyā patanty
anu spṛśa dhṛṣatā śośucānaḥ ।
tapūṃṣy agne juhvā pataṅgān
asaṃdito vi sṛja viṣvag ulkāḥ ॥ 2
Your swirling [weapons: cakra-s implied] fly swiftly;
touch down on [the foes] impetuously blazing.
O Agni, with your tongue [hurl] blasts of heat, flying [sparks]
unstopped hurl forth firebrands all around.

prati spaśo vi sṛja tūrṇitamo
bhavā pāyur viśo asyā adabdhaḥ ।
yo no dūre aghaśaṃso yo anty
agne mākiṣ ṭe vyathir ā dadharṣīt ॥ 3
Send out spies against (the foes). He is the fastest.
Become the uncheated protector of these people.
Whoever wishes us evil from a distance, whoever from nearby,
O Agni, may no one (enemy) evade your meandering course.

ud agne tiṣṭha praty ā tanuṣva
ny amitrāṃ oṣatāt tigmahete ।
yo no arātiṃ samidhāna cakre
nīcā taṃ dhakṣy atasaṃ na śuṣkam ॥ 4
O Agni stand up, stretch your bow against [our enemies],
Burn down the foes, o one with a sharp weapon.
Whoever makes hostile moves at us, o kindled one,
burn him down like dry shrubs.

ūrdhvo bhava prati vidhyādhy
asmad āviṣ kṛṇuṣva daivyāny agne ।
ava sthirā tanuhi yātujūnāṃ
jāmim ajāmim pra mṛṇīhi śatrūn ॥ 5
Rising upwards, jabbing against [the foes, pushing them]
away from us; make your divine powers apparent.
Slacken the taut [bows] of those incited by yātu-s,
be they kin or non-kin slay forth the enemies.

In the transition between the Vedic and Epic phases of the Hindu literary activity, the role of Agni as the commander of the gods was transferred to his son Kumāra, effectively also the son of Agni’s dual Rudra. The stage for this is set deep in the śruti itself. As noted before, the Kaumāra mythology is closely tied to the famous sūkta of the Atri-s we mentioned above (RV 5.2). Already in the fragmentary Vedic tradition of the Bhāllavi-s, we see the hint of this connection in the deployment of the ṛk-s from RV 5.2 by Vṛśa Jāna for the Ikṣvāku ruler Tryaruṇa. The Śatapatha-brāhmaṇa, explaining the duality of Agni and Rudra, explains that the 8 forms of Rudra culminating in the supreme Īśāna, the lord of all, are the 8 transformations of Agni, with Kumāra as the 9th, evidently alluding to the very same sūkta of the Atri-s:

tāny etāny aṣṭāv agni-rūpāṇi । kumāro navamaḥ saivāgnis trivṛttā ॥ Mādhyaṃdina SB
These then are the eight forms of Agni (Rudra, Śarva, Paśupati, Ugra, Aśani, Bhava, Mahādeva, Īśāna). Kumāra (the boy) is the ninth: that is Agni’s threefold state (i.e., 3 \times 3).

This sets the stage for the final ninth form as the son of Rudra-Agni. In this process, Kumāra also inherited to connection to the old equinoctial connection to the constellation of Agni, i.e., Kṛttikā-s – he is also their son, Kārttikeya. Also contributing to the identity of the para-Vedic Kumāra are the aspects of the Vedic sons of Rudra, the Marut-s, who are also seen as leaders of the deva army. For instance, in the Apratiratha Aindra sūkta we have:

devasenānām abhibhañjatīnāṃ jayantīnām maruto yantv agram ॥ RV 10.103.8c
May the Maruts go at the forefront of the shattering, conquering armies of the gods.

Notably, the Maruts are repeatedly referred to as Agni-s (e.g., agnayo na śuśucānā ṛjīṣiṇo bhṛmiṃ dhamanto apa gā avṛṇvata । RV 2.34.1c; na yeṣām irī sadhastha īṣṭa āṃ agnayo na svavidyutaḥ pra syandrāso dhunīnām । RV 5.87.3c; te rudrāsaḥ sumakhā agnayo yathā tuvidyumnā avantv evayāmarut ।RV 5.87.7a; ye agnayo na śośucann idhānā dvir yat trir maruto vāvṛdhanta । RV 6.66.2a). This completes the circle of the connection between Agni and the sons of Rudra, who are the spear-wielding heroes of the deva army.

However, there are rare instances in the itihāsa-purāṇa corpus that furnish descriptions of the martial Agni mirroring his role as commander of the divine army in the śruti. We provide snippets of such accounts below from the third section of the Harivaṃśa, the Bhaviṣya-parvan (sometimes called Appendix I). The first snippet is from the narration of Agni leading the gods in the battle against the daitya Balin, who was subsequently trampled by Viṣṇu:

lohito lohitagrīvo hartā dātā haviḥ kaviḥ ।
pāvako viśvabhug devaḥ sarva-devānanaḥ prabhuḥ ॥
subrahmātmā suvarcaskaḥ sahasrārcir vibhāvasuḥ ।
kṛṣṇavartmā citrabhānur devāgryaś citra ekarāṭ ॥
lokasākṣī dvijahuto vaṣaṭkāra-priyo ‘rcimān ।
havyabhakṣaḥ śamīgarbhaḥ svayoniḥ sarvakarmakṛt ॥
pāvanaḥ sarvabhūtānāṃ tridaśānāṃ taponidhiḥ ।
śamanaḥ sarvapāpānāṃ lelihānas tapomayaḥ ॥
pradakṣiṇāvarta-śikhaḥ śucilomā makhākṛtiḥ ।
havyabhug bhūtabhavyeśo havyabhāgaharo hariḥ ॥
somapaḥ sumahātejā bhūteśaḥ sarvabhūtahā ।
adhṛṣyaḥ pāvako bhūtir bhūtātmā vai svadhādhipaḥ ॥
svāhāpatiḥ sāmagītaḥ somapūtāśano ‘dridhṛk ।
devadevo mahākrodho rudrātmā brahmasaṃbhavaḥ ॥
lohitāśvaṃ vāyucakraṃ ratham āsthāya bhūtakṛt ।
dhūmaketur dhūmaśikho nīlavāsāḥ surottamaḥ ॥
udyamya divyam āgneyam astraṃ devo raṇe mahat ।
dānavāṇāṃ sahasrāṇi prayutāny arbudāni ca ॥
dadāha bhagavān vahniḥ saṃkruddhaḥ pralaye yathā ।
prāṇo yaḥ sarvabhūtānāṃ hṛdi tiṣṭhati pañcadhā ॥
Red, red-necked, the destroyer, the giver, the oblation, the poet;
The purifier, all-consuming, the god, the mouth of the gods, the lord;
The soul of mantras, brilliant, thousand-rayed, full of light;
With black tracks, with wonderful rays, the leader of the gods, beautiful, the sole ruler;
The witness of the worlds, invoked by twice-born, delighting in the vaṣaṭ call, bright;
The oblation-eater, dwelling in wood, self-reproducing, doer of all acts;
The purifier of all beings, the wealth of the gods’ tapas;
The suppressor of sins, with a flickering tongue, full of heat;
With helical swirling flames, bright haired, of the form of ritual;
The oblation-eater, lord of past and future, the partaker of the ritual share, yellow;
The soma-drinker, of good great luster, the lord of beings, the slayer of all beings;
The unassailable one, the purifier, the power, verily the core of beings, Svadhā’s husband;
Svāhā’s husband, the Saman song, the filtered-soma consumer, holding the [soma]-pounding stone;
The god of gods, of great wrath, of nature of Rudra, born of incantations;
Having mounted the chariot drawn by red horses with wind-wheels, the maker of beings,
the smoke-bannered, the smoke-tufted, blue-clothed one, foremost of the gods,
having raised the divine Āgneya missile in battle, the great god
burnt down thousands, millions and tens of millions of dānava-s,
like in the cosmic dissolution, the enraged lord of fire;
He, who is the metabolism of all organisms, situate five-fold in their hearts.

The second short snippet alludes to Agni’s weapons and the manifestation of the other gods with their weapons. This comes from the narration of the fruits of the tapasya of the deities and presents the appearance of Agni within the frame of the old Vedic ritual of the churning out of the fire in the manner of the Atharvan-s of yore:

atha dīkṣāṃ samāsthāya sarve viṣṇumayā gaṇāḥ ।
puṣkarād agnim uddhatya praṇīya ca yathāvidhi ॥
juhuvur mantravidhinā brāhmaṇā mantracoditāḥ ।
haviṣā mantrapūtena yathā vai vidhir eva ca ॥
sa cāgnir vidhivat tatra vardhate brahmatejasā ।
tejobhir bahulībhūtaḥ prabhuḥ puruṣavigrahaḥ ॥
brahma-daṇḍa iti khyāto vapuṣā nirdahann iva ।
divya-rūpa-praharaṇo hy asi-carma-dhanurdharaḥ ॥
gadī ca lāṅgalī cakrī śarī carmī paraśvadhī ।
śūlī vajrī khaḍgapāṇiḥ śaktimān varakārmukī ॥
viṣṇuś cakradharaḥ khaḍgī musalī lāṅgalāyudhaḥ ।
naro lāṅgalam ālambya musalaṃ ca mahābalaḥ ॥
vajram indras tapoyogāc chataparvāṇam ākṣipat ।
rudraḥ śūlaṃ pinākaṃ ca tapasā-dhārayat prabhuḥ ॥
mṛtyur daṇḍaṃ pāśam āpaḥ skandaḥ śaktim agṛhṇata ।
jagrāha paraśuṃ tvaṣṭā kuberaś ca paraśvadham ॥
Having taken the dīkṣā for ritual all the troops [of V1s] imbued with Viṣṇu, churned out Agni from the [lotus/] pond [Footnote 2] and led him forth as per the injunctions [Footnote 3]. Invoked by the brāhmaṇā-s as per the Vedic instructions, and inspired by mantra incantations, and [fed with] offerings purified with mantra-s verily as per the injunctions, he, Agni, as per tradition, blazed forth there with brahman luster. The lord, having become manifold with rays, became anthropomorphic. He is known as the rod of brahman, appearing as though burning [all] with his body. With weapons of divine form, indeed holding a sword, shield and bow. With a mace, plowshare, discus, arrow, shield, battle-pickax, trident, thunderbolt, he is sword-armed and wields a lance and an excellent bow. Viṣṇu is armed with a discus, sword, pestle and a plowshare. Nara of great might is armed with a plowshare and a pestle [Footnote 4]. United with tapas, Indra strikes with the thunderbolt of a hundred edges. The lord Rudra by tapas has taken up the trident and Pināka [Footnote 5]. Yama took the rod, Varuṇa (literally waters), the lasso and Guha the lance [Footnote 6]. Tvaṣṭṛ took up an ax and Kubera a battle-pickax.

While the paurāṇika tradition presents accounts of several of the ancient battles between the deva-s and the daitya-s, all surviving versions aim to downgrade the pantheon as represented in the old Aryan layer of the religion for magnifying their sectarian deities. Nevertheless, we believe that the paurāṇika tradition preserves relatively unmodified fragments from an older layer of narratives. This is supported by the sharing of phrases with the old tradition (e.g., vajraḥ śataparvaṇaḥ or kṛṣṇavartman) and the fact that above snippets are replete with Vedic allusions and metaphors. Both these snippets present Agni in his old martial form; the first might be seen as mirroring and augmenting the presentation of Agni in the famous kṛṇuṣva pāja iti pañca incantations, while the second mirrors his emergence, upon being churned out, presented in the above-mentioned sūkta of the Atri-s. Thus, even with all the religious turnover, some of the primal imagery from the old Aryan past continued relatively unchanged in the paurāṇika tradition.

Footnote 1: Notice the Vedic device of ring-linking in structuring sūkta-s with old IE roots. The words kṛṇuṣva…vidhya from the first are repeated in reverse order in the fifth of the kṛṇuṣva pāja iti pañca as vidhyādhy…kṛṇuṣva to complete the classic ring. Then an intricate network is formed by other linkages; for example: in the first ṛk, the two hemistiches are linked by prasitim. In the second ṛk, they are linked by the root patan+. Then 1 and 2 are linked by the root tap+; prati links 3, 4, 5; 4 and 5 are linked by the root tan+ and so on. For a complete graph, see Figure 1. Such intricate weaving is especially typical of magical incantations, c.f. the Apratiratha Aindra.


Figure 1.

Footnote 2: This is modeled after the action of the primordial Atharvan, which is mentioned to in the śruti:  tvām agne puṣkarād adhy atharvā nir amanthata । RV 6.16.13a. This is a likely allusion to the fire within water found in the regions closer to the homeland of the early Indo-Europeans.

Footnote 3: The Vedic ritual of Agni-praṇayana with the recitation of the Hotṛ and the carrying forth of the wooden sword along with the fire to the mahāvedi.

Footnote 4: The coupling of Nara and Viṣṇu situates this narration with the Nara-Nārāyaṇa tradition of the epic Vaiṣṇava religion where it is often juxtaposed with the more prevalent Sātvata tradition. This verse hints at their “merger” by furnishing Nara with the iconography of the Sātvata Saṃkarṣaṇa.

Footnote 5: The Pināka should be correctly understood as the bow of Rudra.

Footnote 6: This hemistich has many readings. e.g., The Gītā press text reads:  mṛtyur daṇḍaṃ pāśam āpaḥ kālaḥ śaktimagṛhṇata ।; The Pune reading from Parashuram Lakshman Vaidya reads: mṛtyur daṇḍaṃ sapāśaṃ ca kālaḥ śaktim agṛhṇata । We take an uncommon southern reading which more congruent with regard to the weapons and the gods.

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The birth defects of Dhṛtarāṣṭra and Pāṇḍu and related matters

This note has its origin in a conversation with Sharada. We originally intended to incorporate the core of it into one of our usual fantastical stories. However, following a second conversation with her, we decided that it might be best to present it as a note of its own.

Vyāsa Pāraśarya was the author of our national epic, the Bhārata, in more than one way: he first sired the protagonists, and then he recorded their history as it played out. The queen Satyavatī had extracted a promise from her husband Śantanu that her son and not his older son Bhīṣma would take the throne of Hastināpura. However, to her bad luck, both her sons via Śantanu, Citrāṅgada and Vicitravīrya, died shortly after ascending the throne — an eponymous gandharva slew Citrāṅgada at Kurukṣetra and Vicitravīrya contracted tuberculosis and perished before fathering any children. He left behind his widows Ambikā and Ambālikā. Distraught, Satyavatī first asked her stepson Bhīṣma to father children on the widows; however, he refused to break the vow of celibacy he had taken for Śantanu to marry her. Before her marriage to Śantanu, Satyavatī was a boat-woman from a fisher clan who ferried people across the Yamunā. In course of her duties, she once had to ferry the great brāhmaṇa, Parāśara of the Vasiṣṭha clan. He was smitten by her beauty and started wooing her with sweet words during the boat ride. She was caught in the dilemma of her father’s wrath if she went with him and the brāhmaṇa’s curse if she refused. However, the boon he offered convinced Satyavatī to consort with him. He enveloped the region in a mist with his magical powers, and they engaged in coitus. As a result, they had a son, the illustrious Vyāsa Pāraśarya, the editor of the Veda-s and the composer of the Bhārata. Thereafter, Parāśara restored her virginity and gave her the boon by which a pleasant perfume replaced her fishy odor. Vyāsa went with his father, and Satyavatī continued as a boat-woman until her marriage to the king Śantanu. Now in this hour of need, she summoned her first son Vyāsa and asked him to sire children on her widowed daughters-in-law. Vyāsa agreed but stated that the Kausalya princesses should undergo an year of preparatory rites before engaging in coitus with him. However, fearing the dangers of a kingless state, Satyavatī pressed her son to inseminate them immediately.

Evidently, from his being a yogin performing tapas, Vyāsa was in an uncouth state with yellowish-brown dreadlocks, unshaven face and body odor. Thus, when he had intercourse with Ambikā, she closed her eyes not to see his grim visage. As a result of this “impression” of hers, Vyāsa told Satyavatī that Ambikā’s son would be duly born blind despite having the strength of 10000 elephants. Satyavatī beseeched Vyāsa to father another child, as a blind child could not be a king. This time he had intercourse with Ambālikā, who, looking at his dreadful appearance, turned pale. Accordingly, she gave birth to a hypopigmented child. Satyavatī then asked Vyāsa to bed Ambikā again. But remembering his terrifying appearance, she instead sent her slave whom she had decked with her own ornaments. The slave engaged in comfortable coitus with Vyāsa, and he manumitted her and said that she would have a brilliant son who would be the most intelligent of the men of the age. Thus were born the blind Dhṛtarāṣṭra, the blanched Pāṇḍu and the wise Vidura. Pāṇḍu’s troubles did not end with the absence of pigment. After his marriage, while living a sylvan life with his wives, he shot a brāhmaṇa and his wife while they were having intercourse in the form of deer. The brāhmaṇa duly cursed Pāṇḍu that he would too die as soon as he has sex with one of his wives and that wife too will meet her end with him. This is the platform story for the unfolding of the Bhārata, with the birth of Pāṇḍu-s through the intercession of the gods.

One may ask what is hidden behind the mythologem of the birth defects of Dhṛtarāṣṭra and Pāṇḍu? If one were to take a strongly historical position, one could argue that they probably were afflicted by a genetic defect. They could have had something like a variably expressive version of Waardenburg’s syndrome or the Hermansky-Pudlak syndrome, which are associated with both blindness and hypopigmentation. However, the language of myth has many layers. Beyond the historical layer, the Bhārata clearly conceals divine archetypes. These are best seen in the case of the Pāṇḍu-s but as the final parvan mentions, it applies more broadly to the other characters. We suspect that the background of Āditya-s was implied to be present in Dhṛtarāṣṭra and Pāṇḍu, with Vidura representing the joint Mitrāvaruṇā manifesting as dharma. This suggests that the blind Dhṛtarāṣṭra perhaps represents Bhaga (a continuation of the ancient Indo-European blind deity motif also seen in the Germanic Höðr) and the white Pāṇḍu combining the pale solar aspect of Vivasvān and still-born Mārtāṇḍa, which comes forth in greater Germania as the opposite of the blind Höðr, the white Baldr.

Finally, this mythologem also preserves a peculiar “para-medical” motif, namely the “maternal impression”. While not seen as a real thing in modern biology, there is a widespread belief that experiential impressions on the mother during pregnancy might translate into birth defects or birthmarks in her child. At some point, when we were re-reading the Bhārata, we realized that the legend of Dhṛtarāṣṭra and Pāṇḍu was embedding within it this prevalent pre-modern belief in maternal impression. Briefly, this view holds that ghastly sights of amputations or deformities seen by the mother in real life, or in a dream, or bodily transformations of the mother (like Ambikā’s closing of her eyes or Ambālikā’s blanching) from fear or dohada (dauhṛda)-s (satisfied or unsatisfied pregnancy cravings) might on occasion transmit congruent or similar defects/marks to the developing fetus. We did not pay much attention to it, but noted a parallel to the dohada of the mother of the great Chāhamāna hero Hammīradeva reported in the Hammīra-mahākāvya: she had a craving to have a bath in the blood of marūnmatta-s when pregnant with him and that is said to have conferred on him the fury that he manifested when manfully facing the monstrous Army of Islam.

However, beyond that, we mostly set aside these mythic motifs until we had a discussion with a late German professor in graduate school. He brought to our notice a peculiar story reliably narrated by the great Russo-German biologist Karl Ernst von Baer, a pioneer in the evolutionary theory and embryology. Von Baer’s sister saw a fire when looking out while she was about 6-7 months pregnant and thought that her house in the distance was burning. She became obsessed with that fire and kept seeing a vision of it constantly before her eyes until she gave birth to a daughter in due course. Interestingly, her daughter had a red birthmark on her forehead that took the shape of a flame that lasted until she was 7 years of age. I was bemused by this case of “maternal impression” close to our age, that too presented by a pioneer in embryology, whose own work should have suggested that this is unlikely. The late German professor mentioned to us that it is possible that it had some connection to von Baer’s peculiar ideas of a “teleological” force that influenced his version of the evolutionary theory.

Our conversation with Sharada brought to our attention the presence of contemporary belief in “maternal impression” relating to birth deformities and also reminded us of one of the morbid tales that our late grandmother and her relatives liked talking about, which involved such a motif. The possibility of a cross-cultural and temporal presence of this idea made us explore it a bit more. This led us to the work of the famous Canadian physician Ian Stevenson, who is well known for investigating strange things like ghosts, reincarnation, and the like. He had performed a detailed comparison of the published cases of such “maternal impressions” and uncovered some strange cases himself. One such notable case that he details is from Lankā, where a certain Siṁhala rowdy was killed by his adversaries, who chopped off both his hands. It was said that the rowdy’s mother then invoked Viṣṇu and Skanda to bring such a fate upon the child of his killer and repeatedly cursed the killer’s wife that she would have a deformed child. Subsequently, the assailant’s wife gave birth to a son lacking arms – a birth defect closely paralleling the amputations the rowdy was subject to. The deformed child died within an year or two of birth. Stevenson stressed that the deformities, of which he produced a photograph, were unusual and that there was no history of genetic defects in the family as far as his extensive investigations could tell.

In a detailed study, Stevenson gathered at least 50 cases that he considered reliable (they corresponded to rare defects, and in his estimation, in 46 of the 50 cases the similarly rare maternal impression corresponded very closely to the lesion in the neonate) and analyzed them for general tendencies. In 41/50 cases, he reported the pregnant mother directly seeing or hearing about a lesion (which may be a wound, surgery, or something else) in another person. In 6 cases, she experienced something on herself — these are more comparable to those of the Kausalya princesses of our national epic. Stevenson describes some of the cases in greater detail. One was reported in the British Medical Journal in 1886. Here a woman in the 4th month of her pregnancy dreamt that a rat had bitten off the great toe of her right foot. Consequently, “she awoke screaming, and narrated the cause of her fright to her husband, who corroborated her statement”. When she delivered her child, it lacked the very same great toe she lost in her dream. This case may be compared to the examples of Kausalya princesses, in that the impression was not the sight of a lesion in someone else but in the mother. However, the one distinguishing feature of the epic example is that the premonitory impression happened at conception rather than pregnancy proper.

By considering an additional set of less reliable cases, Stevenson drew up a more extensive list of 113 cases of maternal impressions. In this set, he found 80 cases with the impression in the first trimester, 20 in the second, and 13 in the third, which, as he noted, is a statistically significant difference (\chi^2= 72.018, p = 2.299^{-16}). The significant over-representation in the first trimester is aligned with that period being the most susceptible period in pregnancy for birth defects from biological causes. For example, it is well-known that severe alcohol abuse during the first trimester can disrupt the development of the head, resulting in birth defects. Similarly, studies on the teratogenicity of high (>10000 IU) doses of vitamin A suggest that birth defects were concentrated among the babies born to women who had consumed it before the seventh week of gestation. Importantly, it maps to the period when morning sickness is most prevalent, which itself seems to be an adaptation to protect the fetus in early development from potentially teratogenic compounds in the food. Thus, it also corresponds to the peak period of defects from the ubiquitination-activating drug thalidomide that was wrongly used to treat morning sickness (we shall return to its action below).

From Stevenson’s cases, which are mostly from Occidental cultures and relatively close to our times (1700-1900s), it became clear that the idea of maternal impression is indeed a widespread cross-cultural one with arborizations into other “mystery” phenomena like abhicāra and reincarnation. This made us revisit the old Hindu medical tradition to consider the positions they held in this regard. As a representative, we may consider one of the great authoritative texts of early Hindu medicine: the Śarīra-sthāna of the Suśruta-saṃhitā, which starts of with an ancient theory of being. Following the sāṃkhya tradition, it lays out that from the primordial matter prakṛti, various organs constituting the body arose. While these organs are seen as being made up of matter (bhautika \to adhibhūta), i.e., transformations (vikārāḥ) of prakṛti, they are said to have mapping on the realm of consciousness (adhyātma) and the sphere of the gods (adhidaivata; which cuts across the material and conscious realms in a “Platonic sense”). This mapping to the gods derived from the Yajurveda (with some parallels to the puruṣa-sūkta-s and central to the nyāsa-s of later tradition) is stated thus in Suśruta:

sva svaś caiṣāṃ viṣayo .adhibhūtam | svayam adhyātmam adhidaivtaṃ ca | atha buddhe brahmā | ahaṃkārasyeśvaraḥ | manaś candramā | diśaḥ śrotrasya | tvaco vāyuḥ | sūryaś cakṣoḥ | rasanasya+āpaḥ | pṛthivī ghṛāṇasya | vaco .agniḥ | hastayor indraḥ | pādayor viṣṇuḥ | pāyor mitraḥ | prajāpatir upasthyeti | tatra sarva evācetana eṣa vargaḥ | puruṣaḥ pañcaviṃśatitamaḥ sa ca kārya-kāraṇa-saṃyuktaś cetayitā bhavati ||
The mapping is thus: Brahmā: intellect; Īśvara (Rudra): I-ness (personal identity); Moon: mind; directions: hearing; Vāyu: skin; Sun: eyes; waters: tongue; Earth: nose; Agni: vocal system; Indra: prehensile organs; Viṣṇu: locomotory organs; Mitra: excretory organs; Prajāpati: reproductive organs. The final sentence clarifies that while the evolutes of prakṛti are by themselves unconscious, the 25th tattva, puruṣa, enters the primal cause (prakṛti) and its evolutes and endows them its nature consciousness. We layout this sāṃkhya foundation of Hindu medicine because it is via that intersection of the realm of consciousness and matter that it tries to explain things that would be seen as “supernatural” in a modern sense.

The Hindu medical tradition (like that recorded by Suśruta) has a proto-biological understanding of specific issues generally pointing in the right direction:
(1) Unlike the folk idea prevalent in the Indo-European world of the sperm being a seed sown in the vagina/uterus, it understood that there was a biological contribution from both the parents. In the case of the male, it saw that as coming from the semen. That contribution was not visible in the female, but it postulated an ārtava — a theoretical ovum.
(2) It did recognize that there was some sex-determining principle coming from the semen and the ārtava, though the exact nature of it was imperfectly understood.
(3) It recognized a “proto-genetic” principle wherein the parents’ postulated contributions gave rise to different organs in the embryo.
(4) It presented a primitive theory based on “biochemical expressions” in the developing embryo for various birth defects (including those comparable to Pāṇḍu and Dhṛtarāṣṭra), atypical sexuality and pigmentation differences.

However, superimposed on this essentially biological foundation (sometimes with pioneering insights) is a belief in different forms of extra-biological impressions. The roles of the various processes involved and their effects are voiced by Suśruta thus:

san niveśaḥ śarīrāṇāṃ dantānāṃ patanodbhavau |
taleśv asaṃbhavo yaś ca romṇām etat svabhāvataḥ ||
The development of the organs in their proper locations, the fall of [milk] teeth, and the growth [of permanent teeth], non-growth of hair in palms and feet all are [examples] of development as per the natural law [for that organism].

Thus, Suśruta and other authorities acknowledge that basic human development is as per a natural law — i.e., a purely biological process typical of a given species. However, immediately thereafter, he cites śloka-s that goes on to describe a very different hypothesis regarding mental traits:

bhāvitāḥ pūrvadeheṣu satataṃ śāstra-buddhayaḥ |
bhavanti sattva-bhūyiṣṭhāḥ pūrva-jāti-smarā narāḥ ||
Those constantly conditioned in the former bodies by the study of the śāśtra-s become [even in the current birth] men endowed in sattva remembering the former birth.

karmaṇa codito yena tadāpnoti punarbhave |
abhyastāḥ pūrvadehe ye tāneva bhajate guṇān ||
Impelled by acts which he has performed [in the former birth], a person attains his [state] in the reincarnation. Those activities which were repeatedly practiced in the former body are also shared by the [current one].

Thus, there was a belief that mental traits transmitted via reincarnation from the previous birth, like an inclination toward good learning, and behavioral tendencies acquired by constant practice, were superimposed on the basic biological development (mentioned above). Combined with the reincarnational effect (which parallels beliefs in most human cultures across the world) were various beliefs that may be considered as belonging to the domain of maternal impression. One of these is believed to act at the time of conception or just before that, as was the case with Ambikā and Ambālikā. An old verse cited by Suśruta records such a belief:

pūrvaṃ paśyed ṛtu-snātā yādṛśaṃ naram aṅganā |
tādṛśaṃ janayet putraṃ bhartāraṃ darśayed ataḥ ||
Whoever is the first man the woman may see after her purificatory bath following her menstruation, the child she births resembles him; hence, she must see her husband.

The commentators add that if her husband is not around at the moment, she should see the sun. Thus, the impression of the first man she sees is said to determine the child’s appearance. The old Hindu medical tradition also records a variety of alternative causes for birth defects, some biological and other “impressional”:

garbho vāta-prakopeṇa dauhṛde vāvamānite |
bhavet kubjaḥ kuṇiḥ paṅgur mūko minmina eva vā ||
A fetus suffering insults from the derangement of vāta (one of the three basic bodily processes of old Hindu physiology \approx humors of Greek physiology) or due to the [unfulfilled] maternal cravings (dauhṛda), may indeed become hunchbacked, defective in the arms, defective in the legs, dumb, or nasal-voiced.

mātā-pitro .astu nāstikyād aśubhaiś ca purākṛtaiḥ |
vātādīnāṃ ca kopena garbho vikṛtim āpnuyāt ||
From the mother or father being counter-religious or due to their inauspicious ways or from their misdeeds in a past incarnation or from the derangement of the vāta and the like the fetus acquires birth-defects.

Thus, while a proximal physiological cause (i.e., the derangement of the doṣa-s) is offered, meta- or “supernatural” causes are also suggested in the form of unfulfilled maternal cravings and the “impressions” of the inappropriate ways of the parents in the current and past incarnations. For the maternal cravings, the Hindu proto-scientists presented a purely physiological hypothesis within the context of embryological development:

tatra prathame māsi kalalaṃ jāyate | dvitīye śītoṣmānilair abhiprapacyamānānāṃ mahābhūtānāṃ saṃghāto ghanaḥ saṃjāyate | yadi piṇḍaḥ pumān strī cet peśī napuṃsakaṃ ched arbudam iti | tṛtīye hasta-pāda-śirasāṃ pañca-piṇḍakā nirvartante | aṅga-pratyaṅga-vibhāgaś ca sūkṣmo bhavati | caturthe sarvāṅga-vibhāgaḥ pravyaktaro bhavati | garbha-hṛdaya-pravyakti-bhāvāc cetanā-dhātur abhivyakto bhavati | kasmāt? tat sthānatvāt | tasmād garbhaś caturthe māsy abhiprāyam indriyārtheṣu karoti | dvihṛdayāṃ ca nārīṃ dauhṛdinīṃ ācakṣate | dauhṛda-vimānanāt kubjaṃ kuṇiṃ khañjaṃ jaḍaṃ vāmanaṃ vikṛtākṣam anakṣaṃ vā nārī sutaṃ janayati | tasmāt sā yad icchet tat tasyai dāpayet | labdha-dauhṛdā hi vīryavantaṃ cirāyuṣaṃ ca putraṃ janayati ||
There (in the womb), in the first month, a bag-like structure emerges. In the second, starting with metabolic action of the three (śītoṣmānila) physiological processes, the molecular combinations of the primal elements comprise a condensed structure. The presence of a lump-like structure indicates a male; a bud-like structure a female; a tumorous mass, an intersex. In the third, 5 lump-like forms of the hands, legs and head develop. The incipient divisions of the various organs and their subdivisions come into being. In the fourth, the substructures of all the organs become clearly apparent. From the full development of the fetal heart, the substance of consciousness becomes apparent. How so? From the heart being the receptacle [of consciousness]. Thus, in the fourth month, the fetus displays agency for the organs to apprehend/elicit their stimuli/actions. There are two hearts (one of the fetus and one of the mother), and from that the pregnant woman is known to be two-hearted [thus, the maternal cravings]. Hence, unfulfilled maternal cravings result in the woman giving birth to an offspring that may be hunchbacked, defective in the hands, defective in the legs, mentally defective, dwarfed, with deformed eyes or eyeless. Therefore, one must satisfy her cravings as she wishes. Indeed, the woman with satisfied cravings births a virile and long-lived son.

Thus, the Hindu hypothesis of maternal cravings stems from the old belief that the heart is the seat of consciousness [Footnote 1]. Thus, the full development of the heart causes the organs of the fetus to seek their stimuli (jñānendriya-s) or actions (karmendriya-s). These are expressed via the mother resulting in maternal cravings. The hypothesis further posits that non-fulfillment of these results in defects in the fetus. We still do not fully understand the causes for maternal cravings or their adaptive logic in their entirety. The “spandrel” hypothesis suggests that they might arise from the uterine nervous connections activating the neighboring taste-related regions in the part of the brain known as the insula. However, we find it highly unlikely that the phenomenon is a spandrel. On the other hand, modern experiments suggest that, at least in childhood, there might be a liking for the gustatory stimuli the kids were repeatedly exposed to via their mother’s food during their fetal development. It is possible the Hindu proto-scientists made similar observations and accordingly extrapolated and expanded them to propose the above hypothesis. Indeed, tradition holds that such maternal cravings in themselves have a prognostic character. We cite a few examples of these below:

āśrame saṃyatātmānaṃ dharmaśīlaṃ prasūyate |
devatā-pratimāyāṃ tu prasūte pārṣadopamam ||
darśane vyālajātīnāṃ hiṃsāśīlaṃ prasūyate
The woman with a craving to visit a dwelling of sages births a self-controlled child committed to dharma. Indeed, she who desires to see an image of a god births a child who would grace a council. She who wishes to see a carnivorous animal births a child prone to violence.
godhā-māṃsā .aśane putraṃ suṣupsuṃ dhāraṇātmakam |
gavāṃ māṃse tu balinaṃ sarva-kleṣa-sahaṃ tathā ||
She who wants to eat the meat of a Varanus lizard births a child who will sleep well and cling to material possessions. Similarly, she who craves for beef births a strong child capable of enduring all manner of hardships.
māhiṣe dauhṛdāc chūraṃ raktākṣaṃ loma-saṃyutam |
vārāha-māṃsāt svapnāluṃ śūraṃ saṃjanyet sutam ||
She who craves for buffalo-meat births a brave child with reddish eyes and endowed with hair. She who longs for pork births a sleepy though a brave child.

More generally, the tradition holds that the nature of the child would mirror the nature of the animal whose meat the pregnant woman desires. Finally, tradition also holds that there is a direct mapping between the mother’s body and that of the fetus to account for the classic maternal impressions:

doṣābhighātair garbhiṇyā yo yo bhāgaḥ prapīḍyate |
sa sa bhāgaḥ śiśos tasya garbhasthasya prapīḍyate ||
Whichever part of the pregnant woman’s body is afflicted by physiological derangement or by injury, the corresponding part of the child in the uterus is afflicted.

In conclusion, we can summarize the old Hindu medical tradition’s position on fetal development as involving: 1) natural laws expressed as biological processes with parental “genetic contributions” as the primary drivers of development; 2) impressions of past incarnations of the child; 3) impressions from deeds of parents in current and past incarnations; 4) fulfilled or unfulfilled dauhṛda-s, which are physiologically explained as the influence on the mother by the fetal organs exercising the apprehension of their objects; 5) maternal impressions from the mother’s visual images post-menstruation and the mapping of the insults to the mother’s body onto the fetal body. The widespread presence of the impressionist components of these beliefs across cultures suggests that they go far back in history. Apparently, in some Romance languages, the word for birthmark and craving are the same and reflect a belief that the unfulfilled dauhṛda-s spawn those marks. The yavana physician Galen believed in the classic maternal impressions, and Empedocles held that women who fell in love with certain statues produced offspring who looked like them. Similarly, in Greek literature, the dark Ethiopian is said to have given birth to the fair Chariclea because she kept looking at the image of the white Andromeda while pregnant.

A version of such beliefs played an important role in the history of biology closer to our times. Had the French soldier Jean-Baptiste Lamarck (1744–1829 CE) been blown to smithereens by the German guns bombarding his position that he held with great valor, we might not have had one of the famous debates in biology that is somewhat artificially presented in textbooks. Having survived his stint in the French army Lamarck went on to propose one of the early modern evolutionary theories. Lamarck’s theoretical framework lay at the transition between archaic and modern scientific thought — one could say the transition between proto-science and science. His chemistry was more primitive than that presented in early sāṃkhya, subscribing to a four-element model of the universe and opposing the leap towards modern chemistry pioneered by his compatriot Antoine-Laurent de Lavoisier. Aspects of his biology also remained primitive: on the one hand, he subscribed to spontaneous generation, apparently disregarding the work of the Italian Fransisco Redi that showed it to be a myth. On the other, he subscribed to the existence of a “life force”, which among other things, “tends to increase the volume of all organs”. Nevertheless, he was a keen biologist who, through his investigations, realized that organisms of one form must have evolved from those with another. A part of his explanation for this process involved a hypothesis based on a certain proto-biophysics of fluid flow. He argued that the rapid flow of fluids within the tissues of organisms “will etch canals between delicate tissues” much like a river erodes its bed. He then postulated that the differential flow rates that will ensue would lead to the origin of distinct organs. Simultaneously, he saw these fluids themselves becoming more complex, giving rise to a greater diversity of secretions and organ diversification. He combined this proto-biophysics with two so-called laws to explain the evolutionary process. The first law postulated that different organs of a given organism were either augmented or diminished depending on the degree of their use or disuse in the course of its life. The idea was based on the observation of real somatic adaptation to environmental pressures happening in the course of an organism’s life. The second law posited that these characters acquired during the life of an organism by the first law are passed to their offspring — the inheritance of acquired characters. In proposing this, he was more or less following the broad class of ideas coming down from the ancients that were similar to maternal impression in the general sense.

Charles Darwin’s grandfather Erasmus, who was one of the inspirations for his evolutionary theory, also accepted the inheritance of acquired characters, and Charles himself acknowledged this aspect of Lamarckism and incorporated it as a subsidiary component of his own theory. The shock waves from Darwin’s hypothesis resulted in a sizable body of biologists falling back to pure Lamarckism or some variant thereof as a counter to Darwinism that deeply disturbed them. One such was the goal-seeking evolutionary theory of the German biologist, Theodor Eimer, which incorporated the Lamarckian mechanism. Attempts claiming to demonstrate Lamarckian inheritance of acquired characters went on for a long time, including that of the famous Ivan Pavlov, but none of these studies produced reproducible or reliable results. Despite this, Lamarckism remained popular, especially in France, down to the second half of the 1900s. A variant of it had tragic consequences in the Soviet Empire in the form of Lysenkoism. Starting with the embryological work of Johannes Müller, serious questions were raised about whether embryonic development was affected by somatic characteristics acquired by the parents. After Darwin, August Weismann experimentally showed the isolation of the reproductive germplasm from the somatoplasm of the adult body in mammals casting doubt on any mechanism that allowed the Lamarckian changes acquired by the somatoplasm of an organism to be transmitted to the offspring. The subsequent advances in genetics, biochemistry and developmental biology over the 1900s made Lamarckism look less plausible until it was more or less consigned to the history of science among serious biologists.

However, by the late 1990s, a version of the “inheritance of acquired characters” started making a comeback within the Darwinian framework. Microorganisms, especially prokaryotes, show rampant horizontal gene transfer. The horizontally transferred genetic material integrated into the genome or a plasmid can then be transmitted to the offspring. Its subsequent maintenance would be subject to natural selection. In fact, the selection could act even before the transmission to the progeny; for example, the transfer of a gene confers immunity to a virus or resistance to a toxin (e.g., antibiotic). Thus, one could see it as a process of acquiring an adaptation that is then transferred to the next generation. Indeed, several organisms, like bacteria, have sophisticated mechanisms for acquiring foreign DNA. This could be in the form of the competence system that allows DNA uptake in certain phases of their life cycle or domesticated viruses that can act as transfer agents for DNA. Less understood, but potentially in a similar vein, are the lipid vesicles derived from cells that might also carry nucleic acids. Thus, the acquisition of characters encoded in the transferred DNA from the environment has been institutionalized in many bacteria. Indeed, the commonly cited prokaryotic immune mechanisms, namely the CRISPR and PIWI-based systems and other systems we have discovered, can be seen as variants of this process of controlled acquisition of genetic information (in this case, from invading elements like viruses) that is then inherited by the offspring. However, this is entirely within a Darwinian setting with selection post-acquisition explicitly taking the place of use or disuse as an augmenting or diminishing agent.

In prokaryotes, this transfer of DNA in part plays the role of sex. Eukaryotes have evolved an institutionalized sexual mechanism that is decoupled from ambient DNA transfer. Nevertheless, they too have been extensively acquiring “ready-made” adaptations through horizontal DNA transfer. That said, from relatively early in eukaryotic evolution, there have been repeated adaptations for “setting aside” a germplasm from the somatoplasm. In the unicellular eukaryotic world, we see this in the ciliates, which set aside their germplasm in the micronucleus while running their cells with a somatoplasmic macronucleus. Indeed, the macronucleus loses a good part of the genetic information maintained and transmitted to the next generation in the micronucleus. Of course, this separation of the germplasm and the somatoplasm is the norm in eukaryotes like the animals. Moreover, a similar phenomenon to the ciliate macronuclear DNA loss is seen in several animals like the Ascaris nematode worm or the lamprey, where part of the genome is shed in the somatic cells. In vertebrates, in cells like lymphocytes, the DNA is again lost or mutated as part of the generation of immune antigen receptors. In the neurons of at least some vertebrates, the genome is edited by the jumping of transposons. However, all these “acquired” somatic changes are kept out of the germplasm segregated early on from the somatoplasm. The origin of this separation seems to be due to the genetic “addiction” pressure exerted by genomic selfish elements against their loss by excision, as seen in the macronucleus or the somatic genomes of some animals. This has gone hand in hand with mechanisms that suppress the expression of these genomic selfish elements in the germline. This suppression process is achieved by a class of mechanisms collectively termed “epigenetic” or transmission of information over and beyond that transmitted by the genome. Along with this, pressures to safeguard the genomic integrity of the germline have also resulted in strong blood-germline barriers in various animals.

On one hand, these discoveries have been the strongest strike against impressionist information transmission, including Lamarckian acquired characters — a consequence of the strong shielding of the germplasm in several eukaryotes from the somatoplasm. On the other hand, the epigenetic mechanisms have revived a version of this transmission because they do seem to transfer non-genetic information inter-generationally. There have been claims that such epigenetic mechanisms might be behind the intergenerational transmission of the effects of trauma. However, the evidence for such claims is rather questionable, and at best, they remain uncertain to date. Nevertheless, at least in some animals, like insects, there are other forms of epigenetic information like endo-parasitic bacteria transmitted via the germline (e.g., Wolbachia), which influence the sex ratios by killing a subset of the offspring produced in matings that disfavor their transmission. These bacteria often encode toxins to enforce their “addiction”, several of which attack the genomic DNA of their host, including, as we discovered, by mutating it. Thus, there is a possibility of transmission of acquired effects through epigenetic mechanisms that have developmental consequences in a more limited sense.

One of the major discoveries of modern developmental biology, including ones we have contributed to, is the role of protein ubiquitination in regulating development. A major aspect of ubiquitination is its action on protein stability, i.e., tagging of key developmental regulator proteins for their degradation. Thus, both the disruption and enhancement of various ubiquitination pathways can result in a diverse array of birth defects. As noted above, the mechanism of action of thalidomide is via the enhancement of one of the ubiquitination systems, which in turn results in the degradation of a transcription factor causing birth defects. It is conceivable that other than toxic compounds, like thalidomide, certain other stresses (or, if true, transmitted epigenetic information) impinge on the ubiquitin system to affect the stability of various developmental regulators resulting in birth defects.

Coming a full circle, with an improved understanding of biological processes, beliefs in maternal impressions in the broad sense, which lay within the domain of mainstream medicine from the days of the ancient Ārya-s to the late 1800s, were gradually excluded from it. As Stevenson, perhaps, the last intrepid believer in it in the western academe noted, these reports more or less vanished in the 1900s. We are not sure if it is a useful avenue to revisit. Nevertheless, being cognizant of it being a tenacious feature of the cross-cultural belief landscape, we believe that it is something that can be better investigated with the sharper tools that are currently at our disposal. We have a range of mature technologies that span nucleic acid sequencing, biochemistry, and developmental biology, allowing us to more directly probe birth defects than ever before. These could be brought to bear more systematically on the suspected cases of impressions — in the least, the conclusions might contribute in a more humdrum way to our understanding of developmental processes.

Finally, if one were to see a case where one is inclined to bypass biological explanations for the “supernatural”, then one may ask if the relationship between the impressing stimulus and the impression is really causal or a manifestation of the mysterious “synchronicity”. While those inclined toward the rationality of the age might abhor the very mention of synchronicity, it may be simply something they have not experienced.

Footnote 1: As an aside, we may note that Suśruta records that Śaunaka proposed the head to be the controller of the organs and director of the development of other organs rather than the heart:  pūrvaṃ śiraḥ sambhavati +ity āha śaunakaḥ | śiro mūlatvāt pradhānendriyāṇām | However, this more correct apprehension seems to have been dropped in parts of Hindu biological tradition for the more primitive heart theory proposed by Kṛtavīrya

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The rise of Navyonmāda, the subversion of the Mahāmleccha-s, Cīnānusāra and beyond

The past
The dynamics of the establishment of the counter-religious unmāda-s are of some interest. The pūrvonmāda of pharaoh Akhenaten arose from the moha in his own head and was imposed on the populace due to his imperial power. It was quickly erased by the actions of his successors Tutankhamen, Ay and Horemheb and it never was able to take root. The mūlavātaroga seems to have spread rapidly within a tightly-knit ethnic group and soon evolved into an in-group marker. Hence, there was selection against it breaking out into the rest of the populace because its value to the in-group would then be lost. However, by being a strong in-group marker, it tenaciously persisted by maintaining in-group cohesion during military expeditions and raids against out-groups and became the incubator for the future unmāda-s. An infective variant of it first broke out into the general population in the form of pretonmāda. In the early days, pretonmāda appears to have been comparable to the mūlarug, providing an in-group marker for some “good-for-nothings”. However, it acquired several memes from other natural religions in the environs during this period, making it a stronger competitor. Armed with this expanded memome, it conquered the Romans through a less-appreciated strategy of subterfuge, martyr-creation and elite capture (see below). Once in power, it quickly expressed that anti-outgroup militant tendency inherited from its parent strain on a much larger scale. After a period of expansion via this virulent mode, it oscillated between explosive virulence (e.g., South America, Africa and Goa) and its earlier mode of creeping person-to-person infection (e.g., Andhra).

The same meme-complex was transferred to the marusthala resulting in marūnmāda, which codified “spread by any means” as a duty of every adherent. Indeed, the learned māhāmada Ibn Khaldun clearly explains in his Muqaddimah that it is the obligate duty of a marūnmatta to convert everyone to the unmāda by missionary activity or force. He then goes on to add that whereas this is an explicit duty of the marūnmatta-s, it is only an incidental or localized activity among the pretonmatta-s or ādivātūla-s (M 1.473-474). While he might be correct regarding the latter, the former, in reality, are closer to his own. Nevertheless, the fact that he explicitly mentions this as a feature of marūnmāda distinguishing it from the other unmāda-s suggests that it has always been more “in the face” with them. Ibn Khaldun had also emphasized the need for a strong power, like the ruler, to enforce the sharia once the Adyunmatta dies because it is this enforcement of sharia that keeps the people doing the “right thing” from which their natural tendency is to lapse (M 1.472). Ibn Khaldun himself furnishes the example of a North African woman from the Jewish priestly caste, who, distant from the ekarākṣasa core, reverted to a semi-natural religion and as a magician queen. Uniting various tribes, she valiantly blocked the advance of the rākṣasa-senā until the Ghāzī-s eventually beheaded her and her skull was gifted to the Khalif. Thus, the marūnmāda thinkers clearly understood that unmāda-s are maintained by internal and external enforcement programs, without which they can break down and disappear.

Coming to the present
These aspects of marūnmāda reemerged in the secular rudhironmāda that in turn arose out of the matrix of “enlightenment values”, dear to the deluded secular mleccha-s, which itself was a counter-religion to the older prathamonmāda-pretonmāda matrix. While rudhironmāda inherited its claims for secularity from the “enlightenment values”-roga and turned its back to the ethnoreligious mūlavātula-śūlapuruṣīya roots, in its attempt at universality, it followed marūnmāda and pretonmāda. On the one hand, this allowed it to be easily transmitted and embraced by people as diverse as the prathamonmatta-s, śūlapuruṣa-s, the Rus, the Vāṅga-s, the Cera-s, and as an outer coat by the Cīna-s. On the other, as we shall discuss below, its secular universality proved to be its undoing. At its peak, rudhironmāda was held up by the Soviet Rus in the great conflict with the Anglosphere and its vassals. However, at the end of that struggle, the Soviet empire collapsed, and rudhironmāda failed in its grand geopolitical objectives. While it failed as a geopolitical force, it quietly thrived for more than half a century in the fertile breeding grounds of the Anglospheric academia, especially among the Mahāmleccha. There, it underwent a series of mutations before eventually emerging as navyonmāda.

What rudhironmāda lacked was the religious facade of pretonmāda and rākṣasonmāda, and this was a serious downside. While both rudhironmāda and the overt unmāda-s struggled against natural religions, the latter were at least able to offer themselves as the ultimate alternative, i.e., the “only true religion”. However, rudhironmāda was not able to give anything of that kind. As a parallel, one could consider the case of Mustafa Kemal Ataturk — he took the Turks out of marūnmāda. Had he reintroduced the religion of the Tengri-s, the Turks of Turkey might have been a cured people. However, he failed to offer a different religion in its place after removing marūnmāda, and sometime later, they returned to it with a vengeance. A similar phenomenon happened to varying degrees in the marūnmatta states conquered by the Soviets. While it is not apparent to the casual observer, navyonmāda eventually corrected this critical fault of its rudhironmāda parent — the religious vacuum. Navyonmāda neither set out to do this consciously nor did it converge on it right away or in any conscious way. Instead, these features emerged organically via selection within the movement to fill in the vacuum of religiosity created by the earlier rudhironmāda within the Mahāmleccha academia. Navyonmāda gradually evolved this feature over the period from the 1980s to the 2020s. In many ways, navyonmāda resembled the religiosity of the saṃgha of the sugata with its veda-virodhaka tendencies when it exploded as a potential religion for a section of the anglospheric elite and their arborizations in vassal states of the larger leukosphere.

While the religious facet of navyonmāda helped it replace pure rudhironmāda, this aspect also created the foundations for their seizing power by creating both in-group solidarity and the ever-correcting “purge of deviations or moderation” as in marūnmāda. Coming out of academia and originally filled with lots of socially low-ranked, physically unfit individuals, outright warfare was not an option for it to capture power. Instead, it tread on a path very similar to pretonmāda in its early days of elite capture. When Gucchaka was the emperor, and the marūnmatta-s attacked the Mahāmleccha in their big strike, there was a huge surge of military and nationalist sentiment. The result was an attack on several rudhironmatta academics, as well as moderates, who now organized under the banner of navyonmāda. Gucchaka’s neo-con handlers, headed by his deputy duṣṭa Vakrās, the matta Āṇi, and the prathamonmatta-s, squandered the swelling nationalism of the mleccha-s by engaging in the useless invasion of Iraq. That was followed by throwing away a chance to normalize relations with the marūnmatta-occupied Iran, and more troubles emerged for the mleccha-s. Duṣṭa Gucchaka also set up a police state within state, and to date, unironically rationalizes it to the Mahāmleccha by saying that it is a “reminder” of the comfort they can feel. This police state, along with the financial crisis which followed, strengthened the hands of the rising navyonmatta-s when Ardhakṛṣṇa became emperor of the Mahāmleccha confederation. While in part unhinged or even evil, the cara-s like Himaguha, Harijaṅgala (anti-H), Mānavīya (navyonmatta) and Asaṅga, have revealed the depth and the power of the systems, which Gucchaka and Vakrās put in place for the pañcanetra-mleccha-s to closely monitor and control their own citizens. While not in the face, like that of the Han, it has the comparable capacity and only needed a willing player to exploit it to its fullest.

Against this backdrop, when Ardhakṛṣṇa became the lord of the mleccha-s, under his sympathetic rule, the navyonmatta-s started creeping forward aided by his dala. A key factor was the coming of age of the young students indoctrinated by professors in the universities into the newer strains of navyonmāda with prominent religious (c.f. the marūnmatta students from Gandhara in TSP madrasas — The Students). They were available to work for Ardhakṛṣṇa’s campaign and entered the śāsana upon his victory. In his kalevarā, they found their much-needed bodhisattva. With the first of these navyonmatta-s in the śāsana, they could now play filter and amplifier with their professors in the universities. They got more of their kind, while they punished and purged those who failed to fall in line in an alliance with the vālūkavatūla-s (see below). This creep leading to the control of the mainstream media, the academia, internet-social media-based big tech, and offices of the government (the deep state) was rather deep by the end of the reign of Ardhakṛṣṇa. They thought that they could easily usher in their preferred candidate Jārapatnī and establish their regime with their power. However, Mahāmleccha democracy actually worked, and Vijaya, also known as the Nāriṅgapuruṣa, became the mleccharāṭ. He tried to push back, but he was no Julian and full of his own insufficiencies and inexperience. Alarmed by his victory, the navyonmatta-s now went into high gear and used the Nāriṅgapuruṣa as an excuse for quickly pushing in their excesses. They used all their instruments to constantly hector the Nāriṅgapuruṣa and facilitated his ultimate uccāṭana by Piṇḍaka and his deputy Aṭṭahāsakī, who was an opportunistic adopter of navyonmāda and their front-end person.

Analyzing the rise of navyonmāda
Now, let us take a closer look at some of the aspects of the rise of navyonmāda among the Mahāmleccha. As noted before, when the Mahāmleccha became a superpower after their victory in WW2, the world was never more starkly divided into winners and losers. The natural resources of Krauñcadvīpa, together with the victory in war, paid dividends to them like nothing in recent history. As a result, the Mahāmleccha started building sprawling suburbs and zoomed around their vast frontier lands in their gas-guzzlers. This prosperity made them more and more detached from reality, and the myth of infinite “economic expansion” took root in them, making them blind to how unsustainable this was. This softened their elite, worsened their health, and also precipitated the decline of conventional ekarākṣasa religion in a section of their elite. This left them in need of other avenues for feeling a sense of “virtue” and piety. In our opinion, this psychological complex plowed the soil for navyonmāda to take root eventually.

As this was taking place over half a century, there were parallel developments in academia. The victory in WW2, the triumphs of the nuclear bomb, the shift of the epicenter of frontier physics from Śulapuruṣīya to Krauñcadvīpa, and James Watson’s role in the founding of molecular biology, made the Mahāmleccha academia emerge as a potent accreditation system. But over time, the myth of infinite expansion took root in these systems, turning them into pyramid schemes. One consequence of this was what some mleccha academics have termed “elite production”. Having spent a lot of time with mleccha academics, we can say that there are some very tāthāgatan qualities therein — a mix of a tendency for plagiarism and lack of attention to detail and subtlety. The result is the dominance of mediocre or half-baked models of understanding over more complete and nuanced ones. This is nowhere more starkly illustrated than the fake studies in psychology churned out by the dozen, replete with manufactured statistics. “Cancer biology” comes a close second furnished with photoshop artistry. This was mainly because a part of the expanding Mahāmleccha academia divorced itself from a test against the real world due to the fetish of “peer-review”. Here, research gets acclaimed due to publication in “high-profile” journals reviewed by peers rather than being a realistic description or nature or making predictions that proved to be correct. This was yet another fertilizer for the growth of navyonmāda — a system where one can get away with fancy beliefs without checking if they hold in the real world.

Indeed, this blindness to reality in navyonmāda’s priorities can be seen at many levels beyond the obvious. However, it is a kind of māyāvāda that not just takes the jagat to be mithyā but creates a mithyā-jagat of its own. Most plainly, it is seen in the denial of biological reality in the form of sex and race and its replacement with a celebration of pseudo-biological shape-shifting. However, this permeates more subtle matters too. On the economic front, the classic rudhironmatta-s went on endless harangues about economic unevenness in society and acted on it by killing or banishing the prosperous elite. In contrast, the navyonmatta-s do not rail much about the material; instead, they vent against something immaterial termed “privilege”, and their actions are aimed at eliminating those whom they perceive as privileged. The elite in normal societies either signaled their status or were accorded status due to either distribution of material possessions or their praxis of rituals. However, in navyonmatta society, the elite signal it via their beliefs and claims of purity therein (elite capture). In this aspect, they are remarkably similar to vālūkavatūla-s. A critical flashpoint for the māyāvāda of the navyomatta-s came with computers masquerading as phones and the internet becoming ubiquitous. This allowed commerce and social life to transition to a virtual electronic world, completing that sensation of a decoupling with reality, which the navyonmatta-s desired.

In such a system, the holders of the allowed beliefs rather than producers of work that passes real-life tests are magnified by shallow vanity articles as though they are the next bodhisattva-s on the block. Recently, we saw such an article on a navyonmatta human geneticist, who thoroughly mixes her science with the Nicene creed of navyonmāda, to provide that facade of objectivity to those in that pakṣa. A mix of such boosterism of particular types as role models and the sense of virtue from these beliefs takes the place of a key aspect of religion — the sense of belonging among the pious and a concomitant disdain for the impious. Thus, the new votary of navyonmāda, usually from a deracinated or shallow background, suddenly finds a new purpose in life by professing it. It provides for beliefs that touch the human need to feel justified and virtuous. These are often reinforced by actual mimicry of religious activities, such as rhythmic chanting of slogans against enemies (c.f. marūmatta-s stoning the “devil”), internalization of concepts such as the “power of the word”, great fear of taboo words, sacralized ethnicities and “martyrs”, marches resembling a religious procession, and community-building with opportunities for sexual encounters and “group therapy”.

Thus, the American university system slid from being institutions of accreditation to those of degeneration, furthering the capacity of navyonmāda to take over the systems. It was still not straightforward because there were still many academics who still had to give up common sense to adopt it. It breached this barrier through a series of steps. First, it evolved obfuscation of language, wherein common words were reused such that they meant something specific to the insider, whereas the outsider saw them in their ordinary sense as words of virtue. In old religions, for example, in the yoga tradition, words like gomāṃsa did not mean what the commoner might take them to mean. Here, the intention was to conceal secrets and repel the uninitiated (e.g., he may think a yoga text is recommending that he consume gomāṃsa and keep away from it). However, the intention of the navyonmatta-s was not repulsion but outright deception — it was to purposely obfuscate the uninitiated and unwittingly get them onboard with the navyonmāda project. This is rather comparable to Tathāgata redefining common religious words like trayi or ārṣa or sūkta (=sutta). Thus, the lay votary would not realize that the Buddha is subverting the religion and get sucked into the śaraṇa of the saṃgha. Thus, driven by the need to feel virtuous and pious, the mleccha liberal academics vigorously adopted and supported navyonmāda, even as unsuspecting lay ārya-s adopted the cult of the Buddha — after all who would want to reject something termed the way of the ārya-satya-s or dharma?

As they began winning votaries, they transitioned from a bauddha mode to the marūnmatta/pretonmatta mode of counter-religion. The academics who resisted navyonmāda, even left-liberals who had some commonsense left, were attacked by the mobs of navyonmatta-s, typically recruited from the frenzied student body. Some academics were kicked out of their institutions, others who were too powerful to evict were sidelined in public discourse, yet others converted under this pressure. The campus riots, like those of the American Spring (e.g., in Berkeley, a hothouse of navyonmāda), pushed the growing body of university administrators into submission to the demands of the self-righteous navyonmatta-s. The monoculture in academia allowed the capture of institutions founded on it like scientific and medical journals, which now amplified the message and pushed the credo onto young and impressionable minds. Those who sought accreditation from these bodies (academia and journals) had to now submit to the new religion, and the rest had to go silent like Thabit ibn Qurra.

In the next step, these newly minted young navyonmatta-s gushing out of the big-name universities flooded corporations and the government. Those who took navyonmāda lightly used to jocularly remark that when these indoctrinated students get a real job, they would be jolted out of their unmāda into reality. This might have indeed happened in the past with rudhironmāda, as we saw with our own classmates. However, here the reverse occurred. The navyonmatta-s captured and transformed the institutions they invaded. In the days of old rudhironmāda, they presented themselves in opposition to the “capitalist” corporations and “oppressive” government. However, navyonmatta was more agile and worked with the corporations — if the corporations paid them a jaziya and supported their demands they were more than happy to play along and support the corporations. This was an easy way for corporations to appear virtuous too. Like a king who might have committed sins in war, building a temple to expiate those, the corporations were more than happy to absolve themselves of capitalistic excesses by adopting navyonmāda themselves. The mahāduṣṭa-s like guggulu, Dvāra, Mukhagiri, Jaka and Bejha all adopted navyonmāda to differing degrees as a prāyaścitta while continuing to commit “capitalistic sins”. Thus, came about the marriage of these with navyonmāda, which would have been impossible under the old rudhironmāda, which would have attacked them for their wealth. A very real exhibition of this was the repainting of “the occupy movement” working under the old rudhironmāda premises to its new navyonmāda colors.

More sinister than even these was the māhāduṣṭa Sora, who had always channeled his money into navyonmāda being philosophically in union with it. He saw it as an opportunity to enact his grand political plans both among the Mahāmleccha and abroad, like in Bhārata. Among the Mahāmleccha, he saw navyonmāda as a potential private army to put his pakṣa headed by the vṛddha Piṇḍaka on the āsandi. In this, he was mostly aligned with other mahāduṣṭa-s. They got their golden chance when the pandemic lock-downs followed by some egregious acts of violence by daṇḍaka-s on kṛṣṇa-s caused a janakopa. This janakopa was quickly channelized by kālāmukha militant wing of the navyonmatta-s (c.f. the Mahāmlecchīya gardabha-pakṣa’s earlier militant — the ka-trayam deployed against kṛṣṇa-s). This provoked quick and correct action against them by the then mleccheśa, Nāriṅgapuruṣa. However, his effort was rendered toothless as the navyonmatta-s had subverted the deep-state and the Sorādi māhāduṣṭa-s used their riches to provide legal impunity to their kālāmukha rioters aided by the likes of Aṭṭahāsakī.

“Rus! Rus!” and Cīna-capeṭa
As navyonmāda’s priorities are not aligned with the real world, its adoption by the elite would eventually be hit by the real world. However, this does not mean a correction will be immediate, as illustrated by the Dark Ages in the Occident brought on by pretonmāda. With the sailing being smooth for the Mahāmleccha, the deep-state actors kept themselves busy going blue in the face, shouting, “Russia! Russia!”, even as the storm of the Middle Kingdom Corruption was brewing in the Orient. As the said corruption broke out of China, the Mahāmleccha were oblivious even to their own intelligence agencies; instead, they were busy in an internal conflict, with the deep-state and the gardabha-pakṣa trying to overthrow the Nāriṅgapuruṣa. Finally, when the Middle Kingdom Corruption reached the shores of Krauñcadvīpa, the gardabha-pakṣa was trying hard to prevent the appropriate steps from being taken to counter it. The compromised deep-state could not mobilize a proper supply of masks for the citizens or even guda-pramṛja-s for the mleccha-s to sanitize themselves. The result was a massive death toll, the full extent of which has definitely been under-reported. To top it all up, the mleccha-senānī betrayed his own boss, the Nāriṅgapuruṣa, to the cīna-s. Cīna-s saw an opportunity to hit hard at Mahāmleccha power using navyonmāda. Even as the navyonmatta-s disparaging their own constitution, the Cīna-dūta-s gave the mūlavātūla Nimeṣaka and the Mahāmleccha praṇidhipa as resounding slap right in their own den. However, lost in the world of their own making, they failed to wake up — Nimeṣaka was more concerned about flying navyonmāda’s Indracāpadhvaja-s at mleccha-dūtaśālā-s than managing the proper retreat of the mleccha-s from Gandhāra. As icing on the cake of their delusions, Piṇḍaka and Nimeṣaka declared that they had conducted a great operation to kill dreaded ghazi-s of the Khilafat and avenge the death of their baṭa-s, when in reality they killed a bunch of children of one of their own marūmatta friends.

In contrast, emperor Xi proceeded with strengthening the Cīna-s. He lied his way through the Middle Kingdom Corruption, even as the rest of the world was tied down by it. He delivered a definitive punch to the restive vālūkavatūla-s, whom the Han had earlier subjugated. He was able to move ahead with testing hypersonic missiles supposedly. Notably, he was above to exploit the navyonmatta movements, like the kṛṣṇajīvāndolana, to aid the overthrow of the antagonistic Vijaya-nāma-vyāpārin and bring the pliable Piṇḍaka on the āsandi — it is notable how the Cīna-s colluded with the tech-duṣṭa-s to silence any discussion on Vyādha-piṇḍaka and his yantra. The cīna-s have other assets. First, their penetration of the Occidental academia via Galtonism is incredibly deep. Thus, mleccha academics bat for their interests, provide them with cutting-edge knowledge, and might even obtain funds for their research which might ultimately be to the detriment of the mleccha-s themselves. This also provides them a conduit to slip in their spaś-es as students. Second, they have cultivated assets among the big-tech duṣṭa-s, e.g., Mukhagiri is their jāmātṛ who originally courted them before they broke up with him. Perhaps, even the break was not of his own accord but due to the pressure from the mleccha side. Finally, since the mleccha big-tech depends on delivering cheap opiates to the masses, the cīna-s hold them by their balls by controlling manufacturing. Thus, as big-tech takes control of the Mahāmleccha as the de facto government (something which will accelerate with the strides being taken in machine learning and other areas of computing, which in turn have been made effective by the more than two decades of data the masses have supplied them), the leverage on them that the cīna-s have, coupled with their immunity from them, would allow them to use navyonmāda as a potent weapon.

Onward to Bhārata
All unmāda-s, the old counter-religions, and their new secular mutants see the dharma as their natural enemy — what Viṣṇuśarman would term svabhāvavairam. The one area where this is manifest is the hatred for the brāhmaṇa-s and functional H systems. Those among the mleccha-s and their sipāhi-s who fight for the so-called “enlightenment values” (the older delusion) hate the H as they offer a robust and likely superior alternative to their system. This threatens to undermine their truth-claim that they have found the only true formula that succeeded due to being good rather than being enforced by the smoking end of the nālika. Across the mleccha elite, some of the predictions of the evolutionary theory for Homo sapiens are fundamentally incompatible with their cherished beliefs. Thus, the majority of Occidental scientists (across ideological camps) focusing on its study slip into denialism on one or the other matter. Both the “Western values” types and those who fight for navyonmāda are terrified by the fact that H succeeded because they created a system that instinctively acknowledges the pulls of biology, i.e., human nature, on social structure. Thus, Bhārata is the one frontier where there is an alignment of the enemies. Paradoxically, while navyonmāda fights pretonmāda and “enlightenment values” in the Occident, in Bhārata, these align for the break up the H. In contrast, marūnmāda-navyonmāda alliance will be strong in both India and the West. Hence, we predict that given the intrinsic lack of fecundity in navyonmāda due to celebration of biology-denial it will end up aiding marūnmāda in Bhārata.

Historically, H have had to deal with the preta-maru-rudhira triad — while there have been the classic heathen failures against these, at least they were recognized as such by a large fraction of the elite. In contrast, the capacity to recognize navyonmāda is even lower, as can be seen from the fact that even pro-H government circles so quickly and willingly propagate navyonmāda memes. This has also meant that the H elite have adopted navyonmāda memes to differing degrees. The secular elite constantly bombard the impressionable with navyonmāda packaged within opiates such as spectator sports, product advertisement and cinema, as if they were distributing adulterated heroin. Moreover, navyonmāda is also “safe for use” even for Cīna subversion operations in Bhārata. In particular, government adoption of navyonmāda memes would result in undermining the H army. This is something the Cīna-s badly want because despite all their show of tech, they have a dearth of young men needed to fight wars. On this front, the H still hold an upper hand, but if, like the Mahāmleccha, they decided to adopt navyonmāda, they could easily ruin their senā on top of having a tech gap with respect to the mleccha-s and Cīna-s. We have indeed seen evidence for mass deployment of navyonmāda by both Sorādi-duṣṭa-s and cīna-s — the CAA riots, the various khaṇḍa-jāti riots, the kīnāśa-uśnīśa-riots and Dravidianism — over the past few years. Hence, to cut the chase, we argue that the rise of navyonmāda has not only greatly multiplied the H’s threats in Bhārata and abroad but could be an existential threat.

This essay builds on an earlier one, which overlaps in scope, filing in some historical details.

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Some biographical reflections on visualizing irrationals

In our childhood, our father informed us that, though the school told us that \pi = \tfrac{22}{7}, it was not valid. However, he added that for “small fractions” [Footnote 1] it was a great approximation. Moreover, the numerical problems, which we would encounter in the tests, were designed with radii, etc., as “round” multiples of 7; hence, it was alright for vyavahāra. He also told us that in reality, \pi was irrational and that fractions like \tfrac{22}{7} were only approximations thereof. We asked why \tfrac{22}{7} was the best of the “small fraction” approximations. He asked us to figure that out for ourselves but told us that we could get better approximations with bigger fractions: \tfrac{333}{106} and \tfrac{355}{113}.

Armed with this information, not being mathematically too endowed, we had to think long about it before arriving at an algorithm to understand this. It went thus (written using post facto terminology):
1) Given that we had a lot of sheets of graph- and ruled- paper, we first sought to draw a line with slope equal to \pi, i.e., y=\pi x. This would ideally need the squaring of a circle. However, at that point, we did not know of Ramanujan’s constructions and accurately re-doing the yavana constructions with curves like the spiral and kampyle lay in the future. However, we had those approximations with the “larger fractions” that our father had given us, and with a hand calculator, we could verify that they provided more than the accuracy we needed for testing out \tfrac{22}{7}. Accordingly, we counted out 113 and 355 small squares on the X- and Y-axes respectively on the graph paper and carefully drew the line y=\tfrac{355}{113} x with a pencil that had an expensive, thin graphite refill.
2) By definition, the line passed through (0,0). After that, we tracked every time the line cut the horizontal (red crosses) or vertical (green circles) grid lines of the lattice (Figure 1).
3) We saw that the line first cut the horizontal grid lines at 1, 2, 3 and the vertical grid line for the first time at 1 after cutting the horizontal line 3. The horizontal cut at 3 and the vertical cut at 1 come close to each other (Figure 2). This immediately told us the obvious: \pi \approx 3, a reasonable first-order approximation, seen in several ancient traditions, like the Ṛgveda, where it is embodied by the god Trita. Going along, we saw an even closer approach between the vertical cut at 6 and the horizontal cut at 19 (Figure 1). This yielded \pi \approx 3.1\overline{6}, which again reminds one of the kinds of values seen in old traditions like the construction for squaring the circle of the Maitrāyanīya-s or the popular value of \sqrt{10} seen in some old Indian texts or the value in the Rhind papyrus of the Egyptians. At \tfrac{22}{7}, we saw a near-perfect matching of the vertical cut at 7 and the horizontal cut at 22. Further ahead, we saw another close approach between the horizontal cut at 25 and the vertical cut at 8, but this was worse than the approach at \tfrac{22}{7}. It only got worse from there till the higher fraction \tfrac{333}{106}, though there was a fairly good approach at \tfrac{113}{36} (still worse than \tfrac{22}{7}). Thus, we had seen for ourselves that \tfrac{22}{7} was indeed the best approximation among the “small fractions”. This childhood experiment was to leave us with a lifelong interest in irrationals and their fractional approximations, mirroring the fascination for things like the cakravāla seen in our intellectual tradition.


Figure 1. The cuts corresponding to \pi

The only problem with using such a method for finding rational approximations for numbers like \pi or e is that you should know their value to a high degree of accuracy a priori or be able to construct them using special curves. For example, if you can construct a line of slope e you can determine e \approx \tfrac{19}{7} — not as good as the approximation for \pi but has a nice symmetry with it telling us that \tfrac{\pi}{e} \approx \tfrac{22}{19}. However, for irrationals amenable to Platonic construction, e.g., \sqrt{2} this method can be used effectively to find small rational approximations, if any. Thus, we can arrive at \tfrac{17}{12} as a rough and ready approximation for \sqrt{2} (Figure 2).

irrationals_Fig2Figure 2. The cuts corresponding to \sqrt{2}

A few years later, these experiments led us (independently of any literature on the topic) to the idea that every irrational number can be represented as an infinite string of 0s and 1. We take a line of the form y=mx, where m is an irrational number. Based on the above approach, we write 1 if the said line cuts a horizontal grid-line and 0 if it cuts the vertical grid-line of the lattice. Since the line y=mx will pass through a lattice point only at (0,0), we start the sequence by writing 01. Thus, by moving along the line, we get an infinite sequence representing m as a string of 1s and 0s. One can see that as the number of elements in this sequence n \to \infty the ratio of 1s to 0s tends to m. Thus, for \sqrt{2}, the first 50 elements of this sequence are:


We also represented the same as a “string of pearls” diagram (Figure 3). This kind of representation reminds one of the differentiated cells in linear multicellular arrays in biology, for example, the occurrence of heterocysts in a filament of Anabaena. This led us to realize that the irrationals have a deep structure to them that is not apparent in the seemingly random distribution of digits in their decimal representation.


Figure 3. The “string of pearls” representation of various irrationals.

As we have seen before on these pages, this structure can manifest variously. One place where we found this structure as manifesting was in strange non-chaotic attractors, which were first described by Grebogi et al. in the mid-1980s. These attractors are “strange” because they manifest a fractal structure (see below) but do not exhibit chaos, i.e., the attractor has the same structure irrespective of the values with which the map is initiated. The map which we found is distinct from that of Grebogi et al. but produces similar behavior. It is of the form:

\theta_{n+1}= 2l\sin(\theta_n)\cos(x_n)
x_{n+1}=(x_n+2\pi m) \mod (2\pi)

Here l is a constant for which we can choose some value l \ne 0, but we set it to 1.03 simply for the aesthetics of the “irrational spectrum” (see below) and the manifestation of its fractal structure (Figure 4). The other parameter m is the irrational number under consideration. Irrespective of the initial conditions (as long as both x_0, \theta_0 \ne 0), the attractor takes the same form (Figure 4) that is determined only by the irrational m, as long as l remains the same. We term this the “spectrum” of the irrational. While the attractors appear superficially like curves symmetric about the x-axis with positive and negative limbs, we can show that they are actually fractal. The iterates do not proceed along the curve but jump from lower limb to upper limb and vice versa, and between any two arbitrarily close points on a given limb, one can find a finer structure of points jumping between the limbs, thereby establishing its fractal structure. Indeed, as l increases, the attractor acquires a structure reminiscent of the classic fractal structure seen in maps like the logistic map. The spectrum itself depends on the fractional part of the irrational m (Figure 4). This can be established by comparing the spectra for m=\phi, \tfrac{1}{\phi} (the Golden ratio), which have the same fractional part — they are identical. In contrast, the spectra of \sqrt{2}, \tfrac{1}{\sqrt{2}} are not the same, keeping with the differences in their fractional parts.

irrationals_Fig4Figure 4. The spectra of various irrationals initiated with \theta_0=0.001, x_0=0

While this type of “spectrum” captures the structure hidden in the fractional part, we also designed a second type of “spectrum” that captures the structure relating to the rational approximations we saw in the opening discussion. This is defined as the below summatory sequence:

\displaystyle f= \sum_{j=1}^{n} -1^{\lfloor j \cdot m \rfloor},

Here m is the irrational number under consideration. Figure 5 shows a step plot of f for m=\pi and j=1..300. We observe that it has a modular structure that builds up into a larger “wave”. At the lowest level, we have an up-down step of size 1. Then we see repeats of such steps with a transition between repeats at multiples of 7 (red lines). This continues until the “trough” of that wave is reached at 106 (green line). Then an ascent begins at 113, which is complete after a further 113 steps (violet line). Note that the numbers 7, 106 and 113 are the denominators of the successive best rational approximations of \pi. Thus, these sequences have a fractal structure with larger and larger cycles whose size is determined by the denominators of the successive rational approximations.


Figure 5. Step plot of the summatory spectrum of \pi till n=300.

Hence, those irrationals with multiple small rational convergents tend to show more “overtones” resulting a more complex structure (e.g., \gamma or \sqrt{2}; Figure 6)

irrationals_Fig6Figure 6. Step plot of the summatory spectrum f of various irrationals till n=300.

From the above plots, it becomes apparent that while f for all irrationals will be fractal, at least small scales some show greater “complexity” than others, e.g., compare the plots for \pi or \zeta(3) with those for \phi or \sqrt{2}. We wondered if there might be a way to define this visual impression quantitatively? We did so by defining sequences f_1, the sum of the terms of f in a sliding window of size l. For each irrational, we considered all f_1 computed for l=1..25. Then for each of these summatory sequences f_1 for a given irrational, we counted the number of times the curve changes direction, i.e., changes from going down to going up and vice versa. We then normalized the direction-change counts with each l for a given irrational by the maximum number of direction changes seen for that irrational. The mean of this normalized value gives a measure of the complexity, i.e., the “wriggliness” of the curve. This is shown as c in Figure 7 along with a plot of f_1 for l=12 and shows that the measure c indeed matches the visual impression from Figure 6.


Figure 7. Step plot of the sliding window sequence f_1 of various irrationals with l=12

Footnote 1: Fractions with small numerators and denominators

Posted in Heathen thought, Life, Scientific ramblings | Tagged , , , , , , , | Leave a comment


This note stems from a recent conversation with a friend, where he pointed out that the graph representing all possible positions the horse (knight) can take on the chessboard from a given starting square produces interesting graphs. It struck us that this would indeed be an interesting exploration to introduce neophytes into the graph theory and computational exercises relating to it. Hence, we prepared this elementary note for pedagogical purposes and to show some pretty pictures. The origin of this class of problems lies in the Hindu tradition of citrabandha-s and yukti-s, which have been extensively discussed and illustrated by various medieval authors in kāvya and encyclopedic literature (e.g. Mānasollāsa of the Cālukya emperor Someśvara-deva). In his Śiśupāla-vadha, Māgha states:

viṣamaṃ sarvatobhadra-cakra-gomūtrikādibhiḥ ।
ślokair iva mahākāvyaṃ vyūhais tad abhavad balam ॥
That force was difficult to penetrate being equipped with
the sarvatobhadra, cakra and gomūtrikā and like of formations,
even as a mahākāvya furnished with such verses.

Here, he uses a simile to compare the military formations like sarvatobhadra (comparable to the Roman testudo), etc., with the equivalent citrabandha-s or structural constraints used in kāvya. Several of the early medieval examples of such citrabandha-s in kāvya are related to the description of battle scenes: e.g., the Haravijaya of Rājānaka Ratnākara, the Śiśupāla-vadha of Māgha, the Kirātārjunīya of Bhāravi and the Jānakīharaṇa of Kumāradāsa; thus, it seems likely that the authors were trying to embed images of the yuddhavyūha-s of the yuddha-s under description in their kāvya, as suggested by the above verse of Māgha. However, such devices are also used widely in stotra literature, for instance to depict the weapons of the deities instead of the vyūha-s — the Kaśmīrian Sāmavedin, Rudraṭa Śatānanda, uses such in his Durgāṣṭaka. Since we first learned of the use of such structural constraints in kāvya in our youth from the praise of Durgā by the great Kaśmīrian kavi Ānandavardhana, it struck us that just as they have place in illustrating battle-formations, they also represent an early example of using ideas that intersect with graph theory and symmetry with far more general implications. Indeed, even as the “Kavi-prajāpati (to use Kalhaṇa’s term)” selects for such constraints to bring meaning to his kāvya, natural selection picks such constraints in words formed by the alphabet of nucleic acids and proteins to generate biochemical function.

Keeping with the military connections of the citrabandha-s, one such citrabandha, the turagapada, is based on the horse’s movements in that ultimate “war-game” which spread widely in the Gupta age, caturaṅga, i.e., the steps of a horse on the chessboard. While we do not play chess, we found the abstraction to be of interest. On an infinite chessboard, the horse on a given square can reach 8 other squares (Figure 1).

Figure 1. Possible paths of the horse.

However, some of these are unavailable at squares on the boundary or the penultimate circuit on a board of finite length and breadth. For convenience, going forward, we shall only look at square boards. This is stated for an 8 \times 8 board by emperor Someśvara-deva in his Mānasollāsa thus:

koṇa-pārśvasthitasyāsya turagasya pada-trayam ॥
koṇasthasya pada-dvandvaṃ prānte pada-catuṣṭayam ।
dvitīya-valaye koṇe haye pada-catuṣṭayam ॥
dvitīya-valaye’nyatra pada-ṣaṭkaṃ nigadyate ।
madhye ṣoḍaśa[koṣṭheṣu] sthitasya turagasya ca ॥
padāṣṭakaṃ vinirdiṣṭaṃ caturaṅga-viśāradaiḥ ॥

In the cell next to the corners, the horse has 3 moves; in the corner cells, it has 2 moves; interior to these cells, on the border circuit, it has 4 moves; in the corner cell of the second circuit, it has 4 moves; in the interior cells of the second circuit, it has 6 moves; in the 16 interior cells, it has 8 moves each. Thus it has been expounded by the chess experts.

We can consider the cells of an n \times n chessboard, labeled from 1 to n^2 by rows, as the nodes of a graph. An edge connects two nodes in this graph if they can be reached from each other by the move of a horse — the turagapada. The horse can make no moves on boards with n=1, 2. Figure 2 illustrates the graph for n=3. One can easily see that the horse can reach every cell from every other cell except for cell 5. Thus the graph is a simple cycle of 8 nodes with one disjoint node 5.

Figure 2. Moves of a horse on a 3 \times 3 board.

From n=4 board onward (Figure 3), we get single-component graphs with no disjoint nodes. From n=4, all the graphs have 4 nodes with just two connections corresponding to the corners of the board, as mentioned by Someśvara. The nodes are colored as per the number of edges connecting to them. This graph can be rendered as a 3D object, which, in principle, could be the structure of a hydrocarbon. However, it remains unclear to us if such a hydrocarbon exists or can be synthesized in reality.

Figure 3. Moves of a horse on a 4 \times 4 board.

From n=5 onward, we get nodes all the possible connections, namely those with 2, 3, 4, 6, 8. We have a single maximal node at n=5, node 13, with 8 connections. When we render this graph using the Kamada-Kawai force-direct spring algorithm, we get a structure with bilateral symmetry and interesting relationships between symmetrically equivalent neighboring nodes (Figure 4). For example: nodes 1, 19; 7,25, both pairs differ by 18. The nodes 9, 21; 5, 17 differ by 12. The 4 edge nodes 8, 12 and 14, 18 represent another such pair of symmetries.

Figure 4. Moves of a horse on a 5 \times 5 board.

These graphs lead us to the classic turagapada problem of kāvya and its solutions, which simply stated goes as: can you find a path such that the horse visits each cell on the board only once? In terms of the graph, it can be stated as: can you find the path passing through all nodes of the horse graph only once. In modern computational literature, this is called the knight’s tour problem. The above graphs show that no such tour can exist for n=3, 4. For n=3, cell 5 cannot be reached, but the remaining cells can be visited by a closed cyclic tour path (Figure 5). For n=4, though the graph has a single component, the fact that a pair of 2-edge nodes connect to the same pair of nodes means that a tour cannot be completed. However, 15 of the 16 possible cells can be visited on tour (Figure 6).

Figure 5. Incomplete turagapada on a 3 \times 3 board.

Figure 6. Incomplete turagapada on a 4 \times 4 board.

From n=5 onward, one can always find multiple tours that visit every cell. Figure 7 shows such an example on a 5 \times 5 board.

Figure 7. Turagapada on 5 \times 5 board

It is easy to see with the graph that solving the turagapada by brute-force walks along the graph is very inefficient and will explode as n increases. However, a simple algorithm for finding a turagapada exists: (1) Start with a given node and move to a neighboring node from which the fewest non-0 number of further possible moves are available. For the possible available moves, only those neighbors which have not yet been visited are counted. (2) If a tie occurs, then one simply goes to the cell with a smaller index. (3) While these two steps are sufficient to yield solutions from several cells, it is not foolproof. Hence, one may look one level down to see if there are neighbors’ neighbors from which the fewest possible moves are available to find turagapada-s with greater certainty.

Figure 8. Turagapada on 8 \times 8 board

Figure 8 shows a solution for an 8 \times 8 board using the above algorithm starting from cell 1. Given that Someśvara explicitly lists out all the possible moves from a given cell, he implicitly seems to have used a similar algorithm with symmetry considerations to find a turagapada. He specifies it by first providing a chessboard with coordinates indicated by the syllables formed by the first 8 vowel conjunctions (a, ā, i, ī, u, ū, e ai) of the 8 consonants (c, g, n, d, ṭ, r, s, p). He then gives the turagapada asā sequence of 64 syllables shown below (Figure 9).

Chess_09Figure 9. Someśvara’s turagapada.

pa si pu se ṭai ne cai gū । nī cu gi ca nā ṭa sā pī ।
sū pai re dai ge dū gu ci । ga dā ra pā sī pū sai ṭe ।
nai ce nū gai cū gī cā na । ṭā sa pi su pe rai de nu ।
ṭū rī di ṭu ri dī ru ṭi । du ni cī gā da rā ṭī rū ।

Other than this yukti presented by Someśvara, several kavi-s have given their own solutions for complete tours. We have Rudraṭa’s solution in the form of a prosodic pattern with repeating strings of nā and lī interspersed between two distinct syllables, namely se and le (supposedly you can use the principles of saṃdhi and samāsa-vigraha to read this as a Sanskrit verse). His commentator, Namisādhu, provides a mnemonic using the Sanskrit varṇamālā. Ratnākara and Veṅkaṭanātha-(Vedānta) deśika provide pairs of verses with one laying the chessboard and the other providing a solution tour. Rudraṭa, Ratnākara and Veṅkaṭanātha, exploit the fact that the anuṣtubh meter has four feet of 8 syllables each. Thus, they can cover 32 syllables or half a chessboard with a single śloka. This solution is symmetric; hence, it can be reflected to provide a full board solution. The widespread use of turagapada as a citrabandha, from Kāśmīra to Drāviḍa, suggests that a version of the algorithm stated above was imparted in traditional medieval education.

This tradition was transmitted to the Mohammedans, and from them, it appears to have been transmitted to Europe. However, to our knowledge, the first solution appears relatively late in Europe, being provided by Leonhard Euler. William Hamilton considered a related problem: let a dodecahedron represent a graph with 20 nodes and 30 edges. Find a cyclic path that passes through all edges only once. Here every node is connected to 3 edges. A similar question can be asked for the horse-graph — i.e., finding a complete tour that is also a cycle. A solution to this problem is recorded in the late medieval encyclopedia of Nīlakaṇṭha Bhaṭṭa (Bhagavanta-Bhāskara), but I do not have it handy right now. In any case, a vast body of literature exists on algorithms for tours and their use in kāvya; hence, we do not tarry any more on this point.

Finally, few other interesting questions emerge from the horse graphs. First, the diameter of a graph is defined as the longest shortest path between two nodes of a graph. The distance between two nodes is a geodesic — i.e., the shortest path along the graph. The longest such geodesic between any two pairs of nodes in the graph is its diameter. For the horse graph with n=3, 4, \cdots, 14 we can compute this sequence to be: 4, 5, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10. Can one come up with a closed expression for this?

Second, there are m shortest paths between node 1 and node n^2 for a given horse graph. For n=3, 4, \cdots, 8 this sequence goes as: 2, 2, 8, 4, 6, 108. A colleague of ours, a professional mathematician, showed us a complicated formula that can describe this strange sequence. While I am not providing that here, it shows a dramatic jump whenever n \mod 3 \equiv 2. Thus, we see a jump at n=8, making it more than the number of shortest paths found for the n=9, 10. Thus, it appears the inventors of the game chose n to provide a maximal diversity in the movement of the horse for boards of a similar order.

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Asians and Pacific Islanders: The triangle

In our youth, we read with great excitement old books on anthropology obtained from a library with considerable difficulty. The excitement was primarily from learning about the osteology of extinct apes and monkeys, including the closest sister groups of Homo sapiens. Some of those books also had a collection of plates with pictures of stone tools and various extant peoples of the world, especially hunter-gatherers who still lead a relatively archaic mode of life. Those pages too fascinated us, and we spent many an afternoon turning through them wonder-struck by how many different morpho- and eco-types of Homo sapiens were around. Of those tribes, the Melanesian in particular caught our attention with their gripping displays of headhunting, cannibalism and prion neuropathology (there was still a debate about what prions were back then). The images of Fijian tribesmen and a collection of their braining clubs left a deep impression on us (Figure 1). We had a direct experience of the same when we visited a coethnic who had been driven out of Fiji during the attack on the Hindus by the former islanders. However, in his flight back to the subcontinent, he had brought along one of those clubs of the ancestors of his Fijian enemies. We wondered about how the Melanesians and Polynesians reached their distant outposts in the Pacific. We also wondered how these Indo-Pacific peoples might be related to the tribal Indians — it did not escape our attention that some of that ancestry was visibly present in the non-tribal Indians.


Figure 1. Fijian tribal warriors with a club photographed in the late 1800s.

Answers to many of those questions have come from the genomics of extant and prehistoric peoples over the past few years. In its present form, this note is by no means a survey of all that. It is just a very brief account comprised of a few observations sparked by recent discoveries. Unfortunately, due to magazine-fever many molecular paleoanthropology papers while presenting important specimens are poorly written and illustrated. Also, due to the competing groups involved, the speed with which new specimens are piling in, differences in interpretation, and the terminological issues, these works do not afford a synthetic picture of the history of the people under consideration. So, we have had to wade through these presentations, which might ignore each other, to summarize the below points of interest regarding the evolution of the Asians and Pacific Islanders. At the broad level, the ethnogenesis of the Eurasians and Pacific islanders can be summarized by this principal component analysis of the genomic variation of extinct and extant individuals (Figure 2). Some populations are colored distinctly against the gray background of the remaining individuals and labeled. The key prehistoric genomes are indicated by big stars and labeled in the legend.

AsiansFigure 2. A plot PC1 and PC2 showing various Eurasians and Pacific Islanders. LinearBKer: Linearbandkeramik (Early European Farmers: Neolithic); Gandhara: Ancient samples from Northwestern India (what is now TSP); Aus/Papuan: Australians (squares) and Papuans (triangles); Phil/Mal Ngrt.: Aeta, Agta, Jehai and Batak peoples (Philippines/Malaysian Negritos, brown triangles); Cam: Cambodians; Twn Ausn: Taiwanese Hanben site, likely early Austronesians; Karelia HG: Eastern hunter-gatherers from Karelia (Finland-Russia border zone); Iran.Cu: Copper Age people from Hajji Firuz site; Iran.Neo: Neolithic people from Ganj Dareh site; Geoksyur Neo/Cu: Turkmenistan Neolithic and Copper Age people.

One can see that a triangle of clines describes a major fraction of Eurasians and Pacific islanders. The first is the Indian cline extending from the Andamanese populations, like the Onge, Jarawa and Great Andamanese on one end and at the other end terminating in the Sintashta steppe culture that likely corresponds to the expanding Aryans. The Paniya tribe of Kerala and Karnataka represent one of the groups from the mainland that is closer to the Andamanese end of this cline. The second notable cline is the Australasian cline, with the Papuans and Australians on one end (close to the Andamanese) and the East Asians (Hans, among others) at the other end. In between lie the Negrito tribes of the Philippines and the Malay region (brown triangles), and the Austronesians who spread into the Indo-Pacific from Taiwan in a maritime expansion, ultimately reaching New Zealand and Rapa Nui (Easter Island) of the coast of Chile. The third notable cline completing the triangle is the North Eastern Eurasian-First American cline. The Bronze Age Okunevo from Southern Siberia, the North and South American native peoples (including the prehistoric Kennewick man from the Washington state of USA), Eskimos, and Mongols are seen lying on this cline. Based on the prehistoric samples we can summarize some key events in the ethnogenesis of the Asians and Pacific Islanders thusly:

1. Before \approx 50K, there was an unknown number of archaic Homo lineages throughout Asia all the way to the Pacific islands. Of these, the Denisovans were widespread. We have direct evidence for Denisovan admixture in Tibet, Mongolia (the Salkhit woman from 34KYA shows some Denisovan admixture), the Philippines and the Indo-Pacific islands. The ancestral Asian arrived in this landscape after splitting off from the lineage leading to the Western Eurasian in the west. Then the ancestral Asian split up into several far-ranging groups, probably by around 50KYA. These early Asian lineages included:
i. The Tianyuan-like Eastern group prototyped the Tianyuan man from the Tianyuan cave near Beijing, dating to \approx 41KYBP. A recent study posits that the Tianyuan man might have had up to 3% archaic Homo admixture from a Denisovan source. The Tianyuan-like clade was probably basal to the later East Asian clades, which eventually split up into Northeast Asian and Southeast Asian clades ancestral to modern East Asians.
ii Onge-Hòabìnhian group, once extending from at least the Andamans (Onge, Jarawa, Great Andamanese) through Laos, Vietnam and Malaysia. This group has been recently registered in the ancient specimens from Hòabìnhian hunter-gatherers from East Asia from at least 8000 YBP.
iii. An early-diverging sister group of the Onge-Hòabìnhian clade were the hunter-gatherers of the Indian subcontinent from whom tribal Indians on an average get most of their ancestry (e.g., See Paniya in Figure 2). The Onge-Paniya gap in Figure 2 represents this deep divergence between the Onge-like clade proper and their mainland Indian sister group. Varying fractions of this ancestry persist in non-tribal Indians with a mostly south to north gradient (“The Ancient Ancestral South Indians” (AASI) of the Reich group). Today, other than this Indian HG clade, the broader “Onge-like” clade includes the Philippines Negritos, Papuans and Australians (Note their proximity in Figure 2).
iv. The Tibetan hunter-gatherer-Jomon group, which once stretched over Asia from at least Nepal-Tibet to Japan. These people played an important role in the ethnogenesis of the modern Japanese. The old Jomon were first invaded by a Northeast Asian population from the Amur River region, leading to the Yayoi period around 3000 YBP. This was followed by the classical Koreanic-type East Asian invasion of Japan, marking the emergence of the historical Japanese at the beginning of the Kofun period. Pure representatives of this group are extinct now, but their Y-chromosome and some autosomal genome survives in Nepal, Tibet and Japan. It seems the 2800-year-old Chokopani man from the Mustang cave in Nepal had \approx 16% of this ancestry while a 3500-year-old Jomon Japanese individual shows about 44%.

2. The Onge-Hòabìnhian clade proper lack high Denisovan ancestry but their sister group, the Papuans and Australians, show evidence for at least two introgression events with Denisovans. The Philippines Negritos, too, had 1 or 2 Denisovan admixtures. Thus, greater Onge-like clade spreading from India to South East Asia encountered Denisovan races all the way from the Philippines to Sahul (the combined Pleistocene landmass of Papua+Australia) and annihilated them across the Indo-Pacific islands while mating with them on occasion. This raises the possibility that the dwarf Homo (e.g., Homo floresiensis and the Luzon Homo) on the Philippines and Flores were races of Denisovans. The North Asian and Central Asian Denisovans seemed to have had larger body size and a characteristic huge molar.

3. A sister group of the greater Onge-like clade group or alternatively a group branching close to the stem after the Tianyuan-like and Onge-like groups somehow reached America and contributed a small amount to the ancestry of some South Americans. While initially noted by Skoglund et al. in the Amazonian tribes like Surui and Karitiana, recent work by Brazilian researchers also recovered this ancestry on the South American Pacific coast. To date, this ancestry is missing in the North and Central Americans or their Beringian predecessors. This favors the model in which this Onge-like ancestry reached the Pacific coast of South America by sea (see below).

4. Recently, Carlhoff et al. reported the genome of a pre-Neolithic young forager woman from Leang Panninge, South Sulawesi dating to 7.2-7.3 KY from the `Toalean’ archaeological complex. This is the first genome from Wallacea, the Oceanic islands between the Sunda shelf of Indonesia and the Sahul landmass of the Pleistocene. She is modeled as having \approx 50% of Onge-Papuan-like ancestry related to that seen in Sahul peoples along with the Denisovan admixture seen in them. However, the best fit models also suggest a prehistoric East Asian ancestry of \approx 50%. The authors say this can be approximated by Qihe, a Southeast Asian Neolithic individual from \approx 8.4 KYA. This suggests that not just Onge-like groups but also the Southeast Asian clade expanded into the Pacific, mixing with the former. However, this type of \approx 50-50 Qihe-Onge-like mixture is no longer present in Sulawesi or its surroundings. They seemed to have been wiped out in turn by another Southeast Asian expansion, the Austronesian expansion from Taiwan in the past 4000 years.

5. Such a see-saw contest between representatives of the greater Onge-like clade and the Austronesians of Southeast Asian roots played out repeatedly over the Philippines and the Malay archipelago. This is supported by the Southeast Asian admixture seen in the various Negritos and the remarkable the ancient DNA results from Vanuatu. This cluster of about 80 remote islands in the South Pacific is populated by people speaking an Austronesian language but having most of the ancestry from a Papuan-like group. Like Papuans, several Vanuatu tribesmen wear phallic sheaths (koteka). However, the earliest genomes from these islands suggest that they were first occupied by the Austronesians. But they were soon joined by the Papuan-like group. These Papuan-like people seemed to have wiped out the Austronesians on several islands, but the latter seemed to have held out on some of the islands. These Austronesians then appear to have made a return to mix with the former and give rise to the extant Vanuatuans.

6. If the Austronesians expanded primarily via the maritime route to span a vast swath of the globe, another Southeast Asian group, the Austroasiatics, appears to have expanded mainly by land and probably by sea. These include the speakers of Vietnamese and Khmer in continental Southeast Asia, Aslian in peninsular Malaysia and Thailand, the Nicobarese in island India, Khasi in North East India and the Munda in Eastern and North Eastern India. These Austroasiatics seem to have expanded from the Mekong river basin as a group dependent on fisher and some neolithic farming for their subsistence. One group probably arrived in the Indian mainland around 3200 YBP, where they mixed with the original Indian hunter-gatherers to give rise to the Munda-speaking tribal groups like the Santhal (Figure 2). It is likely they also resorted to a maritime route to reach Nicobar relatively early on.

Finally, we shall make a few remarks regarding the implications of these findings for the modes of spread and various language groups. There is little evidence for any clear-cut relationships between the languages of the Andamanese, Papuans and Australians despite some claims to this effect by some of the long-rangers. Nor do they show relationships to Dravidian or whatever is reconstructible of the ancient Indian substrata. This is keeping with their split in the relatively ancient past when Denisovans were still around in the Indo-Pacific region and prolonged existence as hunter-gatherer tribes. The Austronesian and Austroasiatic languages are well-defined families like Indo-European and show the hallmarks of massive, relatively recent expansions. Linguistic investigations suggest that the languages of the Kra-Dai family (e.g., Thai) might be a sister group to the Austronesian languages. The genetic evidence is not inconsistent with this proposal. The Austronesians were the masters of maritime expansion — they probably reached their Taiwanese homeland after splitting off from mainland Kra-Dai speakers. From there, they returned to the Asian mainland and Malay peninsula (Cham in Vietnam and Malay) and spread both East and west. In the East, they first moved slowly, taking the Philippines and then around 3500 YBP covered most of the Malay Archipelago. Over the next 500 years, they took Melanesia and Western Micronesia. By around 1500 YBP, they had swung west to Madagascar off the coast of Africa. Over the next 700 years, they took every remaining Micronesian and Polynesian island all the way to Easter Island and Hawaii. The Southeast Asian admixture in pre-Austronesian Leang Panninge suggests that the Austronesians were not the first to venture into the Oceanic deep East. This is also hinted by the presence of the Onge-like ancestry in South America that likely reached there directly by sea via the Pacific coast. However, Austronesians certainly seem to have been the most successful. This raises the question of whether their boats were the critical factor that allowed the Papuan-like people to reach Vanuatu after the Austronesians got there first. This might also explain why the Austronesian language rather than genetics dominated in Vanuatu by functioning as the link language between the islands. Finally, this brings us to the recent work that has provided evidence for South American admixture from a Zenu-like South American tribe among the Polynesians, supporting the much-maligned contention of Thor Heyerdahl. Here again, it is peculiar that there is no evidence for a greater South American presence in Polynesia if they managed to reach some of the islands and transmit the sweet potato (Ipomea batatas). We suspect the initiative was with the maritime Polynesians who managed to reach the South American coast and bring back some admixture to their islands along with the sweet potato. Perhaps, as Heyerdahl speculated, this contact might have also contributed to some of the iconographic convergences that he noted, like on Easter Island.

Clear monophyletic language families like Austroasiatic and Austronesian are not seen as uniting China, the Korean Peninsula and Japan, though these East Asians are genetically very close. While Japonic and Koreanic have structural similarities, evidence for their monophyly as sister groups or as part of a larger Macro-Altaic assemblage is limited. This suggests that the Yayoi probably brought the Japonic languages to Japan. This might explain the more general structural similarities with North East Asian language families like Koreanic and Tungusic but the absence of a specific relationship. The Kofun, while contributing most of the genetics of the extant Japanese, did not bring the language itself. They instead probably rose up in the Yayoi background as an elite that adopted the Yayoi language while spreading Kofun genetics.

Thus, ancient DNA is making up for the absence of recorded history. Unfortunately, this revolution has not yet touched India. Imagine if we were to know something of Jorwe culture or Ash Mound peoples — they remain archaeological black boxes along with several other slices of Indian history.

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The shape of dinosaur eggs

Readers of these pages will know that we have a special interest in the geometry of ovals. One of the long-standing problems in this regard is: what is the curve that best describes the shape of a dinosaurian egg? While all archosauromorphs hatch from eggs outside their mother’s body, the form of their eggs is rather variable; crocodylians and turtles may lay either leathery or hard-shelled eggs. The dinosaurs almost always lay hard-shelled eggs that tend to be rather uniform in shape in the wild. Being hard-shelled, the shape of a dinosaurian egg can be described by the characteristic curve of its maximal (area) cross-section. The egg itself will be the solid of rotation of this curve around its longest axis. Using this definition, the noted morphometrician and student of Aristotelian zoology, D’arcy Thompson, classified bird eggs into various forms in his famous book “On growth and form”. More recently, this was revisited by Nishiyama, who named 4 shape groups for the eggs of modern (avian) dinosaurs: (1) circular; (2) elliptical; (3) oval; and (4) pyriform. However, an examination of a large data set of eggs from around 1400 extant birds by Mahadevan and colleagues shows no strict boundaries between the shape classes. Hence, in principle, they should all be describable by a single equation of shape with varying parameters. If postmortem distortion and the effects of fossilization have been properly accounted for, the Mesozoic dinosaur eggs featured additional diversity. For example, the eggs of theropods, like the tyrannosaur Tarbosaurus, would not be described appropriately by any of the 4 purported classes. It would rather be a generalized higher-order elliptical egg (see below). Hence, ideally, the equation should not just cover extant dinosaurs but also the extinct ones.

Indeed, there has been a long-standing interest in obtaining that single equation that describes eggs’ shapes, starting with extant birds. One such early attempt was that of the Swiss geometer Jakob Steiner who proposed the oval of Descartes (defined by the bipolar equation: r+mr'=c, where m is the ratio parameter and c the constant sum) as the general equation for avian eggs. However, D’arcy Thomas had pointed out that various avian eggs do not fit this curve. Subsequent explorations of this question have offered a range of solutions. In light of recent work presenting a new potential solution, we consider and compare some notable attempts, including the latest.

\bullet Maxwellian ovals: The great JC Maxwell, while still in his early teens, generalized the ellipses to describe families of ovals. Of these, one class of ovals, the “trifocal ellipse” can be described using three functions in x, y:

f_1(x,y)=x^2+y^2; f_2(x,y)=(x-a)^2+y^2; f_3(x,y)=(x-b)^2+y^2

Then the Maxwellian oval is described by the equation:

\sqrt{f_1(x,y)}+\sqrt{f_2(x,y)}+\sqrt{f_3(x,y)}=c, where a, b, c are parameters.

 Figure 1. A Maxwellian trifocal oval.

We too independently arrived at this curve in our teens but, unlike Maxwell, did not achieve any deep understanding of the physics of these curves. This curve is constructed using the same principle as an ellipse, viz., the locus of points whose distances from the foci add up to a constant, but it has 3 colinear foci instead of 2 of the regular ellipse (Figure 1). For a=1, b=0.2, c=2.2 and close values, we get a curve that even a casual observer will note as approximating the shape of common avian eggs (Figure 1). Indeed, in 1957, such a curve had been used by a certain E. Ehrhart as a possible fit for the avian egg following statistical analysis of real specimens. Unaware of Ehrhart’s work, in course of our own early experiments with this curve, we too considered this as a possible description of the shape of the most common type of avian egg prototyped by those of several galloanseran birds. This has now been borne out by the large dataset of Mahadevan and coworkers, in which the most frequently occurring morphology is close to this curve. However, it is hardly a universal equation of shape as there are several egg shapes lying outside its scope.

\bullet Hügelschäffer’s equation: The the 1940s, a German engineer Fritz Hügelschäffer, derived the equation of an oval that he felt was a good fit for the shape of avian eggs. As we have noted before on these pages, we independently arrived at the construction of this oval and derived its equation while in junior college. Hügelschäffer expressed the equation of this curve in the following form:


Here, L is the major axis or length of the egg with its center placed at origin O=(0,0). B is the minor axis or maximal breadth of the egg. w is the distance between O and the point of intersection of the segments L and B. By taking L=1 this effectively yields a 2 parameter (B, w) shape curve, that accounts for more egg morphologies than the Maxwellian trifocal oval: with L=1, we obtain the circular (B=1; w=0), elliptical (0 \le B <1); w=0) and oval (B \ne 1; w>0) shape classes of extant avian eggs. Moreover, its parameters are entirely intuitive and can be measured easily from real specimens. However, it does not account for the so-called pyriform class (common in shorebirds) or the Mesozoic dinosaurian eggs like those of Tarbosaurus, the caenagnathid Beibeilong, the troodontids, or the ceratopsian Protoceratops.

\bullet Preston’s 4 parameter oval: Incited by D’arcy Thompson’s failure to give a  general equation for the shape of eggs, in the 1950s, Frank Preston, a versatile English engineer (invented a glass-melting furnace and a device to measure avian egg shapes), marksman and naturalist, derived the 4 parameter oval to describe the shape of all bird eggs. He used the following logic: the most symmetric class is the circular class for which one can easily write down the equation: y=\pm \sqrt{1-x^2}. By multiplying this by the parameter a \le 1, the aspect ratio (ratio of minor to major axis) of an ellipse, we get the equation y= \pm a\sqrt{1-x^2}, which can account for both the circular and elliptical classes. Then Preston accounted for the remaining classes using a polynomial function f(x)=1+bx+cx^2+dx^3, where -1 \le b, c, d \le 1, thus yielding the final equation of a generalized oval:

y=\pm a(1+bx+cx^2+dx^3)\sqrt{1-x^2}

Figure 2. Eggs of selected extant and extinct dinosaurs modeled using Preston’s equation.

You can try out various fits with above parameters on a collection of real eggs here

As one can see from Figure 2, Preston’s 4 parameter oval accounts for all dinosaur egg shapes extant and extinct. The egg of the Ural owl is of the elliptical class coming close to the circular class. The emu is nearly a classical elliptical with a very small c parameter that adjusts it to a more generalized ellipse. The song thrush and osprey are very nearly ovals with a pronounced b parameter and slight adjustments, again with a very small c parameter. The guillemot and great snipe are clear pyriforms with both pronounced b and c parameters. The extinct dinosaur Troodon has an egg quite distinct from any extant bird but can still be modeled by Preston’s oval with positive c, d terms. The egg of Tarbosaurs (and others like it, e.g., Beibeilong or Protoceratops) is a generalized ellipse that is again well-modeled by Preston’s equation with just a negative c term with other x powers in the polynomial having 0 coefficients. Preston’s equation can model most extant bird eggs using just the linear and square terms with negative coefficients. The cubic term is only needed for unusual eggs, like in this case, that of Troodon. Since Preston, several researchers have tried to duplicate his approach by using other functions in place of his cubic polynomial (see below). However, recent numerical analysis using a dataset of 132 real eggs from various modern species by Biggins et al. has shown that Preston’s curve outperforms all these other attempts in having the least and an impressively small error. Thus, the Preston 4 parameter oval can be considered a valid, universal description of the shape of the dinosaurian egg.

\bullet In light of the success of the Preston oval, we were a bit surprised when we saw a recent work by Narushin et al. claiming to introduce a universal formula for the egg shape. They acknowledge the success of Preston’s work but state that the parameters in his equation are neither intuitive nor readily determined. The former is indeed a potential criticism; however, the latter is less of any issue with modern graphing software, so long as one has good photographs. Hence, they decided to start with Hügelschäffer’s formula and applied a series of modifications to arrive at a complicated formula for a general oval:

y= \dfrac{B}{2}\sqrt{ \dfrac{L^{2}-4x^{2}}{L^{2}+8wx+4w^{2}}} (1- k f(x))

k = \dfrac{\sqrt{\dfrac{11}{2}L^{2}+11Lw+4w^{2}}\left(\sqrt{3}BL-2D\sqrt{L^{2}+2wL+4w^{2}}\right)}{\sqrt{3}BL\left(\sqrt{\dfrac{11L^{2}}{2}+11Lw+4w^{2}}-2\sqrt{L^{2}+2wL+4w^{2}}\right)}

f(x) = 1-\sqrt{\dfrac{L\left(L^{2}+8wx+4w^{2}\right)}{2\left(L-2w\right)x^{2}+\left(L^{2}+8Lw-4w^{2}\right)x+2Lw^{2}+L^{2}w+L^{3}}}

Here, as in Hügelschäffer’s equation, L is the major axis or length of the egg; B is its minor axis or greatest breadth; D is the breadth of the egg at the point halfway from the center at (0,0) to the narrow end of the egg (Figure 3). However, w is not the same as in the Hügelschäffer equation but is defined as:

w=\dfrac{L-B}{2n}, where n is a positive number.

The landmarks of Narushin et al’s equation for a dinosaurian egg.

One advantage of this equation is that L, B, D can be directly measured relatively simply with Vernier’s calipers and a ruler. The w parameter can be empirically calculated from L, B by adjusting n. While the authors state this equation can account for all extant bird eggs, we found that, unlike Preston’s equation, it could not account for special cases of extinct dinosaur eggs, like those of Troodon and Tarbosaurus, assuming that their reconstruction is accurate. However, we rectified that by using an “inversion” flag that takes 3 values: N, Y, and H. It appears that for all extant birds (at least those considered by Narushin et al.), this flag is N; these can be modeled using their equation as is. For Troodon and related eggs, the flag is Y; here, x has to be substituted by -x. For Tarbosaurus and related eggs, the flag is H; here for -\tfrac{L}{2}\le x \le 0 we use the equation as is and for 0 < x \le \tfrac{L}{2}, we substitute x with -x. This accounts for all dinosaur egg shapes comparable to Preston’s equation (Figure 4). If we normalize it by taking L=1, it effectively leaves us with a maximum of 4 parameters as in the case of Preston’s equation.

Figure 4. Eggs of selected extant and extinct dinosaurs modeled as Narushin et al’s oval with the inversion flag modification.

You can try out various fits with above parameters on a collection of real eggs here.

The relative merits of this equation need to be compared to that of Preston’s using real specimens. Unfortunately, this requires additional work as Mahadevan and colleagues’ large dataset does not have all the necessary measurements for such a comparison. They instead used the oval of Baker, an attempt to duplicate Preston’s work, which significantly falls short of the latter in terms of accuracy while providing a simple 2 parameter space. It is defined by the equation:

y=\pm a\left(1-x\right)^{\frac{1}{1+b}}\left(1+x\right)^{\frac{b}{1+b}}

Here 0 \le a \le 1 is the aspect ratio of the ellipse as in Preston’s equation or the equivalent of B | L=1 in Hügelschäffer’s equation, while 1 \le b \le 2 is an asymmetry parameter similar to w in Hügelschäffer’s equation. This curve has 3 successively tangent lobes with points of tangency at (-1,0) and (1,0). For x \le -1 and x \ge 1 the two lobes have a divergent hyperbola-like form. For -1 \le x \le 1, the curve takes the form of the oval that approximates the shape of a dinosaurian egg. When b=1, the curve becomes an ellipse.

A biologist can easily conceive each of the parameters from Preston’s or Narushin et al.’s curves as being controlled by a genetic factor, with changes in it leading to a change in the parameter. Thus, one can easily account for the diversity of egg morphologies observed among dinosaurs through genetic changes. The parameter \leftrightarrow gene mapping feeds directly into the question of what are the selective forces acting on egg shape. Several of these have been proposed and debated since Darwin. Irrespective of the correctness of some of these, one key point emerging from the suggestions made by Birkhead is that the ovoid morphology is a clear sign of a comprise solution resulting from balancing selection. The compromises themselves might involve very different factors. One such relates to spherical morphology having the smallest surface area for a given volume. Hence, it is ideal for not losing heat quickly. However, eggs also need to be externally warmed, either by direct contact with the mother or exposure to solar or geothermal heat (e.g., in titanosaurs). Here, a less spherical shape would afford a greater surface area to allow quicker external heating. Similarly, a hard-shelled egg would have the greatest strength against external force if spherical. This would be selected for better protection or bearing the weight of the brooding mother or insulating material deposited atop it. On the other hand, it should also be easy enough for the chick to break out. These opposing forces would lead to compromise solutions in the form of deviations from circularity. Mahadevan and colleagues also observed that increased flight performance is often associated with smaller aspect ratios and more asymmetric eggs. Here again, a compromise of sorts might be in play — higher flight performance selects for bigger eggs on one hand and a more streamlined body on the other. Hence, the compromise is achieved by having longer or more asymmetric eggs. A similar effect, albeit unrelated to flight, but body morphology, might have also been at play in the Mesozoic dinosaurs with long eggs. Of course, the shape diversity beyond a simple ellipse suggests that other selective forces beyond the above are also at play.

However, a morphometrician of the bent of D’arcy Thompson would still object that these equations need to be derived ground up from physics — in fact, he voices precisely that problem in his account of bird egg shapes — they need to be derived from an equation which accounts for fluid pressure in a bounded membrane. This was keeping with his wider skepticism towards one of the foundations of biology (natural selection) while emphasizing the other (geometry). Mahadevan and colleagues presented such a derivation a few years back; however, it is not clear if it can actually recapitulate the entire range of ovals seen in real-life dinosaurian eggs.

Further reading:

Avian egg shape: Form, function, and evolution by Stoddard, Yong, Akkaynak, Sheard, Tobias, and Mahadevan

Egg and math: introducing a universal formula for egg shape by Narushin, Romanov, and Griffin

Accurately quantifying the shape of birds’ eggs by Biggins, Thompson, and Birkhead

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Matters of religion: the lesson by the lake

Sharvamanyu was with his preceptors Somakhya on Lootika on the shore of a lake in the midst of the mountains on a full moon night. It was the cremation ground for a lineage of V_4s, who had historically specialized in the arts of preparing medico-recreational substances from Acacias but had risen to the status of warriors during the great Jihād-s of the monstrous Mogol tyrants. Some distance from where they sat, the shore was whitened by the bones of the generations of V_4s who taken the ladder to confront the glowing jaws of the fearsome Sārameyau en route to the realm of Vivasvān’s dark son. Now, those osseous remains gleamed like gold as the bowl of soma climbed over the lake to gladden the bands of Vasu-s, Āditya-s and Rudra-s with a pleasing draught. Mātariśvan, who is also known as Śambhu, drafted a pleasant breeze over the lake making waves lap the banks, even as the music of the sons of Rudra playing on the vāṇa-s sounded in the airy realm. Sharvamanyu sought the mysterious experience of yoga from his preceptors. He had been practicing a mantra for six months but had met with nothing but failure. The ensign of Soma climbing the celestial vault occulted the great mass of stars known as Tiṣyā, marking the day of the great bull-sacrifice of yore to Rudra. That conjunction is indeed a sign of the gods Somārudrā — the Taittirīya-śruti has declared: “rudro vai tiṣyaḥ |”. Somakhya and Lootika had verily received a signal of The god in that regard. It is said that on that day the dānava Maya had built the Tripura-s, and it was on that day that Rudra had destroyed the triple-planets.

They led Sharvamanyu to a small shrine of the awful Vināyaka by the cemetery with an archaic image of the god. An inscription from the Vikrama year 1532 stated that the image had been found in the lake by a Gāṇapatya siddha and installed in the shrine. The siddha was said to have joined the retinue of his chosen god as a phantom upon his death and was believed to manifest occasionally in those regions, especially to V_4 votaries who were uncorrupted by religious degeneration that had passed through the region thereafter. Somakhya and his wife asked Sharvamanyu to worship Vighna with a short stuti and by placing some of the flowers they had collected in course of their trek to the lake at the feet of the image. Sharvamanyu recalled one their teacher Shilpika had taught them:
namas te gajavatrāya namas te gaṇanāyaka |
vināyaka namas te .astu namaste caṇḍavikrama ||
Obeisance to you, the elephant-faced, obeisance to you the lord of the hosts; obeisance be to you the Vināyaka; obeisance to you of fierce valor.

namo .astu te vighnakarte namas te sarpamekhala |
namas te rudravaktrottha-pralamba-jaṭharāśrita ||
Obeisance be to you the creator of obstacles, obeisance to you with a snake-girdle; obeisance to you endowed with a pendant belly, who emerged from Rudra’s face.

Then, they asked Sharvamanyu to do the kara- and aṅga-nyāsa-s appropriately and begin the japa of his mantra 324 times. As he was in the midst of that, Lootika pointed her siddha-kāṣṭha, whose tip bore the carving of the head of Garuḍa, at him. Suddenly, he had a vision of a huddled band of people wearing bone-ornaments singing songs in a strange language. He could catch many words, but it was not entirely intelligible to him — it was neither Sanskrit nor a modern Aprabhraṃśa. At climactic phrases in the lyrics, he felt overpowering emotions seizing him from deep within. From the midst of the band arose the ghost of a V_1 named Kiñjalka. The phantom said: “Having performed great acts of valor, I was beset by a large band of well-armed enemies when my siddhi-s had declined. Hence, I had to do battle as an ordinary man though I was still powered by all the experience I had acquired from my assiduous practice from a young age. Thus, fighting my foes like the sūtaputra in his last fight, I was slain and passed into the retinue of Paśupati Deva.” Then, he saw the apparition of a beautiful woman. As she faded away from his vision, he felt all his strength had ebbed away, and he involuntarily fell to the ground. As he struggled to get up, he saw a dog repeatedly going to the cemetery and waiting for someone who had joined the Pitṛrāṭ. As the apparition of the dog rose up into the sky and merged with the constellation graced by the bright Procyon, he saw a cat descending from a pole and run towards a large bowl with a goldfish. It knocked down the bowl, and the fish jumped out to swim away into the great lake stretching before him clearing the prior apparitions. Then, despite the moon being there in the sky, all went pitch black — he felt he was in a subterranean cave with no lamp. That utter darkness remained for some time during which he lost track of everything except the mantra whose japa he was performing. He then saw his friend Vidrum being dragged away with a lasso around his neck by a buffalo. He wanted to shout out to his companions to aid his friend, but his voice deserted him. Now, to his utter horror, he saw Abhirosha’s corpse being eaten by a voracious dog and crows. Then, he saw himself being dragged the same way but now he felt an utter calm — death no longer seemed to cause him any concern at all and all his emotions were stilled.

Then, he saw a great procession of reptiles and other strange animals, of which he recognized only a few, but the rest still seemed vaguely familiar to him as he had seen Somakhya and Lootika draw them when they were in school. In that utter darkness, he saw a clear vision of the temple of Rudra in the cemetery with the liṅga blazing forth as though self-luminescent — he remembered his teachers telling of the term sadyojyotis — now he realized what that meant. He heard the mantra-s from the Taittirīya-śruti by which the liṅga is installed. Suddenly, the terrifying five-headed god after whom his parents had named him manifested from the liṅga. Each of the brahma-mantra-s from the upaniṣat of the Taittirīyaka-s, revealing the five-fold form of the god known as Mahādeva, sequentially manifested in his mind. The first was that of Sadyojāta with which he saw the face of Kumāra. The next was that of Vāmadeva with which he saw the perfect face of Manonmanī. Then the Aghora mantra manifested, and he saw the dreaded face of the Bhairava. After that came the Tatpuruṣa mantra, and he beheld an aquiline face with a solar orb, glowing like the eagle of the god Savitṛ. Finally, came the Iśāna mantra, and he saw that glorious face known as Rudra-sadāśiva. The apparition of Rudra was more real than any experience he had had. His mind involuntarily remarked over the japa: “He is known as Paśupati; he is Īśāna.”

Thereafter, he could discern the great bow of the god and the missile mentioned in the scriptures as the Pāśupata — the great arrow of Nīlarudra. He then saw the terrifying trident of the god with which he had impaled Andhaka in the vast White Forest; the cakra with which he had severed the head of Jalandhara who had even bound the Vaḍavānala; the Aghora-missile with which he had destroyed the triple-forts; the vajra praised in the śruti; the bhindipāla, the śataghni, the tomara, the prāsa and the terrifying sword with which had dealt death to numerous dānava-s as described in the national epic. He was accompanied by an awful troop of gaṇeśvara-s, the ape-faced Nandin, the skeletal Bhṛṅgin, the many-armed lion-headed Vīrabhadra who had beheaded Dakṣa and flayed the giant daitya Nala, the black Mahākāla, the Kṣetrapāla emitting a meteor that blazed through the firmament and Caṇḍeśa. He also saw Kīrtimukha, of the form of a standalone leonine head, and the Śiva-kṛtyā, who had emerged from the mouth of Rudra to snare the Bhārgava in her vulva, both inspiring a primal fear. He saw a great churn of rapidly moving Pramatha-s holding tridents, axes, and other weapons looking like smaller Rudra-s. Then he saw the Śiva-gaṇa-s with lion-, elephant-, horse-, donkey-, dolphin-, hog-, bovine-, goat- and ram- faces prancing along, brandishing various weapons. He also saw the axe-wielding Gaṇapa surrounded by a great host of ferocious, black elephant-headed Vināyaka-s. For a moment, the realization of the profound gaṇa-vidyā, illuminating the meaning of the multitude of gaṇa-s of Mahādeva flashed in him. He simultaneously saw the import of Rudra in the śruti and the śaiva-śāstra. At that point, he beheld the skull of Prajāpati held by Rudra. At once, he had the realization that he, together with the entire universe, was within that skull gracing the eight-fold god. He spontaneously uttered that Atharvan incantation of The god, manifesting both as himself and his son, believed to have been said by Prajāpati himself as he was decapitated by Rudra:
kāpālin rudra babhro .atha bhava kairāta suvrata |
pāhi viśvaṃ viśālākṣa kumāra varavikrama ||

Thereafter, he spontaneously uttered the devadeveśvara-stuti:
namo viṣama-netrāya namas te tryambakāya ca |
namaḥ sahasra-netrāya namas te śūla-pāṇine |
namaḥ khaṭvāṅga-hastāya namas te daṇḍa-dhāriṇe ||
Obeisance to the odd-eyed one and obeisance to you with the three goddesses. Obeisance to the thousand-eyed, obeisance to you with a trident in hand. Obeisance to the one with the skull-topped brand in hand, obeisance to you bearing the cudgel.

tvaṃ deva hutabhug-jvālā-koṭi-bhānu-samaprabhaḥ |
adarśane vayaṃ deva mūḍha-vijñānato .adhunā ||
You, o god, have the luster like that of the flames of the eater of oblations (Agni) and a crore suns. Before seeing you, o god, we were foolish; now we have been enlightened.

namas trinetrārtiharāya śambho triśūlapāṇe vikṛtāsyarūpa |
samasta-deveśvara śuddhabhāva prasīda rudrācyuta sarvabhāva ||
Obeisance to the three-eyed remover of troubles. O one, who is trident-handed, with a mouth of fierce form, the lord of all the gods, of pure nature, Rudra, the infallible and of all natures, be pleased.

pūṣṇo .asya dantāntaka bhīmarūpa pralamba-bhogīndra-lulunta-kaṇṭha |
viśāla-dehācyuta nīlakaṇṭha prasīda viśveśvara viśvamūrte ||
O destroyer of teeth of Pūṣan, one of terrible form, with the dangling lord of the snakes hanging from your neck, of gigantic body, infallible, the blue-throated one, the lord of the world, whose form is the world, be pleased.

bhagākṣi-saṃsphoṭana dakṣa-karmā gṛhāṇa bhāgaṃ makhataḥ pradhānam |
prasīda deveśvara nīlakaṇṭha prapāhi naḥ sarvaguṇopapanna ||
O one how blew up Bhaga’s eyes, of skilled actions, may you take the foremost offering of the ritual. O lord of the gods, the blue-throated one be pleased. Protect us, o one endowed with all qualities.

sitāṅga-rāgāpratipannamūrte kapāladhārims tripuraghna deva |
prapāhi naḥ sarvabhayeṣu caivaṃ umāpate puṣkara-nāla-janma ||
O god of unattainable form smeared with white ashes, the skull-bearer, the slayer of the Tripura-s, the husband of Umā, and the one born from the lotus-stalk (as Agni or from Prajāpati) protect us from all fears.

paśyāmi te dehagatān sureśa sargādayo vedavarānananta |
sāṅgān savidyān sapada-kramāṃś ca sarvān nilīnāṃs tvayi devadeva ||
O eternal lord of the gods, the root of all lineages of the universe, I see in your body the excellent Veda-s together with the Vedāṅga-s, the various branches of knowledge, and the Pada and Krama recitations of the Veda, everything is inside you, o god of the gods.

bhava śarva mahādeva pinākin rudra te hara |
natāḥ sma sarve viśveśa trāhi naḥ parameśvara ||
O Bhava, Śarva, Mahādeva, the yielder of the Pināka, Rudra, Hara. We all bow to you, o lord of the universe. Protect us, o foremost lord.

Thereafter, Sharvamanyu saw in place of Rudra, his consort, the great goddess Kālarātrī, surrounded by diverse yoginī-s filling the whole field of view up to the horizon with an eight-fold symmetry. At that sight, he spontaneously uttered the Kalarātrī-stuti:
jayasva devi cāmuṇḍe jaya bhūtāpahāriṇi |
jaya sarvagate devi kālarātre namo .astu te ||
Victory be to you the goddess Cāmuṇḍā; victory to you who snatches away all beings. Victory to the omnipresent goddess; obeisance to you, o Kālarātri

viśvamūrte śubhe śuddhe virūpākṣi trilocane |
bhīmarūpe śive vedye mahāmāye mahodaye ||
O world-formed one, the auspicious one, the pure one, the odd-eyed one, the three-eyed one, one of terrible form, benign one, one of the form of knowledge, one of great illusory powers, one of great preeminence.

manojave jaye jṛmbhe bhīmākṣi kṣubhitakṣaye |
mahāmāri vicitrāṅge jaya nṛtyapriye śubhe ||
Victory to you, o one of the speed of mind, the victorious one, the great blossoming, the terrible-eyed one, suppressoress of all agitation, the bringer of great disease, of marvelous body, lover of dance and the auspicious one.

vikarāli mahākāli kālike pāpahāriṇī |
pāśahaste daṇḍahaste bhīmarūpe bhayānake ||
O fierce one, the great goddess of time, time, the remover of sins, wielder of a noose, with a rod in your hand, of terrible form, the fear-inspiring one.

cāmuṇḍe jvalamānāsye tīkṣṇadaṃṣṭre mahābale |
śata-yāna-sthite devi pretāsanagate śive ||
O Cāmuṇḍā, with a flaming mouth, with sharp fangs, you of great might, the goddess riding a hundred vehicles, seated on a corpse, the benign one.

bhīmākṣi bhīṣaṇe devi sarvabhūtabhayaṅkari |
karāle vikarāle ca mahākāle karālini |
kālī karālī vikrāntā kālarātri namo .astu te ||
Obeisance be to you, o terrible-eyed one, frightful goddess, striking terror in all beings, terrifying one, formidable one, the great time goddess and the terrifying one. She is time, the terrible one, striding boldly and the night of dissolution.

Sharvamanyu concluded his japa and rose, wanting to ask his companions about his visions, but they pressed their fingers to their lips, directing him to be quiet and continue the japa as they climbed one of the proximal hills on the rim of the lake. As they made their way up, they paused to take yet another look at the great occultation of Tiṣya nearing its conclusion. Reaching the top of the prominence, they arrived at the little shrine of Śiva. On the wall of the shrine was the relief of a siddhayoṣit Stṛkā performing liṅgārcana. Somakhya and Lootika offered some flowers there and performed a tarpaṇa to the feet of the said siddhayoṣit. They asked Sharvamanyu to do the same and remarked: “This siddhayoṣit, Stṛkā, impelled by Mahādeva arrived from the continent of Śvetadvīpa to become a student of the charismatic siddhayoṣit Indramaṇidevī of the śūdra-varṇa at Śūrpāraka. There having studied the Śambhu-para scriptures and observing the vrata of chastity, she acquired siddhi-s in various Māheśvara-mantra-s. Then, she performed mantra-sādhanā at the cemeteries of Kollagiri, Kilakilārava, where the yadu hero had slain the giant ape Dvividha, and Avimukta, where Kālabhairava had imparted the kāpālika rite to Kubera. She then settled here under the patronage of the lord Jayakeśin. Due to her siddhi-s in the Netra, Koṭarākṣa, Vyādibhakṣa and Aghoreśvarī vidyā-s, she lived a long life free of trouble teaching the Māheśvara-śāstra-s and experiencing the glories of Nīlarudra.”

Then seating themselves on the platform around the shrine Somakhya and Lootika bade Sharvamanyu to ask them questions that he might have regarding his sādhanā. Sh: “The experience I had finally opened me to the possibility of what the lakṣya of the sādhāna might be. I had struggled for months failing with the dhyāna and, after that, with the avadhāna. But the sudden manifestation of a clear vision of the deva and his parivāra has cleared that issue in one stroke.”

L: “Indeed, succeeding at dhyāna is an impediment for many. A small number of people are endowed with sahaja-dhyāna capacity — the same is true for the other steps in sādhanā. Such folks might wonder why others should even raise the matter. Let me be clear, devatā-dhyāna does not come automatically to the majority. However, it can be achieved by multiple means: (i) through the repeated study of mantra-siddhānta-s that give accounts of the devatā-s; (ii) through purāṇa readings; (iii) through study of properly prepared texts known as devatā-citra-saṃgraha-s; (iv) by visits to temples or attending temple festivals or cala-pujā-s to behold the icons of the deities; (v) by an experience induced by teachers or other interested mantravādin-s. We attempted that last option with you, and believe it should help you going forward.”

Sh: “Still, I wonder if ī might succeed at the avadhāna and if the siddhi might manifest at all? Pray tell me what are the rahasya-s of that?”
S: “Abhyāsa is the first and foremost step. As with physical exercise, for most people, the ability to perform something at a certain level does not come as sahaja. They have to use their will often with enormous force to get themselves to practice. At first, the going is hard, the fruits are limited, and there could even be negatives like pain. One has to calibrate the right level the body can take and slowly step up the gradient — you know this well in physical practice. Eventually, one starts seeing the benefits of it and liking the action. Unfortunately, the path to even a modicum of siddhi is littered with sādhaka-s who failed at this step. That is the reason many of our traditions emphasize physical yoga with āsana, prāṇāyāma, and the like. It gives you a tangible object of control — your own body. By observing how you succeeded in that control, you will develop a model that can be imitated at the mental level to acquire the control needed for avadhāna. Lootika, do you want to add something?”

L: “Regarding siddhi, I would remark that svayaṃsiddha-s are rare. The siddhayoṣit, at whose shrine we are now seated, is a case in point. However, she became one due to two points that are often overlooked. First, she had sahaja capacity for abhyāsa — without that, she could not have become a svayaṃsiddha. Second, she was an example of a daivarakṣitā — otherwise, how could she have come safely from Śvetadvīpa to Jambudvīpa — she could have been captured and borne away by any number of men in the process. Then, she managed to perform kṣetrāṭana, reaching difficult śmaśāna-s to perform sādhanā. Hence, one has to be cognizant of one’s condition. The probability of being daivrakṣita is low; one cannot make that a default assumption; likewise with sahajābhāyasa-śakti. Hence, one has to gauge one’s capacity while not faltering at the abhyāsa and set the sādhya accordingly.”

Sh: “Surely the apparition of the phantom of Kiñjalka and the dog that kept visiting the cemetery have a lesson here?”
L: “The case of said V_1 gives an important lesson. A human is a finite being with a relatively small window of opportunity. A person usually begins life with no special endowments. Even a sahaja soul, while endowed with great potential, cannot make that manifest entirely without appropriate abhyāsa. Then he reaches a high point upon honing his sahaja capacity by abhyāsa at some point in his youth. After that, he plateaus. This is because he comes under two opposing forces, one of physical decline with age and the other of the growing wisdom and experience fueled by his constant sādhana. At some point, his physical decline overshoots his accruing wisdom and ability; thus, his downward slide begins. In some sādhaka-s, the fame of their sādhana attracts pupils from the ranks of others seeking glory. They seek para-saṃyogāt mahattvam, which in turn feeds into the sādhaka, for he is now possessed of an army of pupils. As we have told you before, vidyā is enhanced by the resonance with good antevāsin-s. Thus, he may rightly (due to wise counseling of his army) or wrongly (living off the deeds of his students) be buoyed up much longer than his true personal capacity. In the case of Kiñjalka, he was unable to get a troop of any size and had to live off his own capacity. Hence, when confronted by a bhrātṛvya with a larger force, he was killed with his own capacity limited by physical decline. Thus, the lesson is one may experience a plateau in the sādhanā, after which the decline and death follow. Thus, many sādhaka-s have come and gone, and nobody remembers them. Other things can happen. Either when human decline sets in, or when a sādhaka has had an experience like you had today, or when he witnessed the siddhi of his guru from close quarters, he might think that the siddhi is close by and will repeatedly try to grasp it. However, it would be as futile as the dog hoping day after day that its master who has taken the southern path would return from the cemetery.”

Sh: “Somakhya, I note that you are in silent contemplation with almost a tinge of disapproval. Have I done something wrong?”
S: “You know me from so long back that you picked something just from my face that you felt was disapproval. It is not — it is just that I was mentally wanting to head off certain questions that I felt would ensue from you. As for the rest of the apparitions in your experience, spend some days contemplating them in silence, and you will get your answers. You may ask other questions, though.”
Sh: “I now realize that the path can be the climb up the slopes of Kailāsa — some go up to glimpse the god and return to the world of men, but others make an exit midway to join the retinue of the god as a phantom! But in everyday life, man continues with his toils if he sees at least a few positive results else, he gives up. How does one keep toiling in face of repeated failure, when one might be reaching for the unattainable like the dog at the cemetery? Moreover, the tale of the wicks of Bhairavānanda, which Lootika had told us in school, comes to mind — would I end up with wheel on my head for my persistence?”

S: “That is indeed the hardest part. You may have the great svayaṃsiddhi but not be daivarakṣita. Thus, your attainments could be modest. However, due to your svayaṃsiddhi you might have perfect jñāna of what it would be like to be mahāsiddha but still not be one in reality. If you have that jñāna, you may keep trying, though, in the end, your fate would be no different from that of the dog. It might be frustrating, but at least you would have only failed from being a daivahataka. But then take the case of our former classmate Hemaliṅga; he definitely had enormous mathematical capacity and smashed his way to a certain level. But he sought to reach the levels of the great gaṇitajña-s of all times that lay out of his reach. Attempting hard, like trying to leap onto a high ledge, he only hit the cliff-face and dropped down. However, that was not fatal; Vrishcika tells us that his abhyāsa at least brought him comfort in life. Thus, the great failed attempt at mahāsiddhi often builds character and gives you a minor siddhi that might bring pleasure in life. However, we must warn you Sharva, that path to the highest siddhi-s is fraught with much danger even as the fourth wick of Bhairavānanda. You might see warnings as you are headed that way, which you must differentiate from benign failures. Indeed, Lootika and I have seen the ghost of more than one V_1, who persisted through those warnings and fell in such attempts, like the corpses of forgotten climbers on slopes of the Himavant. On the other side, if your sādhya has come in clear vision, keep toiling and change upāya-s upon repeatedly failing. Like in biology, so also here; remember, once one has entered the stream, one sinks when the toil ceases. So you have to do so just to stay afloat. The good teacher should tell those who cannot keep up to exit early.”
Lootika: “Now, let us resume our trek back to reach in time to catch the train back to the city.”

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Twin Āditya-s, twin Rudra-s

This note originated as an intended appendix to the article on Rudra and the Aśvin-s we published earlier. The first offshoot from that work, which we published separately, explored the links between Rudra, Viṣṇu and the Aśvin-s in the śrauta ritual. We finally found the time to fully write down the intended appendix and present it as a separate note. To rehash, we noted an intimate connection between the primary Rudra-class deity (typically in his manifestation as the great heavenly Asura, the father of the worlds) and the twin deities (the Aśvin-class) of the ancestral Indo-European religion. This is preserved in multiple descendants of our ancestral religion, such as in the śruti, the para-Vedic material in the aitihāsika-paurāṇika corpus, in the Roman religion relating to Castor and Pollux, and probably the non-Zoroastrian strains of the Iranian religion. It was definitely there in at least some branches of the Germanic religion, but its destruction by the West Asian mental disease has only left us with the euhemeristic figment of Horsa and Hengist as the descendants of Woden. Likewise, we hear explicitly of the destruction by the Christians of the temple dedicated to the Western Slavic deity Rugiaevit and his twin sons Porevit and Porenut. It is pretty likely that Rugiaevit’s name is derived from the same root ru- as that in the name of Rudra, and these twin sons are the equivalents of the Aśvin-s.

In the śruti, this old motif manifests as the Aśvin-s being the sons of Rudra who follow on his track as he rides his heavenly chariot. As the physicians of the gods, they inherit the medical and pharmacological virtuosity of their father. In a parallel Vaidika tradition, which entered the śruti fold from a group of Aryans distinct from the ṛṣi-s who composed the RV, the twin sons of Rudra are Bhava and Śarva, who accompany their father, like the Dioskouroi of the Greco-Roman worlds. Their worship is prominent in the Atharvaveda and some texts preserved in later Vedic collections; however, in the ādhvaryava tradition, they were absorbed as names of Rudra or those of the multitude of Rudra-s. In the para-Vedic material preserved in the itihāsa-s and purāṇa-s we see them as the twin ectype of Skanda, i.e., as Skanda-Viśākhau, the sons of Rudra.

With this background, we shall consider the enigmatic sūkta of Urucakri Ātreya (RV 5.70), which on the surface is a simple 4-ṛk one in the Gāyatrī meter. The anukramaṇi specifies its deities at the twin Āditya-s, Mitra and Varuṇa. Indeed, the first ṛk of the sūkta is directed to these gods and it is embedded amid the long series of sūkta-s to Mitra and Varuṇa by different Ātreya-s:
purūruṇā cid dhy asty avo nūnaṃ vāṃ varuṇa । mitra vaṃsi vāṃ sumatim ॥
Indeed, now, in full breadth is the aid from you two, O Varuṇa! I have gained the benevolence of you two, O Mitra!

After beginning with an acknowledgment of the help gained from Mitra and Varuṇa, the next ṛk suddenly changes the focal deities:

tā vāṃ samyag adruhvāṇeṣam aśyāma dhāyase । vayaṃ te rudrā syāma ॥
O you two, may we attain you two together, in your benign state (literally: without intention to harm) for our stability. May we be so, o you two Rudra-s.

pātaṃ no rudrā pāyubhir uta trāyethāṃ sutrātrā । turyāma dasyūn tanūbhiḥ ॥
Protect us, two Rudra-s, by your defenses; also save us, since you two are good rescuers. May we overpower the dasyu-s with our bodies.

Notably, the deities remain dual in the above two ṛk-s, but they are explicitly identified as twin Rudra-s. While some students of the Veda have taken this use of Rudra to be merely an appellation transferred to the deities of the first ṛk, there is no support for that. Mitra and Varuṇa are unanimously categorized in the Āditya class, as its leading exemplars, and never placed in the Rudra class. Hence, we have to understand the twin Rudra-s of the above two ṛk-s differently. First, their raudra nature is explicitly indicated in the entreaty to be benign (adruhvāṇeṣam). Second, they are described as sutrātrā, good rescuers, which immediately brings to mind the Aśvin-s who are frequently invoked in such a capacity. In the RV, Rudra in singular denotes the god, in his unitary form, and as the father of his class. Rudra-s in the plural refer to the entire class or the Marut-s. The dual form of Rudra applies only to the Aśvin-s everywhere else in the RV. In particular, the Atri-s repeatedly refer to them as such: e.g., RV 5.73.8, 5.75.3 and in RV 5.41.3 they are invoked together with Rudra as Asuro Divaḥ. Thus, we posit that in RV 5.70.2-3, Urucakri Ātreya implies the Aśvin-s by the dual form of Rudra and not the twin Āditya-s.

The last ṛk of the sūkta goes thus:
mā kasyādbhutakratū yakṣam bhujemā tanūbhiḥ । mā śeṣasā mā tanasā ॥
May you two of wondrous deeds not make us experience some phantom with our bodies. Neither with the rest [of our people] nor with our descendants.

The word yakṣa (neuter) could be taken to mean a ghostly apparition or phantom — perhaps one which causes a disease — a yakṣma. The imploration is to avoid the possession of the ritualist’s own body or that of this people or descendants by such a phantom. On the one hand, this is rather reminiscent of the supplications to Rudra for similar protection, often with the negative particle mā. On the other, it is reminiscent of the supplication to Varuṇa to be relieved from his heḷas (=“fury”; also, a feature of Rudra) for the sins that he unerringly notices. For instance, we have in the śruti the imploration of Śunaḥśepa:

ava te heḷo varuṇa namobhir
ava yajñebhir īmahe havirbhiḥ ।
kṣayann asmabhyam asura pracetā
rājann enāṃsi śiśrathaḥ kṛtāni ॥ RV 1.24.14
We avert your fury (heḷas) O Varuṇa with obeisances,
we implore to avert it by rituals and oblations;
ruling, for us, O all-seeing Asura,
you will give release from the sins that were done.

Thus, both Varuṇa and Rudra share not only the heḷas but are also known as Asura-s (the latter being emphasized for the cognate of Varuṇa in the Iranic branch of the religion, and remembered for Odin (see below) in the Northern Germanic religion). Thus, even though Varuṇa or Mitra are never called Rudra-s, they have a certain overlap of category, particularly in the actions of Varuṇa, and perhaps, to a degree, in the Iranic world in the cognate Mitra. We believe that in the sūkta under consideration, Urucakri Ātreya plays on this overlap in the final ṛk by not naming any deity but simply using the dual epithet adbhutakratū. Thus, we suggest that he is purposefully ambivalent to cover both sets of twin deities referred to in the sūkta — the Āditya dyad or the twin Rudra-s, i.e., the Aśvin-s. The epithet adbhutakratū will transparently apply to the Aśvin-s as they are frequently described as wonderworkers in the śruti. The ṛk could also apply to Mitra and Varuṇa in the sense of supplication to avoid their heḷas. The invocation of the heḷas of Mitra, Varuṇa, and the Marut-s, representing the intersection of their respective functional categories can be seen in RV 1.94.12 composed by Kutsa Āṅgirasa:

ayam mitrasya varuṇasya dhāyase
‘vayātām marutāṃ heḷo adbhutaḥ ।
mṛḷā su no bhūtv eṣām manaḥ punar
agne sakhye mā riṣāmā vayaṃ tava ॥
This one is to be fed ghee [literally suckled],
as the wondrous pacifier of the fury of Mitra and Varuṇa, and of the Maruts.
Have mercy on us! May the mind of these (the above deva-s) be good again
O Agni, in your friendship, may we not be harmed.

This overlap in category has confused some Indo-Europeanists of the Dumezilianist strain. They have split hairs and gone into contortions about whether the Germanic Odin represents a cognate of Varuṇa or Rudra. This ancient functional intersection has meant that one or the other class of deities could have served as a locus for absorption of traits of the other.

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Self, non-self and segregation: a very basic look at agent-based lattice models

In our college days, a part time physics teacher from an old and respected V_1 clan used to chat with us about issues of mutual interest that were beyond that of the rest of the class (or for that matter the rest of the teachers) and well out of the scope of the syllabus. He was the only one among the physics staff with an interest in science for science’s sake. We always felt he had it in him to be a scientist and he was indeed was pursuing a doctoral program at his own pace on the side. However, he clarified to us that he was the big fish in the small pond and that every man’s ambition is like a rocket set off on a Dīpāvalī night — drawing out a parabola on the board he declared with his characteristic smirk: “It will come down; hence, why trouble yourself with a dizzying fall”. In course of one involved conversation spanning thermodynamics, dynamical systems and biological ensembles, he declared to us: “I agree with you that there are several problems where the actual entities are fungible. It doesn’t matter if we are dealing with atoms, cells or animals, they could as well just be numbers. You should explore the Ising models — maybe you will find something there to answer your questions.” We are not really into “proving a few theorems”; however, playing with things on paper or a computer has always excited us. Hence, the next time we could access a computer we began looking into those models and soon realized that it could be used to understand some basic aspects of biological systems.

Here we shall describe some experiments with such models that go no further than the most basic exploration of these systems. In physics, such models were first proposed by Lenz and his student Ising. In sociology, they were introduced by Schelling (of whose work we learnt much later) who carried out the experiments with a graph paper and coins. Today we can do them easily by writing some code on a little computer. The basic rules for the games we shall look at go thus:

1) The games are played on a lattice on the surface of a torus but for visualization we shall cut open the torus and render it as a square board. Thus it would look like this:


2) Each cell of the lattice can be occupied by an “agent” of one of two colors (as above) or be empty.
3) The system has strict conservation laws: the agents can neither be created nor destroyed and the number of agents of each color will be conserved.
4) The neighborhood of an agent is determined by the “span” l which defines a square grid of a centered on it with (2l+1)^2-1 neighbors. l=1 means a 3 \times 3 grid with the agent at the center and 8 lattice positions available for neighbors around it. Thus, the agent marked with a black dot (above) has 5 neighbors at l=1: 2 blue and 3 yellow. l=2 means a 5 \times 5 grid with the agent in the center and 24 available lattice positions for neighbors in two concentric shells around it.
5) The agency of the agent manifests as its ability to read the number and color of its neighbors and either stay put where it is or move over to a random empty cell in the lattice.
6) Beyond this is there is occupancy, o, i.e. the fraction of the total available cells in the lattice that are occupied by agents.

All the experiments described below are played on a 50 \times 50 lattice, i.e. there are 2500 cells available to the agents. In all experiments, the agents move to an available empty cell if the number of its neighbors of any color are \le s, the sociality factor. Thus, if s=0 then the agent will move from their current location if they have no neighbors at all. In the first set of experiments, they additionally sense the the absolute number of non-self neighbors, i.e. those of a different color and move if it is \ge c_n, the non-self count. The movements of the agents are repeated over and over until stability is reached or 30 successive rounds of movement have elapsed. The games are illustrated thus: the plot to the left is the initial configuration where the agents are randomly introduced into the lattice and the plot to the right is the final configuration that is reached as mentioned above. We measure segregation by looking at the \tfrac{n}{s}, i.e. the mean non-self: self ratio in the neighborhood of the agents. As the agents are randomly introduced, the game would start with \tfrac{n}{s} \approx 1. The degree of segregation can be statistically assessed at the beginning and end of the run by means of the t-test to see if the mean number of self and non-self agents in the neighborhood of any given agent are significantly different. At the start of the run the difference would be insignificant.

Game 1 is a run with low occupancy o=0.4 and an equal number of agents of the two colors (blue and orange); l=1, i.e. a neighborhood with 8 available cells; s=0; c_n=5, i.e. the agents tolerate up to half of the 8 available lattice points in the neighborhoods being occupied by non-self agents.

sch_Fig01Game 1

We see that at the end of the run \tfrac{n}{s} remains close to 1 and the mean number of self and non-self neighbors of an agent is not significantly different suggesting that tolerating non-self agents in up to half of the available neighborhoods does not result in segregation at low occupancy and low sociality.

Game 2 is run with the same parameters as above, except that we increase sociality s=3.

Game 2

Notably, the increased sociality results in highly significant segregation. It also results in greater clumping of the agents, resulting in clustered but clearly segregated domains.

Game 3 is run with the same parameters as Game 1 but we increase the occupancy o=0.6.

sch_Fig03Game 3

Here, we see a small but significant reduction of \tfrac{n}{s}. Thus, increasing the population with same level of tolerance for non-self by itself results in some segregation that is not seen at low occupancy.

Game 4 is run with l=2; thus, the agent responds to the status of the 24 available lattice points of the neighborhood around it. The occupancy is low o=.4; c_n= 11 i.e. less than half the available neighborhood positions tolerated as non-self; s=6.


Game 4

In these runs we often seen no significant segregation of the agents despite the relatively low tolerance to non-self; however, we see greater clumping of the agents resulting in a more anisotropic distribution of the agents at the conclusion of the run. Thus, when larger neighborhoods are sensed by the agents, even relatively low tolerance for non-self is overridden under low occupancy leading to paradoxical clumping of self and non-self into spatially restricted domains.

Game 5 is similar to 4 but the sociality of the agents is increased to s=8.

Game 5

This change has the dramatic effect of moving the agents towards strong segregation along with formation closely packed monotypic domains of the two agents, with clear boundaries. Thus, the sociality parameter drives a phase transition from packing with little segregation despite relatively strong non-self tolerance to nearly complete separation into domains with shared borders.

Game 6 is different from the previous ones in that it senses the relative non-self fraction rather than the absolute count of non-self in the neighborhood, f_n=\tfrac{c_n}{c_t}, where c_t is the total number of occupied lattice points in the neighborhood. The sociality factor s is applied just as in the above cases. In this run, we set s=0, i.e. the agent moves only if it has no neighbors. f_n=.7, i.e., the agent move only if the fraction of its non-self neighbors is greater or equal to f_n. Thus, if an agent has 5 neighbors and 4 of them are non-self then f_n=0.8 and it would move. We set it to a low occupancy of o=0.4.


Game 6

We observe that even with a low occupancy, high tolerance for non-self and low sociality, relative sensing drives significant segregation. Thus, with relative sensing the f_n is the primary determinant of segregation.

What might be some real-life scenarios where these games matter? We can easily imagine the system of agents being a set of living cells showing their agency in response to chemotactic signals. At a macroscopic level, we can imagine these as individual animals (e.g. humans). Indeed, Game 6 was the one played by Schelling as a model for the sociology of segregation. In terms of cells, we can conceive of the following mechanisms: 1) cells are primitively motile; hence, they can move. 2) they can exhibit sociality by means of adhesion molecules (usually proteins) that are expressed on the cell surface. Unless these adhesion interactions are satisfied to a given degree, they would keep moving. 3) They can sense self from non-self. In the simplest case this can again happen via adhesion molecules. Indeed, such mechanisms are used by fungi and ciliates, among others, for discriminating self from non-self for mating or hyphal anastomosis. This kind of adhesion-based self-non-self discrimination can be a mechanism of sensing the absolute number of non-self neighbors. Alternatively, it can happen through sensing of diffusible signals. This mechanism is best suited for relative sensing of the ratio of non-self to total neighbors. Either way the cellular agent could respond to nonself agents in the environment.

Thus, these simple models show how the interplay of sociality and non-self tolerance can result in segregation or paradoxical grouping. In the case of relative sensing (game 6), even with high tolerance and low sociality we see segregation. This is the reality of human societies that many modern occidentally oriented social observers find very hard to swallow. But they are simply tilting against a mathematical reality, much like their medieval representative, old Don Quixote, charging a windmill. Similarly, absolute-count-sensing shows the role of sociality and crowding in segregation (e.g. Games 2, 3 and 5). Greater crowding and sociality can lead to greater segregation when compared to the same, relatively high, tolerance for non-self under low crowding and sociality. Similarly, certain middling level of sociality with sensing of larger neighborhoods can trigger long-lasting clustering without strong segregation despite lower than half tolerance for absolute number non-self agents in the available neighborhood spots (Game 4). This scenario could explain the grouping of bacteria in mixed species biofilms or relatively long-lived clustering of distinct groups in a social setting, e.g. the jāti-s in an Indian village. In conclusion, some social phenomena can be accounted for by simple lattice models of agents which are entirely agnostic to the actual mechanism of agency and sensing.

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Bhairavānanda’s wicks: a retelling

It was early in the school year and the last class for the week, the English class. The students were restless as the sea at the time of the tide from the pull of the impending weekend. For the first time in his life, school seemed to hold something of interest to Somakhya as he had just made acquaintance with the clever girl of the Aṅgiras-es. However, at that point he was lost in working out the geometry of a new chaotic map he had discovered. Vidrum, who sat beside him, peeked into his work, trying to follow it, but soon lost track of the scribbles of his algebra. Just then the English teacher entered the class and surveyed the mood in the room, twirling his chastising rod. The more rowdy students were in a raucous mood. Mahish and Gardabh were singing a cyclical song using their desks, lunch boxes and water bottles as percussion surfaces. The song ran thus in a local apabhraṃsha:
There was a man.
The man had a wife.
The wife birthed a boy.
The boy became a man
The man had a wife.

Episodically, Mudgar decorated the song with single-word interjections comprised of salacious apabhraṃsha words. The English teacher swung his cane and charged towards them: “If I hear anymore of this I’ll have you all stand outside the class.” Sensing the students’ mood, as he did once or twice a month, he decided to provide some exit for the steam building in his restless class: “Today we shall not have any lesson. Instead, I will let two of you all tell the rest of the class a story. You must hear the story quietly without interrupting and I will give you’ll five minutes at the end of each story to ask questions and discuss it. I’ll be choosing one boy and one girl to tell the story; however, be warned, I will not tolerate any lewd story and shall punish you if you narrate one. Today, I’m selecting Sharvamanyu as the boy who shall tell us the first story. Come to the front of the class.”

Sharvamanyu did so and narrated a tale of a Marāṭhā or a Piṇḍārī evading a wild chase from the English army and finally ending up eaten by a mugger crocodile. After a lively discussion for five minutes the teacher said: “As most of you’ll would have noticed, we have a new girl in the class this year, Lootika. She did extraordinarily well in the surprise entrance test we administered to verify her candidacy for admission to our school. As you know, our institution strives for high standards and hopes to send students to the best colleges by ranking in the highest decile in the school-leaving certificate. She shows promise to be one of our good students in final reckoning three years from now. Hence, I select her as the girl who will tell the next story.”

As Lootika came to the front of the class, Mudgar whistled loudly, even as Mahish made an obscene gesture and shouted, “Four eyes!”, alluding to to Lootika’s spectacles. Lootika retorted: “Hey buffalo dung-skull!”. The English teacher struck Mudgar with his cudgel, even as Mudgar himself might have played a square cut on the field of cricket, and swinging it menacingly he and went up to Mahish: “Lootika, stop counter-swearing and tell your story. I’m the one supposed to maintain discipline here. Mahish, I’d already warned you. Now, you shall stand outside class, next to the door. The next time you do something like this, my stick will mightily kiss you.”

With Mahish evicted from the class, Lootika told them the following story:
Friends, I will tell you a story that is believed by some to have been composed by a certain Viṣṇuśarman. However, it is possible that it was a folktale that was later inserted into the older work of Viṣṇuśarman. I am going to follow the plot of the original closely but tell it in my own words. In a certain town in central India lived four brāhmaṇa youths, who were great friends. They were very poor and, as is usual, the richer folk shunned them. They felt that nobody appreciated them for their good qualities and no woman showed any interest in them due their poverty. Tired of their destitution, they decided to seek a fortune elsewhere. They declared that it was better to live in the jungle wearing bark-fiber vestures than to be among their kin while being indigent. Wandering a long way, they eventually reached the river Śiprā in the environs of Avanti and took a dip in the tīrtha. Having ritually purified themselves, they set out to the great temple of Mahākāla and worshiped the god sincerely. As they were making their way out, they ran into the great Śaiva yogin Bhairavānanda. He was a profound master of the Śaiva lore. Having toiled through the voluminous tantra-s of the tradition of Īśāna, Tumburu and the four sisters, Garuḍa and the Bhūta-s, he attained the pinnacle of his mastery through the study of the Bhairava-tantra-s. Thus, he became a renowned teacher with a school at Avanti.

Having fallen at his feet, the four youths accompanied Bhairavānanda to the hall of his school: “Young men what do you seek?”. The four: “Sir, we are very poor. We have decided that we shall either become lords of wealth or die in the attempt. We have heard of your great renown in the magical arts, the nidhividyā by which a man finds subterranean wealth or the kauberīvidyā by which one can locate the great stores of wealth that the lord of the yakṣa-s has provisioned at secluded spots on this earth. We are exceedingly brave and committed, and are willing to perform bold or arduous acts to succeed in this regard. Please inform us of the right way — whether we should seek deep cavernous mines, or perform Bhairava-sādhana in the cremation grounds, or worship the goddess Śākini, or sell human meat to attain our goal.” Bhairavānanda took pity on the youths and brought out four cotton wicks from his sacristy: “Young men, proceed north to the Himālayas towards the source of the Sindhu river along the the route I shall lay out. These wicks will act like dowsing rods. As you climb the slopes of the great mountain, if the wick accidentally drops from your hand dig at that spot. You will definitely come upon some great source of wealth. Collect it and return home.” Having consented, the four youths hurried along on their Himālayan expedition, pregnant with excitement. They made a pact that they would share whatever wealth they might obtain upon the dropping of the wick. After a few days of wandering in the regions where the Sindhu river descends from the great mountain glacier, the wick of one of the men dropped. They quickly excavated the spot and soon hit a lode of copper. The man whose wick dropped said: “Let us take this wealth and return. We can make quite a gain from selling this copper and establish ourselves.” The three other friends shot back: “You are welcome to return with all the copper you want. We have three more wicks and hope to discover even greater stuff than this red metal.”

Thus, the three proceeded; in a while one of their wicks dropped. This time their excavation yielded a lode of silver. The youth said to his friends let us take this silver and return — we have done much better than a mere mass of copper that our poor friend settled for. The two others retorted: “We hit copper first and now silver. Do we need to tell you what lies ahead?” The man who found the silver said: “I’m content with this silver and will take it and return.” The two others pressed on and neared a dizzying cataract on the Sindhu. There the third wick dropped. They dug up that spot and caught the gleam of huge nuggets of gold. The lad whose wick had fallen said that they their quest had truly ended as there was nothing more to find and urged that they split the gold between themselves and go back home. His companion thundered. You are a fool. We still have a wick left. Why should we waste this blessing, which has come to us from the great Mahādeva himself via the mantravādin Bhairavānanda. Each time around we have greater success. You may return with your gold if you choose, I am going to press on alone — I believe I’ll find diamonds and sapphires, which will yield us more wealth than gold. The youth who found gold said that he would wait for his friend to return at that spot and they could go back home together.

The fourth lad pressed on but soon realized that he had lost his way as saw none of the landmarks which Bhairavānanda had indicated to them. He found himself going in circles and ran out of water. To break the cycle and find some water he just marched ahead on a straight line. Soon he came upon a ghastly sight. A man stood in the midst of a cold plateau with his body bespattered with gore. On his head spun a wheel on a peg that seemed to trepan his skull and blood freely spurted out of that perforation. The fourth youth asked this unfortunate man: “What the hell are you doing to yourself? Is there someplace I can find water?” As soon as he spoke that wheel flew away from the man’s head and dropped on the head of the fourth lad and began drilling into it. He yelled: “Ouch, what is this!”. The man who had just been relieved of the wheel said to him: “I thank you from the bottom of my heart for relieving me. I don’t know how long I have been waiting for your arrival.” The fourth lad: “I don’t understand, please get this wheel of my head”. The original wheel-bearer: “Young man, like you, I too had sought wealth from the dropping of the final wick. Thus, did I reach this spot and had the wheel jump on to my head from another unfortunate soul from Tibet who stood here bearing wheel before me. When another man with a wick comes this way, you too will be eventually relieved.” The fourth lad:“How long have you been here?” The former wheel-bearer: “It seems like eons. Who is the emperor of India now?” The fourth lad: “When we left on our expedition, queen Mṛgāvati had just handed over the throne to her son Udayana Vatsarāja, who mightily lords over the land from Kauśāmbī.” The original wheel-bearer: “Oh, I’ve never heard of him. When I left home, Rāmacandra Aikṣvākava held sway over the land from the city of Ayodhyā, like an Indra on earth.” The fourth youth shrieked in pain: “What! that is terrible — it must be more than a 1000 years. How is that even possible!”. The old wheel-bearer: “Well, this is the punishment that has been set up by the great yakṣa Naravāhana. Indeed, it has been said that the man who seeks to plunder that which belongs to the great deity of the North is penalized thus. As you would have noticed, your thirst should have left you — deathless, hungerless, thirstless but full of unrelenting pain, like me, you too shall suffer this wheel of Kubera till someone else comes by with a wick. Thank you and I shall now take leave.”

Having waited long for his friend the third youth followed his footsteps and eventually found him suffering the torture of the wheel. Having heard his tale the third youth said: “Friend, I told you that we had met our ultimate aim in gold. But lacking sense and fueled by false extrapolations you went ahead. Now I’ll have to leave you and return.” The fourth lad: “You are such a heartless fellow to betray your friend in suffering and return to a life a wealth. You can only be called a traitor who will go to naraka.” The third lad: “That would be so if I were in situation where I could have helped you. No human can ever free you from this punishment which has been inflicted by none other than the great Vaiśravaṇa. Seeing the pain on your face from the boring of your skull, my instincts tell me that I should leave this place right away — I sense the deepest dangers lurking in this spot from the dreadful agents of Kāmeśvara. Goodbye.”

As Lootika finished her tale, some of her classmates, Tumul, Vakraas, Skambhakay and Muhira started loudly vocalizing: “boo…boo… what a boring story! Lootika is a bore!” The English teacher intervened: “Stop it or I’ll whack you. Lootika, I want you to ignore them and quietly go back to your desk. It was an excellent story, capped with a didactic flourish, which introduced the class to trivia of history, geography and quaint superstitions reminding one of the notorious Rasgol-bābājī. Lootika also introduced few words and phrases that some of you’ll might not be familiar with. Note those, check their meanings in the dictionary, and write down sentences with each of them. Now, Tumul, stand up and tell the class what you thought was the message of the story?” Tumul: “Sir, it is was just a long-winded way of saying a simple sentence: it is bad to be greedy. Other than that, the story does not make much sense, and as you said, master, it talks about all kinds of superstitions that cannot be true — Lootika doesn’t seem to know even basic physics — a wheel cannot spin by itself forever. it seems she just wants to show off that she knows some words from the dead language Sanskrit in an English class. Moreover, how much copper ore can a single man take back home from high in the mountains. Surely, it will not be enough to even pay for his food and boarding on the way back. As a Brahmin girl, Lootika is simply following the example of how her kind has operated since time immemorial — obfuscate simple, common sense things with elaborate stories packed with mumbo-jumbo.” The English teacher: “Tumul, you are unable to get the device of a metaphor. Now it is you who are are being long-winded to be mean to your classmate. We need to be more welcoming to our new students. Hence, I want you to write out the whole story in your own words properly explaining its significance and bring it over to me next fortnight. Now, Somakhya stand up and tell the class what you think about the story.”

Somakhya: “I think there are two ways, not mutually exclusive, to look at it. First, due to conservation laws and the principle of entropy growth can never be infinite. Hence, if you keep on the growth curve you will go up and eventually down. Second, you could look at it as the problem of the gambler’s ruin at a casino. Even if the gambler makes some gains, eventually he will be ruined if the probability of loss is lesser than or equal to 0.5. In both these, which have bearing on real life, the big question is when exactly to bail out. If you bail out too soon you may catch much less of the growth or gain than those who stuck on longer but stay too long and you will go bust. That’s the metaphor of the wheel of Kubera. But making that decision can be extraordinarily difficult.”

The English teacher: “That is a practical take. Vidrum, what do you have to say?” Vidrum: “Based on what Somakhya has said, I think the whole story is a metaphor for participating in pyramid schemes like cryptocurrency. If you know when to quit you can retire with good money but if you don’t you can sink into misery like that fourth chap. Hence, I think persistence might be an overrated quality and quitting often might after all help.” E.T: “That last sentence is a rather negative take and I believe a fallacious inference from the story. Hemling, I see you gesticulating and getting excited. Did you understand the story and have anything to say?” Hemling: “ The problem looks hard because the gain function in Lootika’s example is discrete. If it were a continuous function that were differentiable throughout then I think we could for most part use the derivative of the function to reach some conclusions regarding when it would right to bail out. For typical gain functions in real life you have decrease in the magnitude of the derivative as you are nearing the peak. That should give you a reasonably precise indication of when to quit.” The English teacher seemed confused: “Hemling, you better keep that stuff for your math class.”

ET: “Sumalla, what do you have to say about the story?” Sumalla smiled and puckered her lips and face, uttering a lot of filler words to make up for the lack of anything substantial to say. Finally, she managed to string a few grammatical sentences: “I love diamonds and also sapphires but I would be happy to settle for gold. Lootika always has something more to say any topic. Sir, please ask her if there is some secret message in this story.” ET: “Alright, Lootika, I’ll permit you to have the last word on your story.”

L: “First, I think Tumul is a fool. He for one should brush up his physics by reading about photovoltaic cells — something like that could have kept the wheel spinning.” E.T.: “I don’t want you to go there Lootika. Stick to your story.” L: “I think the boys have more or less covered the basic message of the difficulty that we constantly face in life with respect to the quitting problem. However, I’m not sure if Vidrum’s strategy of quitting often is the winning one in an unqualified setting. Maybe it can be explored by means of computer simulations. I’d also add that these problems are made more complex by hidden variables that we cannot see or whose causal chain is too complex for us to trace — like what decided whose wick would fall when. Hence, as a general lesson, it is good to think in terms of the nature of probability distributions and perform cost-benefit analysis accordingly while making life decisions. That said, life in the margins with some copper is much better than getting trepanned.”

Just then the bell rang announcing the end of the class and the school closed for the weekend. Lootika realized that there were classmates who hated her and might want to have a go at her for the punishment they had suffered. Hence, she quickly scurried over to Somakhya and his gang. Thus, shielded by her friend she exited the school premises.

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Pandemic days: bālabodhana

As the pandemic grinds to a close or at least to a pause in some parts of the world, there is a certain fear from new mutants threaten that threaten break current the status quo. The strain that arose in the deś is a case in point. This short note is some bālabodhana on how understand some of the basics of the mutational process.

At the most fundamental level, biology is written in a 4 letter alphabet — the four nucleotides (A, G, C, T/U). A RNA virus, like SARS-CoV-2, has U, whereas cellular DNA genomes have T instead. Any biological word, i.e. string of nucleotides, occupies a node in a graph (network). This graph might be seen as multi-layered where in each layer l_i contains all words of length L. In the subgraph corresponding to any given layer 2 nodes are connected by an edge if they differ by a single letter, i.e. a single substitution can change the word corresponding to a node to that corresponding to the node to which it is connected by an edge. Thus, there are 4 words of a single nucleotide (l_1) which are all connected to each other, i.e. a tetrahedral graph.


With 2 letters (dinucleotides) l_2 we have 4 \times 4 = 16 possible words and the graph is way more complicated as each node can be connected to 6 other nodes.


One can easily see that l_1 will define a tetrahedron in 3D Euclidean space. However, any biological word of L \ge 2 cannot be faithfully visualized in our everyday 3D space, as it will require many more dimensions to render it with real edge-lengths. Thus, as Martin Nowak stated, biological words are rich in dimensions but short on distance. For simplicity, we draw our graphs in whatever dimensions are easily grasped by us (i.e. 2D as above) and simply take each edge of the graph to be measured as a non-Euclidean length. In reality, not just the topology of the graph but also the length of the edges matter. Nucleotides are more likely to mutate to the same type ( pyrimidine (U/T) \leftrightarrow pyrimidine (C), purine (A) \leftrightarrow purine (G)) rather than a different type (i.e. purine \leftrightarrow pyrimidine). Thus, the lengths of edges corresponding to heterotypic substitutions are longer than those corresponding to homotypic edges. However, for convenience we shall simplify the situation by taking all edges to be of length 1. Thus, the distance between two nodes will be length along the graph: in the dinucleotide example shown above the distance between AA and AG will be 1, while that between AA and GG will be 2. Thus, for L=2, while paths of various lengths are possible, from one node you can reach every other node with path of at most length 2. This shortest distance along the graph, D, between 2 nodes is no different from the so called Manhattan metric or Hamming distance.

Some of the nodes on a given layer l_i are connected nodes on l_{i+1} by an edge too because by the addition or subtraction of a nucleotide you go from a sequence of length L to L+1 and vice versa. However, given that you can have in a single step such additions and deletions of arbitrary length you can also connect sequences of various lengths by these length 1 edges coming from the so-called deletions and insertions. For this simple examination we shall ignore those types of mutations.

The basic, necessary process of life may be defined as the copying of an biological word, in its maximal form a genome, by a nucleic acid polymerase. All polymerases are prone to error when they make new copy of the genome from the existing template. We may define this error by u, the probability of single nucleotide substitution at a arbitrary position in the genome. Then 1-u is the probability of the genome being copied correctly. This leads us to a key equation that measures mutation in the genome, i.e. probability p_{ij} that the copying of genome i results in a mutant genome j:


Here, D_{ij} is the shortest distance along the graph between sequences i, j and L is the length of the genome. Wrapped into this are two simplifying assumptions: 1) u is constant throughout the genome and 2) it is independent of mutations at other sites.

I could not find a proper estimate of u for SARS-CoV-2. However, a closely related coronavirus, with a similar-sized genome, the Mouse Hepatitis Virus RNA-dependent RNA polymerase has u=10^{-6}. The same may be safely used for SARS-CoV-2. Hence, the probability that the viral polymerase makes a copy with no mutation at all, with L \approx 3 \times 10^4, is given as:


For a comparison, the HIV-1 virus reverse transcriptase has u=3\times10^{-5} and L=9400; thus (1-u)^L=0.75. Therefore, HIV-1 is a far more mutation-prone virus, which copies its genome without a mutation only 3/4th of the times. The higher fidelity of replication of the coronavirus is a consequence of its distinctive proofreading 3′-5′ exoribonuclease, which the HIV-1 reverse transcriptase lacks. This increased fidelity is keeping with its 3.19 \times larger genome, coding for several more proteins than HIV-1.

Conversely, consider the probability that a specific point mutant arises upon replication of the coronavirus genome. For example, the mutation in the Spike protein E484K can confer resistance to some of the typical antibodies made against the wild type Wuhan strain. This is a substitution of K for E which can arise from a single A \to G point mutation. This probability can be calculated using the above formula with D_{ij}=1; hence,

u(1-u)^{L-1}=9.7\times 10^{-7}

When a virus infects a cell, it makes numerous copies of itself and these “burst” out eventually resulting in the death of the cell. The number of such copies that emerge out from the cell on an average is termed the burst size. To our knowledge, there are no recent studies on burst size estimates for coronaviruses. However, a study in 1976 by N. Hirano et al estimated it to be about 600-700 virus particles, again using the Mouse Hepatitis Virus in a tissue culture system. By taking a burst size of 650, one would need \approx 1585 successfully infected cells producing bursts of this size for a specific point mutation, like the above mentioned one in the spike protein, to emerge. During peak SARS-CoV-2 infection, an individual is estimated as carrying \approx 10^{10} virus particles based on calculations of Sender et al. Hence, a particular point mutation can emerge in an infected individual \approx 9704 times.

If a point mutation confers some selective advantage, like the above-mentioned immune escape mutation, then even with the low error replication of coronaviruses relative to HIV-1, they have ample potential for developing escape mutations. Consistent with this estimate, we saw the E484K mutation repeatedly emerge in different lineages that showed antibody escape, such as the B.1.351 variant that arose in South Africa, the P.1 variant that arose in Brazil and the within the B.1.1.7 lineage in the UK. Finally, a serial passaging experiment by Andreano et al of the virus with plasma from a recovered patient found that for 7 passages the plasma neutralized the virus; thereafter point mutations emerged that allowed escape and eventually complete resistance to the plasma. One of these was the E484K. The evolutionary history of SARS-CoV-2, assuming that it broke out in Wuhan, China, in November 2019 was one of relative stasis for about an year followed by emergence of several mutants that allowed immune escape. The among these were the multiple emergences of E384 point mutations. This suggests that for the first year the virus was rampaging through a relatively immunologically naïve population with little advantage for specific point mutations. However, as pandemic response measures and the virus load in the population greatly increased, there was an advantage for specific mutants. The above numbers show that point mutations were the easiest path to this, as seen with the emergence of variants with mutations such as D614G, E484K etc.

Yet, we see that the vaccination programs have played a big role in bringing the pandemic under control in several parts of the world. Why has it worked, given the above? For this let us take a closer look at the antibody response to SARS-CoV-2.


Roughly 90\% of the antibodies against this virus are directed at the Spike (S) protein. The above picture shows the spike with the top part being the surface which it contacts the ACE2 receptor on the host. Within the spike protein the residues that are targeted by 5 distinct classes of antibodies are marked in different colors on a single monomer colored cyan, while the other two monomers of the trimer are shown in transparent light yellow. The majority of antibodies target the Receptor Binding Domain (RBD), while the minority target the N-terminal galectin-like domain (dark violet). First, since, there are at least 5 distinct classes of antibodies, the escape via a point mutation could be compensated by the binding of one of the other classes. Second, the titer of antibodies seems to matter a lot in terms of immunity. Individuals with high titer seem to be able to overcome much of the escape by single mutants like E484K. Third, there is the cellular immunity. Thus, the vaccine are in most part likely generating high enough titers of antibodies of different classes to make up for escape by single point mutations and a reasonable cellular immunity.

Now, to escape a whole class of antibodies one might typically need 3 or more point mutations. We can compute the probability of 3 point mutations arising from one replication of the virus as u^3(1-u)^{L-3}=9.7 \times 10^{-19}. This means it is very unlikely to ever arise in a single person in single replication (p \approx 9.7 \times 10^{-9}). For a comparison, the probability of a round of replication producing a triple mutation in HIV-1 is 2.03 \times 10^{-14}. On a given day, an infected person carries about 2 \times 10^{10} HIV-1 particles; hence a person has only a 4.1\times 10^{-4} of developing a triple mutant in a single replication. However, 1 in every 2455 persons infected with HIV-1 can develop such a mutant in single round of replication. Hence, it has not been possible to vaccinate against it. However, as the serial passage experiment illustrated, in successive rounds of selection for individual point mutations one could eventually get to total resistance with SARS-CoV-2. The B.1.617.2 (\delta) variant has already shown the capacity to partially break through the commonly used Pfizer and Astra-Zeneca vaccines in the least. In theory it is possible that a strain that is entirely resistant to the antibodies generated by the vaccine could arise in the relatively near future. Fortunately, antibodies are not the only aspect of immunity as they can also trigger cellular immunity. Hence, at least for the near future, with all aspects of immunity put together, the vaccines are likely to provide some level of protection. However, the strong selection pressure they are imposing on the S protein could result in the emergence of more consequential escape mutants. Hence, there is a lingering danger of the disease persisting in some form.

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Some further notes on the old Mongol religion-2

O fire mother,
whose father is flint,
whose mother is pebble,
whose meal is yellow feather grass,
whose life is an elm tree.
An incantation to the Fire Goddess Ghalun-eke; translation from the Mongolian by Yönsiyebü Rinchen

This note revisits some themes relating to the Mongol religion gathered in the 1950s and 1960s by the Mongol scholar Yönsiyebü Rinchen from the Mongolian Academy of Science, Ulaanbaatar. He says that he descends on his father’s side from an ancient Hunnic clan founded by a certain Yöngsiyebü, who was the lord of a tümen. He records an oral chant preserved by the clan on this ancient ancestor of theirs. On his mother’s side he claims descent from Chingiz Khan via Tsoktu Taiji (1581-1637 CE), the chief of Kokonor, who aided the practitioners of the ancient Bon religion of Tibet before they fell to the bauddha-s backed by the Oirat Mongols. Rinchen, with his connections to the old Mongol religion prior to its fall to the bauddha-s, records several notable features of its practice. As we have noted before on these pages, the fall of the old religion to the bauddha-s was neither smooth nor complete. In addition to the material collected by Heissig, we have deprecations such as this one from the old shamans against the religion and followers of the tathāgata invoking at the black (qara) “ghosts” or “spirits”:

O you, you who come to eat 90 bhikṣu-s,
and returns to eat 100,000 bhikṣu-s,
O you, you who come riding the frenzied wolves,
and feed the fire with the Kanjur and the Tanjur.
Translation from the Mongolian by Yönsiyebü Rinchen

The old Mongol religion was organized thus:

Mongol_divinitiesFigure 1

As one can see from the diagram their world is heavy in what might be termed “ghosts” or “spirits”, which are incorporeal presences of ancestors. Of the gods themselves, there were the 99 tengri-s who are mentioned in the famous kindling fire hymn of the clan of Chingiz Khan. They were headed by the tengri Qormusta Khan Tengri, who was also known as Köke Möngke Tengri, and associated with the great blue sky. The latter name of his seems to have been the original Mongol name that is encountered in the Chingizid epic. The former name seen in texts like the Mongolian Geser Khan epic is transparently a tadbhava of the great Iranian Varuṇa-like deity Ahura Mazda. His later iconography closely converged to that of the ārya Indra, paralleling tendencies on the Indo-Iranian borderlands. Of these tengri-s, 55 are seen as benevolent and white in color; 44 are black in color, wrathful and destructive, but their fury is directed at the enemies of the Mongol nation. We had earlier discussed some of the other Tengri-s. We know much lesser of the 77 Earth Mothers, the natigai, with the exception of the fire-goddess Ghalun-eke, whose elm tree “samidh-s” are well-known from multiple surviving Mongol kindling incantations, including the aforementioned one of Chingiz Khan. These high deities are common to the Mongol peoples, and are worshiped by the elite (tsaghan yasun or the white bones) and the high shamans in special community rituals.

Rinchen recognizes two levels of shamans. The high shamans involved in worship the tengri and the great ghosts or spirits and are known as jhigharin (shamans) and abjhiy-a (shamanesses). The lower ghosts are invoked primarily by a lower grade of shamans known as böge (shamans) and idughan (shamanesses). The former word is related to the Turkic bögü, who was a shaman-magician of the pre-Abrahamistic Turks. The words might be related to the Iranic Baga (Sanskrit: Bhaga), as the name of an Āditya god or a respected one with divine capacity (e.g. Skt: bhagavat). In this regard, it may be noted that, at least since the Kirghiz Khaganate, the Turkic shaman was more commonly known as the kham or the kham khatun (female). It was explained in the Sogdhian Iranic dialect as the prophet of Baga (Sogdhian: faghīnūn, c.f. faghfur for Bagaputhra used similarly to the Chinese title of Tianzi by Eastern Iranic emperors). The lower shamans were deployed for commonplace religion and for the quotidian needs of the lay populace (qara yasun or the black bones). For special occasions, the qara yasun might call upon the high shamans for more involved rituals. One of these was the mysterious weather magic that was shared by the Turks and the Mongols, done with what was known as a “rain stone” or a “snow stone”. In times of peace, this shamanic magic was used to help during droughts and was observed closer to our times by Russians and Russified Germans during their exploration of the Mongolian east. However, there are several accounts of such as a tactic in warfare, some of which we shall describe below.

From the Pre-Mongolic times we have the account of a Zoroastrian Iranian encyclopedist, who among other things compiled a version of the Pañcatantra, preserved via Gardīzī. He recorded that such a rain-stone magic was in the possession of the ancestral Turk and its inheritance was contested among the Khazar (the Judaistic Turks), the Oghuz (from whom descend the Black and White Sheep Turks, the Khwarizm Shahs/Qangli Turks, the Osmans and Seljuks, who may have originally been a Judaistic branch of them before becoming Mohammedans) and the Khalji-s (from who descend the monstrous tyrants of India like Jalal al Dīn and Alla al Dīn). The Oghuz are said to have obtained the stone by giving their cousins fake versions. Isma’il ibn Ahmad the Sāmānid Sultan mentions that during his slaving jihad on the heathen Turks, their high shamans deployed the rain-stone magic stirring up a hailstorm. However, the Sultan grandiosely claims that he deployed his Mohammedan Allah magic and backhurled the hailstorm on the Turks. This was perhaps an old motif in Turko-Mongol tradition because it makes its reappearance in the Chingizid epic, when the great Khan was facing facing the confederation of the Naiman Turks. Their shamans raised a blizzard against the Mongols but the Khan’s invocation of Köke Möngke Tengri turned the blizzard against the Naimans. However, the Mongols too described their shamans successfully deploying the rain-stone magic in war. During the sack of Khwarizm, the Mongols spared the life of a Qangli Turk who still remembered the old heathen ways and incorporated him into their shaman contingent for weather magic. When Chingiz Khan’s youngest Tolui was leading the Mongol army against the Jin, he was ambushed surrounded by them. He is said to have had his shamans, including the said Turk, deploy the great blizzard magic, which caused confusion among the Jin, and allowed the Mongols to cut them down. Later after the fall of the Mongol Khaganate, when the belligerent Han under the Ming emperor invaded Mongolia, Biligtü Khan Āyuśrīdhara the son of Toghon Temür organized the defense of his homeland. In the fierce battle in Orkhon, when it looked like Mongolia might fall, the Mongol shamans are said to have deployed the “snow-stone” magic, resulting in many Han freezing to death in the holes they dug to keep themselves warm on the steppe. Biligtü Khan is said to have then rallied the Mongols to save Mongolia from the Cīna-s drive them beyond the wall.

This use of the rain-stones and snow-stones continued even after the Islamization of the Mongols of the Chagadai Khanate in the West. In great battle near Tashkent, between the Chagadai Khan Ilyas Khoja and the alliance of Timur and Mir Hussain, the former first attacked the Mongols and gained some success and called on Mir Hussain to attack the other flank of the Mongols. At that point, the Mongol shamans were called to deploy the rain-stones and a thunderstorm is said to have struck the side of Hussain who was then smashed by the Mongols. Timur tried to rally the forces but he too was hammered by the Chagadais and forced to retreat losing several thousands of men. Finally, the Timurid Mogol Abu Sa`id himself is said to have had Özbek Mongols in his retinue perform the same magic to obtain rain to alleviate their thirst when they ran out of water on the steppe in 1451 CE. Interestingly, the English agents at Madras note that Chatrapati Śivājī sent a brāhmaṇa Mahāḍjī Pant to obtain the same kind of stones from them. However, it is not clear if the Chatrapati wanted them for some magical purpose or as medicine. While there have been records of this ritual in inner and outer Mongolia in the last 150 years with a smooth white stone the size of a pheasant’s egg and a ceramic bottle in which it is placed, unfortunately, we know little of the incantations.

We know more of the traditions relating to the genii, which are an amalgam of ancestor worship, apotheosis and reverse euhemerism. Rinchen holds that the distinction between the different types of genii follow the status of their living progenitors. The ghosts of the great Mongol lords of clans and great Khans are said to become the “lord spirits”, who are invoked in special rites by the entire clan or nation. These usually require the great shamans and shamanesses for the invocation ritual and have survived the bauddha takeover surviving within the tāthāgata pantheon. The spirits of the noted shamans, i.e. jhigharin and abjhiy-a become the “protector spirits”, while those of the lower grade böge and idughan shamans become the “guardian spirits” who are usually genii of loci. The loci themselves, usually in the vicinity of their graves, were marked by heaps of stones known as obugh-a, where the Mongols might make offerings of food or horsehair or alcoholic drinks. The three types of lower genii were collectively known as the jhalbaril-un ghurban. These were pacified with an offering of tea from China or some strong ferment and in modern times, cigars (c.f. the cheroot offerings made in the Drāviḍa country to comparable deities such the horse-riding Mūtāl Rautan depicted like a medieval cavalryman in the retinue of gods like Ārya). These lesser spirits are important in daily life for ghost-magic to attack enemies, to avert accidents while foraging on the steppe, and to protect an individual animal or child. The lesser genii are more in line with ghost-lore from other parts of the world. With appropriate agreements before their death for pacification, otherwise inimical but notable persons might become protector spirits, like Jamuqa in the Chingizid epic. The commoners who lived a bad life upon death might become vengeful or resentful evil spirits. These might need pacification with a lower grade shaman’s assistance or could even be directed for causing harm on their enemies in life and their families.

Shamanstone1Figure 2: A “deer stone”-like enshrinement of an ancient shaman

Some of the lord spirits often straddle the line between the tengri-s and the genii. As we have noted previously, the most notable of these are the Sülde associated with the yak or horse-hair standard known as the tuq (c.f. the Indo-Aryan symbol of royalty the cāmara or yak-tail whisks). Regarding these, in later tradition a peculiar tale, clearly inspired by the ancient ārya brāhmaṇa narratives, is told: Qormusta Khan Tengri instructed the other tengri regarding the Sülde when they were defeated by the Asura-s. This custom brought them victory. From the Chingizid times we know there were two distinct Sülde: the white one (tsaghan), which was used to protect the camp in an apotropaic deployment and the black one (qara), which was used to bring harm to the enemies. That one was planted on the holy fire hearth of the enemy once their camp was taken. Sometimes, an enemy might be sacrificed to the Sülde; as of recently even bauddha ritualists sacrificed goats to some venerated Sülde. It is not clear if the followers of the ekarākṣasa cults who were sacrificed for refusing to bow before the Mongol divine symbols were killed before a tuq for the Sülde or the lord spirits of the Khans.


Figure 3. A depiction of the qara and tsaghan tuq of Chingiz Khan at the Mongolian Hall of Ceremonies.

This latter point brings us to the worship the lord spirit of the Khans. As noted by the Jewish chronicler, Rashīd al-Dīn, who was employed by the Mongols in Iran, the lord spirits of the dead Chingizid Khans were worshiped at the Yeke Qorig (the Great Forbidden Sanctuary) that is believed to have been located in the Hentii mountain range. Here the idols of the Khans received a continuous burning of incense sticks and was restricted in access. Khan Kamala, the grandson of Quibilai built the temple of the Chingizid lord spirits at Burqan Qaldun, which Igor de Rachewiltz associates with ruins found on the bank of the Avarga river. The Japanese researcher Shiraishi Noriyuki holds that the icons mentioned by Rashīd al-Dīn and the site of Kamala’s temple were the same as this Avarga river ruin. The Mongol chronicles explicitly mention that the idol of Chingiz Khan had a golden quiver with real arrows in it. Even the Manchu, during the Ching dynasty, still maintained a temple for the youngest son of Chingis Khan, Tolui, at Ordos housing an idol of his. These life-sized stone Mongol ongon icons for housing the lord spirit follow in the long tradition of Altaic steppe peoples as seen in the form of the stone images of the old Blue Turk and Uighur Khans and lords. The Khitan Khans’s spirits were worshiped in golden idols. Similarly, we have smaller metal idols among the Chingizid Mongols, which some believe might have been for the worship of the lord spirits of leaders of clans, like those of Boghorju, Muqali and Subedei.

Turkic_Mongolic_stonesFigure 4. 1 and 2 are Turkic balbal-s housing the ancestor spirits of ancient Turk Khans/lords. 3 is a similar stone of an ancient Mongolic lord from Mongolia recorded by Russian researchers.

MOngol_era_StoneFigure 5. A comparable Chingizid era stone from the 1200s of CE housed at the Mongolian National Museum of History.

During the initial bauddhization of the Mongols, ancestor worship of the Mongols was brought into the maṇḍala-s of vajrayāna. One notable case is the placement of the pictures of the Khans in the maṇḍala of the deity Vajrabhairava. As we have noted before, since the Chingizid period the lord spirit of Chingiz Khan and his prominent successors became national deities. From lord spirits they were raised to a higher divine status, who within the bauddha system was seen as a yakṣa associated with Vaiśravaṇa — the great king in the Vedic tradition or as an incarnation of Vajrapāṇi. One such incantation that worships him in his aspect as an incarnation of the great yakṣarāṭ Mahārāja goes thus:

Chinggis Khan, who has the power of three thousand people
His body was wrapped by the ten thousand white moon rays.
He has one face, two arms, and three eyes.
He was smiling wryly,
Brandishing to the center of the sky a white spear in his right hand.
In his left hand he was holding close to his heart a plate full of treasures.
He got rid of poverty in the samsara and nirvana.
His white garment was fluttering in front of his chest.
-translation by N. Hurcha from Inner Mongolia.

Some later Chingizid lord spirits also appear to receive prominent worship in certain localities. One such as is Altan Khan, who famously reunified the Mongols to defend them against the resurgent Han belligerence under the Ming, and launched a raid on Beijing. The lord spirits of him his and of some royal women of his family are worshiped in large paintings. The spirits of Abtai Khan (A Chingizid lord of one of the Khalkha Khanates) and his family were also actively worshiped before their suppression by the Marxists. This was almost like karma visiting him as he had actively suppressed the shamanic cults upon the calling of the third Dalai Lama.

The Geser Khan epic (to be treated separately) and the work of the Heissig, and more recently that of Elisabetta Chiodo and Ágnes Birtalan, suggest that some lord spirits from a pre-Chingizid period have attained deity or near deity status. Most notable of these is Dayan Degereki (Deerh), who has survived the bauddha action and was even incorporated into their framework. His enshrinement in a stone ongon icon with a bronze casing is clearly mentioned in the litany used by the shamans invoking him. What is notable is his opposition to both the founder of the Mongol nation, Chingiz Khan and the later Dalai Lama. The latter is rather understandable given the above-noted tension between the tāthāgata-s and the shamans — we had described this in an earlier note. However, Dayan Deerh’s opposition to the great Khan hints that he might have come from a clan that was subjugated by the Khan or his successors. One possibility is that he originally belonged to the Oirats, given that he is also worshiped by them. Indeed, the litany to him mentions that after he was enshrined in a stone ongon at the Örgöö river, the warriors of Chingiz Khan tried to smash his “his unruly, damned skull” [translation by Birtalan] with their swords and scimitars. However, their weapons were blunted, and they fled. Eventually, he is said to have accepted the overlordship of Chingiz Khan and became a Sülde and a wide-ranging protector of the Mongol people along with his son Saraitan and daughter Saraimoo (who appears to be a reverse-euhemerized Sarasvatī). Saraitan appears as a healing deity as indicated in the incantation recited to him in a shamanic ecstasy:
You protect every orphan,
You enrich every poor man,
Your knowledge is perfect,
You have healing powers in your thumb,
Your index finger heals,
You know everything that is hidden,
Saraitan, you are a healer
To the seventieth generation.

Similarly, the incantation to Saraimoo invokes her vīṇā and seeks the blessing of progeny:
You put a curb on the reckless,
With sounds of music on the strings of your lute,
You show what the mountain hides,
You grant fine offspring to all
Who yearn for them.
You, Saraimoo.
[translated from the Mongolian by Birtalan]

In terms of iconography, Saraimoo is depicted exactly like Sarasvatī. The iconography of Dayan Degereki closely parallels that of two other martial deities Dayisun tengri and Dayichin tengri, who seem to have been invoked along with the Qara Sülde while proceeding for a battle. This raises the possibility that this complex of deities had evolutionary connections linking them to the lord spirits to the tengri-s. However, the exact processes involved remain unclear — euhemerization versus its reverse. Among other things, Dayan Deerh’s key pre-bauddha cultic stone image was destroyed by the Soviet-backed Marxist terrorists during the ascendancy which poses impediments to our current understanding.

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Some talks at the Indic Today portal

We had a chat with with C Surendranath, Contributing Editor and (in part with) Yogini Deshpande, Editor in Chief of Indic Today. It is divided into four parts:
A few clarifications for this part: 1) We do not as personally identify “trad”, “alt-right” or whatever. However, Hindu, brāhmaṇa, Vaidika smārta (with a degree of parallel adoption of tāntrika practice) are part of our identity. 2) The name of the German philosopher Schopenhauer was mysteriously blanked out twice! 3) We did say gulag but it sounds like kulak. 4) The first German Jewish professor we were thinking about was Moritz Stern, who succeeded Carl Gauss. Moritz Cantor also Jewish was Stern’s student. Related to this part is our essay on the Lithuanian (Baltic) heathen tradition.

This part covers issues which we have presented in the writing here: e.g. 1) Early “free-thinkers” in the Abrahamosphere (especially see second part). 2) Further details on the extra-military aspects of the Islamo-Hindu confrontation. 3) More focused discussion of aspects of counter-religions and their interactions. 4) Military labor entrepreneurship and related issues in the last days of the last Hindu empire. 5) Some Hindu polemics against the preta-mata.

This part covers: 1) A basic introduction to legalism (fajia) & its manifestations in old & recent Cīna thought. 2) Comparisons between the imperial political frame in fajia and the arthaśāstra. 3) “Fads for people” as a mechanism in legalism. 4) About half of Cīna history Hans were ruled foreign powers: the consequences and responses. 5) Counter-religions and Cīna responses: some comparisons with India. 6) Hui and Cīna little brother of the preta and their suppression
Overall you can take it prolegomenon for a H analysis of one of our civilizational rivals. In the oral medium some little points can slip through the cracks: We should have explicitly mentioned that eunuch Zheng He was a Hui descending from those brought to Cīna by the Mongols.

The final part of this chat covers the Rus. It meanders along touching on: 1) the pagan Rus and their Christianization to the Orthodox church; 2) The Mongol conquest of the Rus. 3) The Rus fight back with Dimitri. 4) The see-saw struggle with Khan Toktamish burning Moscow. 5) Closer to our age the attempt by the Rus to present themselves as the chief of the preta world. 6) Exploration of the East – Siberia. 7) Conflicts with the Western powers and Japan. 8) Marxian subversion of Russia. 9) WW2 and the attack on Japan. 10) Rus as a Superpower. 11) Decline and demographics. It is peppered with some other excursions and a discussion on the movie on Alexander prince of Novgorod who fought the Germanic invaders by Surendranath. We should have explicitly stated that he was a feudatory of the Mongols and aided by Khan Sartaq of the Golden horde.

In our opinion the oral medium is best suited for a discursive exploration of “big themes” along with interesting trivia as as raisins in the pudding. It is not the best for “technical” or detail-oriented presentations especially when not accompanied by other aids, like figures and maps. This could compromise accuracy to a degree and also the sequence in which events are treated. Hence, these should be heard with those caveats in mind.

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The cicadas return

Seventeen years after we first saw them emerge, like a great horde of Cīna-s invading Tibet, the cicadas of Brood X reemerged. 17 years is a good amount of time, making one pause to reflect on what has passed by life, in addition to the cicada-s themselves. This coming of Brood X was not very successful in our area. We first started noticing them around May 17 and surveyed them in a 2 sq km region that that we walk through by foot. They emerged alright after their 17 year underground larval development at night from holes in the ground. The 17- and 13-year cicadas seem to emerge after their fifth instar molts.


Then the crawl up to reach trees or posts.


Most of them then proceeded to molt. However, right at this stage about 1/5-1/4 seemed die even as they are emerging from the molt. In the first image below one can see a specimen that has died while molting. In the second picture one see another such being scavenged by ants.



After that, several underwent proper melanization but failed to properly inflate their wings and died.


Those that did survive started their famed song.


This emergence was already a bit shaky relative to 17 years ago. From my records they were already going well by May 22 of 2004. I examined about 100 or so and did not see any obvious signs of Massospora mycosis. It was relatively cold for several days from May 17 onward (low <15 C). However, by May 21 the temperature was pretty good (low >15) but they still struggled and hardly any of their noise was heard as of May 23. While our friend reported a similar situation in his site about 15 km away from mine, others further away reported high densities at this time. The cause their poor performance in our regions remains unclear. Was it just the temperature or some other unknown pathogen or the insecticide use by residents? We saw a couple moles scurrying around in the twilight in one wooded area as also their predator a fast-running fox. Moles are known predators of their larvae but we doubt they are numerous enough to make a difference. In any case, much of the death which we saw was post-emergence. The cicadas finally hit their stride around June 2 and the wooded paths were reverberating with their tymbals. All the noises — the choruses, females clicking the wings, the coupling noises and the distress screeches as birds attacked them — could be heard. On June 18 a precipitous decline in their calls was noticed and they were gone by June 20. However, their final act was registered in the wilting of tree shoots as the females slit the terminal branches to lay their eggs.

cicada_eggs (2)

Thus, it was just about a month of activity with a weak start and even the peak afflicted by several days of heavy rains. How exactly this would affect their prospects 17 years down the line remains unknown. This brings us to hypotheses regarding the long periods of these cicadas. Cicadas are unique among insects in having long lifespans, most of which is spent in larval stages. A study in Ohio where there was an unusually warm January followed by a freeze resulted in maple trees producing two sets of new leaves in the same year; during that event the 17 year cicadas came out one year earlier once the late spring soil temperatures stabilized at around 18 C (their preferred emergence temperature). This suggests that they have a mechanism to track the cycles of leafing in the trees whose sap they suck deep underground and thus count the years. In this regard we propose a dendrochronological exploration wherein tree ring records are examined to see if a periodicity relating to cicada emergence can be discerned in them.

More broadly, several cicadas come out every year. However, there are those, such as Okanagana in North America, which can have lifespans in the range of 9 to 19 years. At least the 9-year ones exhibit a 9 year “proto-periodic” cycle, where they are abundant in 4 of these years and relatively rare or absent in 5 of them. This indicates a degree of synchronization among the broods of the 9-year Okanaganas. One of the Okanaganas from Canada, Okanagana synodica, and Tettigades “chilensis” from Central Chile have 19-year life cycles and could very well represent transition to the next highest prime number cycle beyond 17. The Japanese cicada, Oncotympana coreana, might have converged to a shorter prime cycle of 7 years; there are several other Japanese cicadas with even shorter 3-year cycles. There are also cicadas with 4- or 8-year cycles from India, Japan, Fiji and Australia, most of which are likely proto-periodicals with abundant years and rare years. However, of these, the so-called “World Cup Cicada”, Chremistica ribhoi from the Ri Bhoi District, Meghalaya, India, with a 4 year cycle, and Raiateana knowlesi from Fiji with an 8 year cycle, appear to be truly periodic with non-prime cycles.

One argument was that the relatively long prime cycles were selected to evade predators and parasites that might take advantage of their periodic emergences to coordinate their own generations to divisors of the cicada cycle. But long primes could throw this off. However, the alternative hypothesis has been that the long cycles are to evade prolonged periods of harsh climate and that the prime cycles are likely to throw off mating with cicadas with shorter periods that may be divisors of the longer cycle. Thus, prolonged harsh climate would segregate broods which do not mate with each other favor long prime periods. However, the discovery of the even-period cicadas from India and Fiji raise questions about these prime periodicity proposals and suggests that prime periodicity is not hard and fast in cicadas.

Whatever the case, there is support for predation being a potential selective pressure for synchronicity once a period longer than a couple of years is established. It has been proposed that a mass, synchronous emergence overwhelms the predators with satiation. There is some evidence for this from field observations. We have ourselves noted that while the initial emergents are eaten by dinosaurian and mammalian predators, they are quickly overwhelmed by the huge numbers in the case of the prime periodic cicadas. More recent observations, that need further study, indicate that once they establish a high intensity chorus, they inhibit birds by driving them away from the areas with high levels of noise. This has been observed with both tropical cicadas in Central America and 17-year periodic cicadas in the USA. Very loud cicadas are seen all over the world and their noise can be damaging to mammalian ears, like those of humans, at close range. Hence, the synchronous emergence with a chorus likely to be adaptive against predators irrespective of the period.

However, for this strategy to evolve first a relatively long life has to be in place. Most insects have annual cycles and several cicadas are no different. Hence, this was likely the ancestral condition from which early on a long-lived version emerged. The origin of such a long-lived version could have been selected by harsh climate because by skipping an year or two before emergence they could tide over period of drought or cold. It is conceivable that it first arose close to the tropics in response to drought and it allowed colonization of higher latitudes as it provide a means of tiding over cold. Per say, climatically driven selection for long life is unlikely to favor synchronicity because by hedging bets and distributing the emergence one is more likely to get a good year eventually. We see the longer periods to be more common in higher latitudes like Japan, New Zealand and North America suggesting that cold weather might have been a selective pressure for increasing the length of life cycle. Once long life was selected, it is likely that a degree of synchronicity was selected next by predator pressure. It is possible that this happened several times independently in different parts of the world giving rise to the several proto-periodic cicadas and that the transition to long periodic cycles was likely via synchronization of the proto-periodic intermediates. Yet, it seems to us that we don’t still have the strongest or cleanest hypothesis for the emergence of prime periods.

A cycle of 17 years is a big span even in human reckoning. Hence, we could not avoid looking back at the ebb and flow of life in the 17 years since the last emergence. One thing has ironically remained the same both at this emergence and that last. However, when we looked at that thing during the last emergence, we were still quite hopeful. Having seen a lot of life, we are inherently pessimistic about things that need near miracles, but at that time we were still hopeful of victory in the battle of the blind-spots — one where we had to shoot the target without being able to see it. That conquest remains as elusive now as 17 years back. In those 17 years, we did see some glimmer of the hiding foe but whatever we saw inspired no confidence whatsoever that we could beat it. A strange thing happened somewhere roughly midway in those 17 years though. Tied down by the lassos of Varuṇa and the darts of Rudra, we knew not at some point how long we may trudge on. It looked like the climb of Himalayan slope while being low on supplies. At that point, like at few other points in our life, we took a big but carefully calculated gamble. There are some such junctures when the probabilities can be relatively precisely computed, and you can make an “informed” bet. However, this bet was rather different from the rest in that we made it in a niṣkāmya manner — one where we had properly steeled ourselves for the negative outcome. There we were in a three-front war but only two of them mattered at all. The gods aided this time, unlike at the time shortly after the last emergence, and we scored several outright victories on both those fronts — like the Aśvin-s and Indra aiding emperor Trasadasyu in the demolition of forts. But that third front was a mysterious experience. We neither won nor lost. However, we got a fairly clear glimpse, late one evening, of what true victory in that front would look like. We had started doubting if there was even such possibility — maybe it was just a figment of our imagination — the quest for something that did not exist in real life. That glimpse showed that it was real; we were not chasing a gandharva-nagara — we could almost get there — but the chance that we would rule over that city was perhaps not going to come. Such are the ways of the gods — they sometimes show you after a long trial that something you thought should exist really exists, but it might indeed be out of reach, like a man yearning to reach a planet going around another star.

In the 17 years that have gone by the gods took us to many deserved victories against powerful foes — indeed, the wielder of the thunderbolt raises the Ārya yajamāna against the dasyu. At some point in the last third of those years we made another calculated gamble. We were in a position of relative power and we knew that if we did not make it our enemies would gain a complete advantage. A lot more depended on our allies than on us in that samarya. Our allies tried their best, fighting to the utmost of their abilities, but they lost, and our enemies made away with all their riches. We ourselves won most individual battles, barring one where we were betrayed by expectedly flaky fellow travelers. But the advantage our foes had gained and the flagging morale of our pakṣa placed us at the foot of yet another mountain fortress that seems formidable as that of emperor Jarāsaṃdha of Magadha. Time will tell if we might be able raise an army that will accompany us in new campaigns at a time when the physical virus from China and the mental one with ultimate roots in West Asia has widened the gulf between winners and losers.

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Matters of religion: Varuṇāvasiṣṇavam, Agnāvasiṣṇavam and the vyahṛti-s

Like the clouds lifting after the monsoonal deluge to unveil the short-lived comforts of early autumn, the metaphorical pall over the nation cast by the engineer’s virus was lifting. Somakhya and Lootika were at the former’s parents’ house, relieved that they had survived and overcome the tumultuous events. Somakhya’s parents asked them to offer the Varuṇāvasiṣṇava and associated oblations as ordained by the Bhṛgu-s and Āṅgirasa-s of yore. Vrishchika and Indrasena were also present as observers of the rite. Somakhya donned his turban and identified himself with the god Indra to initiate the rite, for indeed the śruti has said: tad vā etad atharvaṇo rūpaṃ yad uṣṇīṣī brahmā । — that brāhmaṇa who is turbaned is indeed of the form of the Atharvan. He explained to Indrasena that the śruti holds the Indra took the shape of the Atharvaveda in his turbaned form to protect the ritual of the gods from the dānava-s. Indrasena: “Indeed, even the primordial śruti records that form of Indra in the ṛk of the Kāṇva-s:

yajña indram avardhayad yad bhūmiṃ vy avartayat । cakrāṇa opaśaṃ divi ॥ 
The ritual magnified Indra [with praise] when he made the earth rotate, making [himself] a turban in (= of) heaven.
One may note play on the word opaśa; by taking it as neuter one could also interpret is as the pillar or the axis of heaven.”

Then, Somakhya and Lootika took their seat before the fire on the hide of a reddish brown ox strewn with darbha grass. Thereafter, they performed an ācamana and prokṣaṇa with the incantations: apāṃ puṣpaṃ mūrtir ākāśaṃ pavitram uttamam । indra jīva sūrya jīva devā jīvā jīvyāsam aham । sarvam āyur jīvyāsam ॥ [The flower is the form of the waters, the empty space [and] that which the most pure. Enliven, o Indra; Enliven o Sūrya. Enliven, o gods. May I live. May I complete my term of life].

Thereafter, Somakhya meditated on the special connection of the founder of his race to god Varuṇa and uttered the incantation establishing his connection to the founder of his lineage, great Bhṛgu: tad bhṛgor bhṛgutvam। bhṛgur iva vai sa sarveṣu lokeṣu bhāti ya evaṃ veda ॥ [That [connection with Varuṇa] is the Bhṛgu-ness of Bhṛgu. He who knows thus shines in all the worlds like Bhṛgu]. He recited the formula: OṂ sarvair etair atharvabhiś cātharvaṇaiś ca kurvīya॥ [OṂ May I perform [this rite] by means of all these incantations of Atharvan and the Ātharvaṇa-s]. OṂ mantrāś ca mām abhimukhībhaveyuḥ [OṂ may the [AV] mantra-s face me [favorably]]. Somakhya then explained to Indrasena and Vrishchika: “The śruti holds that like a mother can be killed by the fetus she bears, the mantra-s can kill the holder if he improperly applied them or has not been diligent in their study. Hence, he must utter this incantation beginning with OṂ. The praṇava indeed protects him from such backfiring.”

He muttered the Sāvitra incantation-s to the god Savitṛ as per the teaching of the great brāhmaṇa Śvetaketu, the son of Uddākaka Āruṇi:
OṂ BHUR BHUVAḤ SVAḤ tat savitur… prachodayāt ॥: This first cycle is done with the 3 mahāvyāhṛti-s.
OṂ BHŪR JANAT tat savitur… prachodayāt ॥: Somakhya touched Lootika with a darbha-bunch and she made an oblation as is appropriate for the sacrificer’s wife in the fire at the utterance of svāhā (idaṃ na mama ॥)
OṂ BHUVO JANAT tat savitur… prachodayāt ॥: Somakhya’s parents stepped forward and made an offering with a silent svadhā call and touched water.
OṂ SVAR JANAT tat savitur… prachodayāt ॥: Somakhya made an oblation with a svāhā (idaṃ na mama ॥)
OṂ BHŪR BHUVAḤ SVAR JANAD OM tat savitur… paro rajase ‘sāvado3m॥: Somakhya made an oblation with a vauṣaṭ uttered loudly (idaṃ na mama ॥)

Then he proceeded to the main oblations:
yayor ojasā skabhitā rajāṃsi
yau vīryair vīratamā śaviṣṭhā ।
yau patyete apratītau sahobhir
viṣṇum agan varuṇaṃ pūrvahūtiḥ ॥

By whose power the domains of space were stabilized,
by whose energy, the most energetic and mightiest,
who lord it unopposed by their powers,
to [that] Viṣṇu and Varuṇa have gone the first offerings.

yasyedaṃ pradiśi yad virocate
pra cānati vi ca caṣṭe śacībhiḥ ।
purā devasya dharmaṇā sahobhir
viṣṇum agan varuṇaṃ pūrvahūtiḥ ॥
vauṣaṭ + idaṃ varuṇāviṣṇūbhyāṃ na mama ॥

In whose direction is that which shines forth,
[whatever] that vibrates and observes with power
from ancient times by the god’s law with might,
to [that] Viṣṇu and Varuṇa have gone the first offerings.

OṂ BHŪH pra tad viṣṇu stavate vīryāṇi
OṂ BHUVO mṛgo na bhīmaḥ kucaro giriṣṭhāḥ ।
OṂ SVAḤ parāvata ā jagamyāt parasyāḥ ॥

Thus, he praises forth his heroic deeds, Viṣṇu is
like a dreadful lion wandering, stationed in the mountains
From the distant realm may he come close.

agnāviṣṇū mahi tad vāṃ mahitvam
pātho ghṛtasya guhyasya nāma ।
dame-dame sapta ratnā dadhānau
prati vāṃ jihvā ghṛtam ā caraṇyāt ॥

O Agni and Viṣṇu mighty is your might;
you two drink from name of the ghee’s secret.
In home after home you two place the seven gems.
may your tongue move here to meet the ghee.

agnāviṣṇū mahi dhāma priyam vāṃ
vītho ghṛtasya guhyā juṣāṇau ।
dame-dame suṣṭutyā vāvṛdhānau
prati vāṃ jihvā ghṛtam uc caraṇyāt ॥
vauṣaṭ + idaṃ agnāviṣṇūbhyāṃ na mama ॥

O Agni and Viṣṇu mighty is your dear domain;
may you two savor the secret enjoyment of the ghee
In home after home you two are magnified by good praise-chants.
may your tongue flicker upward to meet the ghee.

OṂ BHŪH yasyoruṣu triṣu vikramaneṣv adhikṣiyanti bhuvanāni viśvā ।
OṂ BHUVA uru viṣṇo vi kramasvoru kṣayāya nas kṛdhi ।
OṂ SVAḤ ghṛtam ghṛtayone piba pra-pra yajñapatiṃ tira ॥

In whose wide three strides all the worlds are laid down;
stride widely O Viṣṇu for wide lordship; make [that lordship] for us;
Drink the ghee, O source of ghee; prolong the lord of the ritual over and over!

mama devā vihave santu sarva
indravanto maruto viṣṇur agniḥ ।
mamāntarikṣam urulokam astu
mahyaṃ vātaḥ pavatāṃ kāmāyāsmai ॥

May all the gods be at my ritual invocation;
The Marut-s with Indra, Viṣṇu and Agni.
Let the broad realm of the atmosphere be mine.
May Vāta blow for favoring this wish of mine.

yo naḥ svo yo araṇaḥ sajāta uta niṣṭyo yo asmāṃ abhidāsati ।
rudraḥ śaravyayaitān mamāmitrān vi vidhyatu ॥

Whether one of ours or one who is in a truce, a kinsman or an alien, whosoever attacks us
may Rudra releasing a shower of arrows pierce those enemies of mine.

yaḥ sapatno yo ‘sapatno yaś ca dviṣan chapāti naḥ ।
devās taṃ sarve dhūrvantu brahma varma mamāntaram ॥

Whichever competitor or whichever non-competitor and whichever hater curses us,
the gods shall injure him. The incantation is my inner armor.


Having concluded the after-rites Somakhya, Lootika, Indrasena and Vrischika left to savor the fresh air and the natural world, and engage in some brahmavāda on the hills beyond the late medieval temple of the awful Caṇḍikā. They stopped at the quadrangle in the low ground facing the stairs leading to the temple on the hill before a towering bastard poon tree. Vrishchika: “There used to be an old woman with a goat who used sit in the vicinity of this skunk tree. We used to feed her goat as a representative of the god Kumāra. She has likely passed into the realm of Vivasvān’s son along with her aja. Hope Rudra was kind to her when her time came. Indrasena, sometimes, thinking about you, as though seized by Skanda or Viṣṇu, I used to feed her goat hoping that Skanda might be kind to me.” Indrasena: “O Gautamī, after all the meanderings, it seems, Skanda has brought you to your destination as he did to the Kāṇva and his goat.” Lootika: “I also recall that Somakhya’s family observes a Kaumāra rite on the Āśvina fullmoon, where they make a rare dish from payasya (curdled milk cheese; Iranic: paynīr). They would offer some of that to the woman with the goat getting the leaf in which the dish was wrapped” Indrasena: “Is that a folk Atharvan rite?”. Somakhya: “Yes, the folk Atharvan tradition holds that Skanda is the teacher of Paippalāda, one of the promulgators of the AV saṃhita-s, and the paurṇamāsya rite is held in the honor of the enlightenment of Paippalāda.”

It was a quiet time of the day with just a light stream of votaries and gawkers on the stairway to the temple. The four made their way up the steps, mostly in silent thought, to pay their respects to the enshrined parivāra-devatā-s and the wife of Rudra at the main shrine. Even as they were about to exit the circumambulatory path to resume their climb beyond the stair way further up the crag to the plateau beyond, Lootika was approached by a woman who wanted to fall at her feet. Lootika prevented her from doing so and she began pouring out a litany of medical troubles. Lootika signaled to her sister: “This lady seems to have mistaken me to be you.” Vrishchika: “Stepped in and having briefly heard her out gave her some reassuring words and asked her to attend to her father’s clinic.” Somakhya and Indrasena instinctively felt their concealed guns and knives for a dasyu could always be lurking in the shades. Having reached their favored vantage point, the site of an old megalithic stone circle, they looked on at their city below. There seemed to be some hesitancy in returning to the old normal; hence, the air seemed cleaner and the horizon clearer. The, nakṣatra of the day, the eye of Mitra and Varuṇa, had mounted the vault of the cloudless southern sky. Looking into the distance they saw that the fires in the yonder cemetery were far fewer than when Somakhya and Lootika had looked on from the same place during the height of the conflict. Lootika: “The lull between the storms.” Vrishchika: “You think it is not yet over?” L: “The clash with the rākṣasa-mata-s is like the fight between the Daitya-s and the Deva-s — the bigger disease from the mleccha-s is that of the mind — it will play out next with much spilling of blood — but then our people could end in a whisper too.”

Indrasena: “Coming to the rite of morning, I’d like understand more about the AV vyāhṛti-s — both the combination of the mahāvyāhṛtis with the incantations as also the connection of the vyāhṛti-s to Maruta Indravantaḥ.”
Vrishchika: “Could we please also have a broader discussion of role the mahāvyāhṛti-s and their transcendence by other vyāhṛti-s across the Vaidika collections?”

Somakhya: “Alright, Vrishchika, let us lay the groundwork for the exploration desired by Indrasena by first addressing the brahmavāda on the vyāhṛti-s in the śruti-s other than those of the Atharvan-s. Let us begin this discussion with testing your knowledge of the traditions regarding the vyāhṛti-s in the traditions you are familiar with. Why don’t you tell us what you know regarding the three mahā-vyāhṛti-s?”
V: “The śruti of the Aitareya-s holds that the 3 vyāhṛti-s are like connective tissue that holds together the three disjunct parts of the śruti in the form of the ṛk-s, yajuṣ-es and sāman-s — thus they are compared to procedures akin to reducing dislocations of joints or sewing up cut skin. Indeed, this analogy of the Aitareya-s provides early evidence for these medical procedures among the ārya-s, which parallel those surgical and bone-setting procedures explicitly mentioned in the Veda of the Atharvan-s and having echoes among other Indo-Europeans, like in the Merseburg spell of the śūlapuruṣa-s:
(bhūr bhuvaḥ svar) … etāni ha vai vedānām antaḥ śleṣaṇāni yad etā vyāhṛtayas . tad yathātmanātmānaṃ saṃdadhyād . yathā parvaṇā parva yathā śleṣmaṇā carmaṇyaṃ vānyad vā viśliṣṭam saṃśleṣayed । evam evaitābhir yajṅasya viśliṣṭaṃ saṃdadhāti ॥

These are verily the internal bindings of the Veda-s, these vyāhṛti-s. Even as one joins the one individual thing other separated thing; like setting one joint with another joint; like suturing with a cord, skin with another torn one. Even so, verily with these one joins the disjunct parts of the the ritual.

Thus, the suturing role of the vyāhṛti-s is critical for the terminal sviṣṭakṛt-s for fixing the errors in the ritual.

S: “That is good. So, what do you know of the thesis of the transcendence of 3 mahāvyāhṛti-s?”
V: “Well, the Upaniṣat of the Taittirīyaka-s holds that there are three primal or mahāvyāhṛti-s, bhūr, bhuvas and suvar; however, the sage Māhācamasya held that there is a fourth, i.e. mahas. In his teaching mahas is privileged over the remaining three. He establishes four homologies between them and other entities. Those are the following: 1) He sees the three primary one bhūr, bhuvas and suvar as corresponding to the earth, the atmosphere and the space beyond. The fourth, mahas, is seen as the Āditya, the sun, which causes the world material worlds to take form — perhaps in more than one way — by supplying the matter to make them and also the light by which their existence is perceived. 2) The next homology is to the first 3 and the sources of light — the fire, the wind (he implies lightning here) and the sun. The reflected light of the moon is homologized to mahas — here again we might see it as the ambient light that makes perception possible even the source themselves are invisible. 3) He also homologizes the first 3 with the 3 categories of incantations in the śruti, the ṛk-s, the yajuṣ-es and the sāman-s, and mahas with the brahman, which is to be understood here as the praṇava. 4) The next homology is between the vyāhṛti-s and the inhalation, exhalation, and retention in the prāṇāyāma cycle. Specifically, in that context mahas may be understood as the air. However, I hold that from the śruti we may infer that what was meant was more general — the physiological process of nutrient uptake, export of unwanted and secreted compounds and the anabolic processes. The free-energy-providing material in this process, i.e. the nutrients, is the fourth, mahas. Thus, as there are four homologies in each set with a total of 4 sets, the vyahṛti-s are seen as being 16-fold. The summary of these linkages is presented as the understanding that the first three are the limbs of the physical body and mahas corresponds to the consciousness. Thus, mahas in different domains is equated respectively with the link substance, the work-generating substance, the diffuse or reflected light that pervades the universe and the mantra essence — all of these are seen as analogies for the nature of consciousness with respect to matter.”

Somakhya: “Excellent upa-gautamī. Dear Lootika is there something you might want to add to what your sister has just expounded from other Vedic traditions?”
Lootika: “Sure. I actually learnt of the multiple expressions of vyāhṛtyutpatti in the scriptural readings I did with your mother. This theory of Māhācamasya, introducing the fourth vyāhṛti perhaps led up to the theory of multiple vyāhṛti-s in both Atharvan and Yajuṣ traditions. This is clearly a departure from the triple vyāhṛti system expounded in the brāhmaṇa of the Vājasaneyin-s, that of the Aitareya-s and the Upaniṣad brāhmaṇa of the Jaimini singers. There explicitly Prajāpati is described as generating only 3 vyāhṛti-s.”

Indrasena: “Indeed. However, each of those accounts have notable points — one may see a gradual build up of concepts within the 3 vyāhṛti system that led to the emergence of the fourth. In the śruti of the Vājasaneyin-s, we have an account that might be seen as retaining the archaic form which the thesis of Māhācamasya eventually emerged. That account describes the heat (tapas) of Prajāpati as the basis for the emanation of the 3 worlds. Since these worlds were heated by his tapas they emanated the same 3 primary sources of light (the deities Agni, Vāyu and Āditya) mentioned by the TU. That tapas causing those lights to radiate heat spawned the collections of the 3 types of mantra-s of the śruti. His tapas then caused those mantra collections to radiate heat from which Prajāpati extracted three generative substances (śukra-s) that are the vyāhṛti-s. We might trace the origins of the two other traditions, which Lootika just mentioned, from such a foundation — one present in the Sāmavaidika tradition of the Jaiminīya-s and the other in the Aitareya-brāhmaṇa.

In the former, Prajāpati is not presented in a protogonic context, but is competing with the other gods, probably reflecting the tension between the surging Prājāpatya religion among our people and the older Ārya-deva-dharma. Prajāpati conquered the triple-world with the 3-fold mantra collection. Fearing that the other gods might see the same and take over his conquest, he extracted the essence of the ṛk-s uttering bhūḥ. That became the earth and its essence streamed forth as Agni. From the Yajuṣ-es he extracted the essence with bhuvaḥ and that formed the atmosphere and streamed forth as Vāyu. The sāman-s were distilled with the suvaḥ call and they formed the the heaven, from which the essence streamed forth as the Āditya. However, there was one akṣara he could not distill into an essence, namely the praṇava. That remained by itself and became Vāc.

In the Aitareya text, we have a cosmogony closer to that the of Vājasaneyin-s, wherein Prajāpati’s heat generated the triple-world. As in the former account, by his heating of those, the 3 luminaries emerged and by heating those the 3 mantra-s collection were generated. By heating those again the generative essences (śukra-s), which are the 3 primary vyāhṛti-s emerged. But in this account those were heated further to generate 3 phonemes: A, U, M, which Prajāpati got together to generate the praṇava, OṂ. Thus, here too, as in the Jaiminīya-Upaniṣad-brāhmaṇa we see that there is something beyond the 3 basic vyāhṛti-s, namely the praṇava. It is the praṇava that Māhācamasya equated to the fourth vyāhṛti mahas in his thesis and that which appears as the final vyāhṛti of the Atharvan-s. Thus, we see an evolutionary process within the śruti which paved the way for the vyāhṛti-s beyond the primary three via the concept of the praṇava.”

Lootika: “There is no mention of the triple vyāhṛiti-s in the oldest layer of our tradition, the Ṛk-saṃhitā. Now, that could be because they are specialized calls that don’t fit into the metrical incantations. However, in all the accounts of vyāhṛtyutpatti, which we have discussed so far, we see the central role of the protogonic god Prajāpati. Is that the vyāhṛiti incantations arose within a Prājāpatya milieu? As Indrasena pointed out, one of the narratives might hint at Prajāpati competing with the gods of the old religion. Moreover, in specifying the deities of the vyāhṛiti-s, Śaunaka mentions in the Bṛhaddevatā that, whereas Prajāpati is the god of all the 3 vyāhṛti-s as a group, individually they have Agni, Vāyu and Sūrya as their deities. Likewise, for OṂ, Śaunaka mentions Vāc and Ka Prajāpati as in the brāhmaṇa narratives, but also Indra and the gods in general as its deity. This is indeed supported by the fundamental teaching of the upaniṣat:

yaś chandasām ṛṣabho viśvarūpaś
chandobhyaś candām̐sy āviveśa ।
satām̐ śikyaḥ provācopaniṣadindro
jyeṣṭha indriyāya ṛṣibhyo
namo devebhyaḥ svadhā pitṛbhyo
bhūr bhuvaḥ suvaś chanda om ॥

After all even in its declining days our tradition was quite unanimous about this teaching and held that: “sa praṇava-svarūpī paramaiśvarya-yukta paramātmā indra upaniṣat pratipādyo bhūtva vyāhṛti-trayātmā oṃkāraḥ ।” Indra, in the form of the vyāhṛti-s and the praṇava, is indeed is the first causal link in the “bringing together” (upaniṣat) of the sambandha-s for our ancestors — fitting well with the Aitareya statement on the vyāhṛti as the internal bonds of the śruti. These hint at a pre-Prājāpatya origin for the vyāhṛti-s and the praṇava within the old religion. So, can we find evidence for the ritual deployment of the vyāhṛti-s in the pre-Prājāpatya layer of the religion?”

S: “A superficial student could indeed reach the conclusion that the vyāhṛiti-concept emerged as part of the linking of the cosmogonic role of Prajāpati with the cosmic origin of the mantra-s, like in the accounts that Indrasena just expounded. However, as your sister noted there is an association of the vyāhṛti-s with the sviṣṭakṛt rite for setting right the errors of ritual. Indeed, as you know well, such vyāhṛti oblations and calls are a general feature of most gṛhya rites and also śrauta rituals such the sāmidhenī incantations where they make up the syllables corresponding to the rest of the year beyond the 365. I would say this pervasive use is an indication of their ancient and pre-Prājāpatya ritual roles, substantiating their conception as the internal fastenings of the Veda. An unambiguous case for their pre-Prājāpatya role is made by their central role in the silent incantations that are inaudibly recited as as part of various śastra-s, as also the similar incantations used in the morning and evening offerings of the Agnihotra. Thus, the śruti of the Aitareya-s states that one concludes the Ājya and Praüga recitation with bhūr agnir jyotir jyotir agniḥ ॥; the Niṣkevalya and Marutvatīya recitations with indro jyotir bhuvo jyotir indraḥ ॥ and the Āgnimāruta and Vaiśvadeva recitations with sūryo jyotir jyotiḥ svaḥ sūryaḥ ॥ as the inaudible incantations. This give us a glimpse of their ancient use in a śrauta context going back to pre-Prājāpatya times. But we see pervasive signs of their place in even more routine rites that reinforce the proposal that their common place use was of pre-Prājāpatya provenance — I guess you might agree Indrasena?”

I: “I was just about to interject in this regard. To build up the context, we may first note the Vyāhṛti-kalpa from the gṛhya appendix of the Bodhāyana-s, which teaches the 12000x japa of the mahāvyāhṛti-s for the attainment of specific goals and purification. It was this tradition of the japa of the vyāhṛti-s that one hand was incorporated into nitya and naimittika rituals focused on Deva Savitṛ — like the very fact that we use either the three mahāvyāhṛti-s or the seven vyāhṛti-s: BHUḤ BHUVAḤ SUVAḤ MAHAḤ JANAḤ TAPAḤ SATYAM coupled the Sāvitrī in our routine daily japa. The same applies to the coupling of the mahāvyāhṛti-s with the Sāvitrī and the Trisuparṇa in the mahat (the great) rite taught by Uddālaka Āruṇi to Yājñvalkya Vājasaneya, the founders of the śukla tradition. Indeed, this coupling of the mahāvyāhṛti-s with the Sāvitrī is extended in tradition of the Jaiminīya singers, wherein two additional vyāhṛti-s are added to the list to make it a total of five, just as the seven in the other traditions. These are SATYAM and PURUṢA, after which comes the gāyatra sāman composed on the Sāvitrī. On the other hand the pure mahā-vyāhṛti japa also developed into the song of vyāhṛti-s in the tradition of the Rāṇāyanīya and Kauthuma singers. In that song, the musically rendered vyāhṛti-s are sandwiched between two musical praṇava-s and interspersed with 3 repetitions of the magical stobha-s with the concluding bhakti-s of the suvar-jyotiḥ, reminiscent of both the śastra incantation you mentioned and the brahmaśiras that closes the Sāvitrī with the ten praṇava-s (OṂ BHUḤ । OṂ BHUVAḤ । OM̐ SUVAḤ । OṂ MAHAḤ । OṂ JANAḤ । OṂ TAPAḤ। OM̐ SATYAM । OṂ tat savitur vareṇyam । bhargo devasya dhimahi । dhiyo yo naḥ prachodayāt ॥ OM āpo jyotī raso amṛtam brahma BHŪR-BHUVA-SVAROM ॥). Similar to this connection to the Sāvitrī, the vyāhṛti-s are also incorporated into the important vyāhṛti-homa-s of the Yajuṣ tradition (bhūr annam agnaye svāhā… ityādi) which culminate in the incantation of Indra as the bull among the meters that your wife just mentioned. Notably, the last of the vyāhṛti-homa incantations include the fourth vyāhṛti, mahas. All these not only indicate the pervasive presence of the mahāvyāhṛti-s but also the other vyāhṛti-s in a range of prayoga-s, like say the offerings to Mahārāja in the Āraṇyaka of the Taittirīyaka-s. These strongly favor the idea that the Prājāpatya-s were merely incorporating a widely use mantra-tradition into their philosophizing.”

S: “Good. In this regard I would note that the special vyāhṛti, PURUṢA, of the singers is also used in other recitations. One such is during the expiatory singing of the Vāmadevya Stotra based on RV 4.31.1-3 composed by the illustrious ancestor of our wives. The last of these ṛk-s is short by 3 syllables from the gāyatrī; hence, he should insert the the 3 syllables of the vyāhṛti PURUṢA and recite this ṛk as a proper gāyatrī before the Vāmadevya Stotra is sung. Thus, we have:
OṂ abhī ṣu ṇaḥ sakhīnām PU avitā jaritṝṇāṃ RU । śatam bhavāsy ūtibhiḥ ṢAḤ ॥

Similarly, during the recitation of the Rājana incantation in the nocturnal ritual of the winter solstice we insert a PURUṢA into the intertwining of the ṛk of my ancient clansman Bṛhaddiva Ātharvaṇa and that of Priyamedha Āṅgirasa:
tad id āsa bhuvaneṣu jyeṣṭham PU
nadaṃ va odatīnāṃ |
yato jajña ugras tveṣanṛmṇo RU
nadaṃ yoyuvatīno3m ।
sadyo jajñāno ni riṇāti śatrūn
patiṃ vo aghnyānāṃ |
anu yaṃ viśve madanty ūmāḥ ṢO
dhenūnām iṣudhyaso3m ॥

Bṛhaddiva Ātharvaṇa:
He indeed was the foremost in the universes,
who was born with fierce, mighty manliness
Simultaneously, with his birth, he melts down the enemies
as all his friends [Viṣṇu, Vāyu and the Marut-s] cheer him on.

Priyamedha Āṅgirasa:
At the roaring bull among the eager females,
at the roaring bull among the coy young ladies,
at the lord of your milk-giving cows,
shoot your arrow [in the form of the chant]

As you can see, the intertwining couples a mantra indicating the manliness of Indra with one indicating him as the bull among the females; thus, the vyāhṛti PURUṢA here becomes the seed that is infused into the incantation.”

I: “Somakhya, having gone so far into the realm of the vyāhṛti-s and their prayoga-s, let us return to my original question regarding the roots of the vyāhṛti-s of the Atharvan-s. Where all are they found and what are their prayoga-s?”
S: “Prājāpatya brahmavāda for the AV vyāhṛti-s given in the Gopatha Brāhmaṇa is closely but briefly paralleled by the Jaiminīya Brāhmaṇa, which states that the local worlds was generated in fluid from the three mahāvyāhṛti-s. In contrast, the higher realms of space are said to have been generated from vyāhṛti-s KARAT, JANAT, VṚDHAT and SATYAM. This indicates that there was a wider knowledge and tradition of most of the AV vyāhṛti-s. In terms of ritual, we can see from the Gopatha Brāhmaṇa itself that the prayoga is in the context of the older Aindra religion, in the offering to the Marut-s and other deities who are in the company of Indra (the Indravant-s). Even as the vyāhṛti PURUṢA is deployed within the incantations that we just discussed, one could make a case for a similar wider presence for most of the AV vyāhṛti-s. You noted the emergence of equivalents of MAHAT and JANAD among other vaidika traditions. To that I would add the use other AV vyāhṛti-s in various traditions. For instance, the Vārāha-gṛhya-sūtra of the Maitrāyaṇīya-s specifies the use of the triad karat, janat and bṛhat (unique to that tradition, and semantically mirroring mahat or mahat) in the Garbhadhāna ritual. Coming to the Taittirīyaka-s, we have Iḍa recitations laid down by my ancestor Jamadagni Bhārgava for the drawing of the milk offerings in the Agnihotra as taught by Āpastamba: BHŪR IḌĀ BHUVA IDĀ SUVAR IḌĀ KARAD IḌĀ PṚTHIG IḌĀ ॥ In this tradition, the first three mahāvyāhṛti-s are, as usual, associated with the Agni on earth, Vāyu in the atmosphere and Sūrya in the heavens. Of the two further vyāhṛti-s, karat, matching the AV tradition, corresponds to the moon moving against the backdrop of the nakṣatra-s and the YV-specific PṚTHIK corresponding to the medicinal herbs discovered by Jamadagni Bhārgava. The Āśvalāyana-s have only four drawings of milk, and use vṛdhat instead of karat, again matching one the AV vyāhṛti-s. Interestingly, the Maitrāyaṇīya-s instead use janat in this context. Thus, these vyāhṛti-s are individually used in the other traditions but come together as a whole in the AV tradition and the svāhā offerings with them are specified in the Kauśika-sūtra-s.

This brings us to TAT and ŚAM. I’d posit that those arose from the ancient opening of the Śamyuvāka incantation which is repeatedly mentioned throughout the śruti. Its ancient use is suggested by opening of the Śamyuvāka being sought from Rudra in the ṛk of Kaṇva, the son of Ghora Aṅgirasa in the RV. In regard to TAT, one might also note that it might have a link to its use in the Yajuṣ incantation of the supreme Vāyu: OṂ tad brahma । OṂ tad vāyuḥ । OṂ tad ātmā । OṂ tat satyam । OṂ tat sarvam । OṂ tat puror namaḥ ॥ Finally, coming to the praṇava as an AV vyāhṛti, it seems to be a natural inclusion given the intimate link the mahāvyāhṛti-s share with it mīmāṃsā and prayoga. We see that, for example, in the Upaniṣat statement that identifies them with Indra as the bull among the Chandas, which Lootika just mentioned. Finally, I should also mention one of the homologies that the Atharvan tradition recognizes between these vyāhṛti-s and the wider horizon of texts. Thus, it has janat as equivalent to the compilation of the Āṅgirasa-s — the Āṅgirasa-veda; similarly we have vṛdhat and Sarpaveda, karat and Piśāchaveda, ruhat and Asuraveda, mahat and the Itihāsa-s, and tat and the Purāṇa-s. This perhaps reflects both the growing corpus of texts and awareness of texts of other traditions — like the Asuraveda — it could be some kind of memory of the Iranian texts.”

V: “What are some of the meditations one should be mindful of when performing a japa or contemplation on the progression of vyāhṛti-s?”
I: “The most important one is the japa of the threefold mahāvyāhṛti-s preceded by a praṇava. During this, as in the Vaiśvadeva songs of the Chandoga-s, one meditates on the Vasu-s associating them with BHUR; them one meditates on the Rudra-s associating them with the utterance of BHUVAS; then one mediates on the Āditya-s while uttering SUVAR. With the preceding OṂ, one meditates on Indra or Viśvedeva-s, i.e. the entire pantheon. While moving from one mahāvyāhṛti to another one perceives the connector deities: SUVAR and BHUḤ are connected by Dyāvā-Pṛthivī; BHUR and BHUVAS by Agnī-ṣomā; BHUVAS and SUVAR by Vātā-Parjanyā. When uttering the five vyāhṛti-s, i.e., mahāvyāhṛti-s + TAPAS and SATYAM one additionally meditates on the primal heat from which all arose and the very nature of existence. May be Somakhya could add more while returning to our starting point of the AV vyāhṛti-s?”

L: “Also, before rounding up this discussion it would be worthwhile if you could touch upon some of the mīmāṃsā-s on the different sets of vyāhṛti-s that are not widely aired by the extant brahmavādin-s focused on Prājāpatya and Uttaramīmāṃsā traditions.”
S: “Sure. Their connection to the Sāvitrī and the god Savitṛ is the most apparent one. The śruti holds that the inviolable laws of Savitṛ, like the probabilities of the draws of the vibhīdaka nuts from the hole, are the ones which run the universe: deva iva savitā satyadharmā: like the laws of the god Savitṛ that hold true. The mahāvyāhṛti-s illustrate their most apparent domain of action: the near realm, the mid-region and the realm of the sun. The more expanded set of seven vyāhṛti-s yoked to the Sāvitrī indicate their broader sphere of action — Mahas: the wider space. Then we move into the temporal axis: janas: the origin of space itself. What drives its emergence? tapas: heat. Finally, the very fact that something exists: satyam: also expressing the inviolable or true nature of the laws of Savitṛ, the ṛta. Indeed, Kṛṣṇa Āṅgirasa states:
ṛtena devaḥ savitā śamāyata
ṛtasya śṛṅgam urviyā vi paprathe ।
By the natural law the god Savitṛ exerts himself,
[by that] the antler of the natural law has spread widely.

The Samaveda adds the vyāhṛti, PURUṢA, after SATYAM. This may be seen as the root of the concept that was later expanded in Sāṃkhyā — the Puruṣa as consciousness. In placing the final Puruṣa, the singers posited a system in which the laws and existence itself might be objects in the conscious experience of the sole reality, the Puruṣa. The next notable mīmāṃsā of the vyāhṛti-s pertains to the way we deployed the mahāvyāhṛti triad with the three feet of the AV mantra-s to Viṣṇu. This connection is declared by the Jaiminīya-s, who state that the vyāhṛti-s were offered to Viṣṇu. The three feet of the deployed mantra-s indeed correspond to the three steps of Viṣṇu, who appeared as a dwarf and suddenly grew to a gigantic size to conquer the worlds from the Dānava-s with his famed triple strides: bṛhaccharīro vimimāna ṛkvabhir yuvākumāraḥ praty ety āhavam ॥ As the founder of your race, O Gautamī-s, states in the primal śruti, that gargantuan form of Viṣṇu is said to measure out the worlds with the ṛk-s — those are the corresponding AV ṛk-s we deploy conjoined to the vyāhṛti-s. The Kāṇva-s further add that those steps were the ones with gathered the atoms — samūḍham asya paṃsure ॥ — from which the universe condenses. Hence, while uttering that incantation the ritualist meditates on the great Viṣṇu stamping out the Asura-s and likewise calls on him to exclude his rivals from his space. This association with the vyāhṛti-s also extends to Viṣṇu’s wife in the incantation seen in the Mahānārāyaṇopaniṣat of the late AV tradition that you all know very well:
OṂ bhūr lakṣmī bhuvar lakṣmīḥ suvaḥ kālakarṇī tan no mahālakṣmīḥ pracodayāt ॥
This incantation is notable in placing Kālakarṇī in suvar. This is from her association with visible time in the form of the apparent movement of the sun in the sky. As you all know well from your Āgamika practice she is none other than the gigantic, dreadful, fanged, death-dealing eponymous goddess, armed with a bow, arrows, axe, sword, cakra, trident and a cleaver, emitted by Rudrāṇī from her mouth to terrorize the gods for their support of Prajāpati Dakṣa. The fourth vyāhṛti mahat is subliminally hinted by her name Mahālakṣmī, encompassing the wider space.

Coming the the AV vyāhṛti-s, the coupling OṂ BHŪR JANAT ॥ expiates ṛk errors and is offered in the Gārhapatya; OṂ BHUVO JANAT ॥ expiates yajuṣ errors and is offered in the Dakṣiṇa; OṂ SVAR JANAT ॥ expiates Sāman errors and is offered in the Āhavanīya; OṂ BHŪR BHUVAḤ SVAR JANAD OM ॥ expiates Atharvan errors and is also offered in the Āhavanīya. Here the JANAT is seen as regenerating the flawed incantations. Vrishchika, it may interest you that the AV tradition uses a metallurgical analogy of fusing metals for the welding role of these vyāhṛti-s, unlike the medical analogy of the Aitareya that you noted. The AV also holds that the same combination of vyāhṛti-s are the incantations uttered by the brahman before he asks the udgātṛ to sing the stoma to the god Bṛhaspati in the somayāga. Likewise he utters the entire gamut of vyāhṛti-s OṂ BHŪR BHUVAḤ SVAR JANAD VṚDHAT KARAD RUHAN MAHAT TAC CHAM OM when urging the udgātṛ to sing the song of the Indravant-s derived from the famous Evayāmarut ṛk to the Marut-s and Viṣṇu that was composed by your illustrious ancestor, O Indrasena (pra vo mahe matayo yantu viṣṇave marutvate girijā evayāmarut । pra śardhāya pra yajyave sukhādaye tavase bhandadiṣṭaye dhunivratāya śavase॥). As an aside that mantra-s is notable in more than one way but you may note the phrase girijā — Viṣṇu emerging from the mountain — a mythologem that presages his emergence from the pillar in the later Nṛsiṃha cycle — but here he emerges with the marching troop of the Marut-s to head for battle, evidently to join Indra in the battle against the Dānava-s.

In the purely AV performance, as we did earlier today (or in the muttered incantation of the brahman), it is deployed with the Indravant ṛk which illustrates the connections to various vyāhṛti-s. With Agni we are connected to Bhūr, with Vāta (Vāyu) Bhuvas, with Viṣṇu, the realm of the Āditya-s. All the special vyāhṛti-s can be seen as having deep connections with Indra and the Indravant-s. As we saw before the two praṇava-s at the beginning and the end are the mark of Indra. Janat indicates the emergence of Viṣṇu and the Marut-s from the mountain, which is a metaphor for the world axis — thus on one hand it represent the origin of time that Viṣṇu manifests as. On the other birth of the Marut-s that made the universe manifest. That manifestation and the growth of the universe, which is how the sons of Rudra manifest is indicated by Vṛdhat. Karat is action of filling the universe, as the ancient Bhārgava, Uśanas Kāvya, is quoted by Agastya Maitrāvaruṇi: karat tisro maghavā dānucitrā : Maghavan made three realms fill with glistening droplets. Ruhat, stands for the ascendance of the gods, manifesting as the rising sun in which they are worshiped. Mahat, as we saw before represent the great expanse of the universe. Finally Tat and Śam are the bliss that one attains from the gods upon the success of the ritual.”

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Two exceedingly simple sums related to triangular numbers

This note records some elementary arithmetic pertaining to triangular numbers for bālabodhana. In our youth we found that having a flexible attitude was good thing while obtaining closed forms for simple sums: for some sums geometry (using methods of proofs pioneered by Āryabhaṭa which continued down to Nīlakaṇṭha Somayājin) was the best way to go; for others algebra was better. The intuition was in choosing the right approach for a given sum. We illustrate that with two such sums.

Sum 1 Obtain a closed form for the sum: \displaystyle \sum_{j=1}^{n} (2j-1)^3

These sums define a sequence: 1, 28, 153, 496, 1225…
Given that we can mostly only visually operate in 3 spatial dimensions, our intuition suggested that a cubic sum as this is best tackled with brute-force algebra with the formulae for individual terms derived by Āryabhaṭa and his commentators. Thus we have:

\displaystyle \sum_{j=1}^{n} (2j-1)^3 = \sum_{j=1}^{n} 8 j^3 - 12 j^2 + 6 j- 1
= 2n^2(n+1)^2-2n(n+1)(2n+1)+3n(n+1)-n= \dfrac{(2n^2-1)2n^2}{2}

The reason we wrote out the final solution in this unsimplified form is to illustrate that the above sums will always be a triangular number of the form:

\displaystyle \sum_{j=1}^{2j^2-1} j, i.e sums from 1 to 1, 7, 17, 31, 49… or triangular numbers T_1, T_7, T_{17}, T_{31}\cdots

Thus, the nth terms of sequence of sums would be triangular number T_m, where m=2j^2-1, j=1, 2, 3.... From the above, one can also see that the difference of successive terms of our original sequence of sums will be 27, 125, 343, 729…, i.e., they are perfect cubes of the form (2k+1)^3 (odd numbers 3, 5, 7, 9…). These cubes are thus the interstitial sums of the indices j of the triangular numbers T_j up to the index m corresponding to the triangular number T_m that is a term of our original sequence. Thus:
j\mapsto 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31
then, 2+3+4…+7=27; 8+9+10…+17=125; 18+19+20+21+22…+31 =343 and so on.

Another interesting feature of the original sequence is its decadal cycle in the terms of the numbers in the last 2 places (written in anti-Hindu, i.e. modern order). They will always end in the following sequence of 10 numbers:
0, 1, 28, 53, 6, 25, 6, 53, 28, 1
Similarly, the index m of the triangular numbers T_m the define our sequence also shows a pentadic cycle in the last place of the form:
1, 7, 7, 1, 9

A comparable pattern is seen if we generate a sequence that is the sum of successive terms of our original sequence: 1, 29, 181, 649, 1721, 3781, 7309, 12881… The last place has a pentadic cycle of the form: 1,9,1,9,1. The last 2 places has a cycle of length 25: 01, 29, 81, 49, 21, 81, 09, 81, 69, 41, 61, 89, 81, 89, 61, 41, 69, 81, 09, 81, 21, 49, 81, 29, 01. Both are palindromic cycles.

Finally, the sum of the reciprocals of the original sequence converges to a constant: 1.04607799646… We suspect there is a closed form for this constant but have not been able to identify it.

Sum 2 Obtain a closed form for the sum of alternating negative and positive perfect squares: -1+4-9+16… i.e.

\displaystyle \sum_{j=1}^n (-1)^j j^2

With the sum involving just square terms it is possible to use a wordless geometric proof along the lines of that proposed by Āryabhaṭa (Figure 1).


Figure 1.

Thus, we get the above sum as -1^n T_n, where T_n is the nth triangular number.

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Pandemic days: Vaccines and war

In American history-writing we come across various attempts to the justify the use of nuclear weapons on Japan in the closing phase of WW2. We often hear the claim that by using the nukes they avoided a large number of casualties that they would have suffered in a long-drawn conventional war to conquer Japan. Neutral outsiders who have studied the matter realize that this is merely the American narrative to justify and positively spin something, which many of their own people (some leaders included) found rather disturbing. A closer look indicates that the Japanese were brought to the brink of surrender by the demolition they faced at the hands of the Rus in Manchuria. Indeed, the Rus were poised to invade the main islands and probably kill the emperor of Japan. Faced with this, the Japanese calculated that surrendering to the Americans might help them save the emperor and perhaps avert a more brutal assault that the Soviet military would have subjected them to. Were the Americans aware of this? While we rarely hear anything pointing in this direction in the many American presentations of these events, it seems very likely to us that the Americans were fully aware of the situation. Hence, we posit that the reason the Americans used the nukes on densely populated Japanese cities was to graphically demonstrate to the Rus what the “super-weapons” in their possession could do and that their leaders were dead serious about earlier hints they had given the Rus. Hence, the intended audience for nukes was likely the Soviets rather than the Japanese. This was one of clearest examples of a technological change of game in times closer to our own. The Rus and other nations eventually developed their own nuclear weapons despite the American attempts to prevent some of them from succeeding. However, we do think that being the first to make and use the nukes contributed in a big way to the American rise to superpower status.

There are indications that the rising new religion of Navyonmāda might be turning that arc of American superpowerdom to towards the horizontal; however, even in these days of late empire the accumulated technological capital of the superpower could make a major difference in the “war of the day”. We had briefly alluded to this in the past note. In this note, we shall expand a bit on that. Right from when the Wuhan corruption exploded among the Cīna-s, it was clear that the disease was being used as a geopolitical tool. With hindsight we can say that the Cīna-s knew early on that it was a respiratory disease likely transmitted by the airborne route. Despite all the show of disinfecting the streets of Wuhan with chemical sprays, early in the epidemic they were wearing masks and practicing social distancing. However, they did little to stop international traffic despite restricting internal traffic between cities with high infection. Moreover, via the WHO, they initially held back information on human-to-human spread instead pointing to wet markets. Then they used the WHO to project messages about washing hands rather than wearing the appropriate masks (Sure, early the cruise ship incidents had a norovirus-like fomite feel to it, but the Cīna-s knew better). Thus, even as they facilitated an information blackout to the rest of the world, within their own borders they took draconian measures of isolation and blockade to curb the disease. As a result, even as the left-liberal occidental media was running paradoxical pieces on how “totalitarian” governments fare worse with pandemics and how Vijayanāma-vyāpārin was being xenophobic towards their Galtonian partners, the virus breached their defenses and established itself in their midst. The result was waves of massacres that really made the occidental powers look limp. At the end of it, emperor Xi had deftly leveraged the epidemic in his lands to hit his rivals while hiding his own losses from it.

vaccinesFigure 1

With the virus established in their midst, both the Cīna-s and the mleccha-s soon realized that lasting victory could only be achieved by an effective treatment — vaccination being the method of choice for the long-term. Here is where a technological race, like the one to make the nukes, came to the fore. It was not easy, given that human coronavirus vaccination programs (like those inspired by SARS) had not really reached their culmination as the disease had been curbed by public health measures well before a vaccine became necessary. Figure 1 shows the popular vaccines in use or close to deployment (by no means comprehensive) classified by method, along with the country that developed them. One can see that the Americans were able mobilize multiple vaccines based on “advanced methods” — i.e. those using artificial mRNA with modified nucleobases, adenovirus vectors and baculovirus expression systems. The most basic of these technologies, i.e., cloning of the gene for the viral spike protein, can be easily mastered. However, to develop a truly successful vaccine, there is a lot more knowledge and technology that needs to be in place. These include: 1) the knowledge of and a repository of vector viruses like the Human Adenoviruses 26 and 5, or the Chimpanzee Adenovirus in the vector-based vaccines. 2) the capacity for chemical synthesis of nucleic acids for producing codon-optimized genes. 3) A knowledge of protein structure and evolution to produce optimal S protein constructs to be used as vaccines. 4) In the case of mRNA vaccines, the knowledge of and the capacity to synthesize modified nucleobases. The de novo development of these vaccines need a long-standing and well-developed culture of molecular biology and biochemistry. The totality of this knowledge is possessed by only a few nations in the world. Thus, developing the vaccine indigenously from scratch is not possible for most of the world. This fact in itself can be weaponized in a pandemic situation to gain a geopolitical advantage. It is in this regard that the superpower capital accumulated by the USA remains unchallenged.

Of the others, the British managed to successfully develop an adenovirus-based vaccine, showing that their accumulated intellectual capital still powers some technological propulsion in crisis. While we do not know as much regarding the success of the Russian attempt from external trials, they too seem to have achieved something comparable to the Brits with their Sputnik vaccine. Their subunit vaccine seems to deploy a rather unusual concept and its true efficacy remains entirely unclear to us. Still the gulf between these and the multiple American successes remains, illustrating the distinction between the great powers and the superpower. Several other nations possess the scientific and technological capacity to develop vaccines by themselves. In the Orient, we have Japan and Korea. In the Occident we have Germany and France. None of these have managed to develop and deploy their own vaccines to date. Some of them are even facing the adverse edge of not having a suitable vaccine that they can use. This hints that the task at hand it not easy in practice, even if a nation were to possess the theoretical know-how.

The Cīna-s have shown great prowess in molecular biology in recent times. A closer look at their research capacity in this regard has shown a tendency for plagiarism, faking and imitation of more original work coming from elsewhere. However, as the Americans say, you can fake it till you make it. Keeping with that, the Cīna-s have recently managed some pieces of high-end original research suggesting that they are coming of age. However, this is not visible in term of the vaccines that they have managed to deploy — to date they have only managed the conventional inactivated viral vaccines. There are suggestions that they have been trying to pilfer more advanced technologies and reverse engineer them, but we are yet to see the results of those attempts. Thus, the head-start the Cīna-s had with the virus has not really translated into vaccinological success. Finally, coming to India, we infer that the leadership correctly realized the danger posed by the virus to a populous country with little scope for urban social distancing and went for obtaining a vaccine as soon as possible. Perhaps, they correctly judged that the Indian biotechnological capacity was not up to the mark of developing any of the advanced vaccines indigenously in time. However, they did leverage the same low-tech solution as the Cīna-s to develop the indigenous inactivated virus Covaxin vaccine. Wisely, in a parallel track they purchased a stake in the AstraZeneca adenoviral vector vaccine developed by the Brits and the American subunit vaccine Novovax for local manufacture.

Next we come to the question of how these vaccines actually fared on the ground. The American and British image had taken a heavy beating at the hands of the virus by early 2021. The US had stacked up nearly a million deaths from the virus in an year (with almost 1 in 10 Americans being infected), while the count in UK is at least 200,000 (probably 1 in 12-15 people have been infected, keeping in mind their poorer accounting of cases ). However, both these nations have flattened their curves and have gone a long way towards mitigating the pandemic in their lands. This is in no small measure from the success of their vaccines — the capacity to develop and deploy them in time. The confidence in this success in the US is reflected in the recent CDC statement relaxing the use of masks among the fully vaccinated. From the viewpoint of both cases and fatalities, France and Germany have done poorly with respect to their island counterpart — to us this is a clear indication of their failure at vaccine deployment. A similar situation is seen with Poland — a western aligned Slavic nation. The Russian situation is harder to assess. Despite their Sputnik vaccine being apparently successful (as per their published papers) they have had no success in bringing down their deaths significantly from mid-February to mid-May 2021. The causes for this remain unclear to us.

Coming to the Cīna-s, they saw immense potential to use vaccine-diplomacy to leverage their head-start with the virus and the pandemic they had helped create. They sent their vaccines to all takers but as of date of this note there are no great results to see. Recently, a good comparison has come up in the form of two countries, the small Israel and the tiny Seychelles. Given the ties the CEO of Pfizer company has to Israel, they were able to obtain that vaccine right away. The latter received the Sinopharm (majority) and AstraZeneca (minority) vaccines. The former managed to control the epidemic within their borders with their mass vaccination program, whereas the later has so far failed to do so despite fully vaccinating 60% of its people. In large part this seems to stem from the lackluster performance of the Cīna vaccine. UAE, which also deployed this Cīna vaccine, is now thinking of going for a 3rd dose to improve immunity. The results of Sinovac in the field are not inspiring confidence either. Further evidence for the Cīna failure comes from the statements of emperor Xi asking for international collaboration on vaccines. Why would he want “collaboration” if his “guns” were firing alright? This generally poor performance of the conventional inactivated virus vaccine raises questions about how the Indian Covaxin would fare in the field — we still await the official publications in this regard.

A recent study by Khoury et al indicates that the modified mRNA vaccines developed in the US provided the strongest neutralizing-antibody response, whereas the AstraZeneca vaccine providse a lower tier response. Moreover, it also appears that the American vaccines are likely to provide sufficient (severe disease/death) protection against the B.1.351 South African strain whereas the AZ vaccine might be far less efficacious in preventing infection by that strain. In conclusion, the vaccine race has left the Americans as the clear winner both in terms of currently possessing the best vaccines (and an abundance of them) and having reasonable success in controlling the disease (As of the date of writing, the US still has a 7 day rolling average of over 500 deaths daily but we suspect in large part this can be contained if more people were proactive in getting the vaccines and observing disease-limiting behavior). This allows them to weaponize the vaccine in geopolitics.

It is precisely this point which brings us to the Indian situation. India began by handling the first wave reasonably well. This was followed by a good start to the vaccination program among elderly people with the AZ vaccine. Then we saw the Indian version of vaccine diplomacy, where the mass manufacture of the AZ vaccine was used to distribute it to several small countries, including those in the Caribbean. The overconfidence and behavioral recklessness (mask laxity and vaccination hesitancy) which ensued, along with ignorance of the function y=ke^{rx}, poorly managed testing and contact-tracing, Khalistani rioting in the Panjab spreading the British B.1.1.7 strain, and the emergence of the (likely) more virulent B.1.617 strain resulted in the brutal second wave. Evidence from both within India and the UK indicates that the B.1.617 strain can displace other ambient strains, making it particularly dangerous. While this strain breaks past the AZ vaccine and causes disease, that vaccine seems to be capable of preventing death/serious disease in most vaccinated people. Thus, it was of paramount importance for India to ramp up vaccine production and vaccinate as many people as possible. While we admit that is a difficult task for a big country, we feel that the inability to keep up the vaccination program as a percentage of the population was a one of failures on the part of the nation.

The question which then arises is what the proximal cause for this might be. In our opinion, a major reason for this was the embargo placed on raw material by the Mahāmleccha led by Vṛddhapiṇḍaka. If one were to game Mahāmleccha geopolitical realism from it is foundational principle, it is obvious they would do everything to limit any rival potentate that aspires “great power” status. Even if it is regional player, it has to be broken if it marginally challenges the pañcanetra mleccha power. Moreover, Vṛddhapiṇḍaka has been placed in power by Big Tech and navyonmāda which has a svabhāva-vairam with all things H. Thus, H power, however, limited in the big picture is not something they tolerate — they were in particular pricked by Indian vaccine-diplomacy in their own hemisphere. Thus, they decided to use their victory in the vaccine war to settle scores with the Lāṭeśvara by limiting resources during the critical phase of the second wave in India. The job has been done as it took quite a bit of the sheen of the nation and showed it to be no better than a third-rate power, leave alone aspirations of “great power” status. Importantly, it has also taken the sheen of the the Lāṭeśvara, even among those generally supportive of him. The evidence for the mleccha hand is further supported by the active subversion program by Big Tech (Jāka-Bejha-Mukhagiri-Dvārādiduṣṭāḥ), Soraduṣṭa and the first responders dove-tailing their action with this wave.

That said, the deeper problem is the failure of the H to learn from history. Such perfidious mleccha action has been seen time and again — for instance the mleccha mercenaries hired by the Marāṭḥā-s or the sale of defective weapons by the English to them. Hence, the H leadership should have been prepared for mleccha action against them, especially with overthrow of Vijaya-nāma-vyāpārin in their land by the navyonmatta-s. They should have prepared to source key materials to keep themselves afloat in the vaccine war. The production failures are seen more generally with things like antifungals (e.g. amphotericin B) which are needed to tackle the ongoing epidemic of mycoses accompanying the Wuhan disease. An even deeper issue is the regard for research in India. Before the pandemic there was a generally dismissive attitude towards doing hard original science (not scientism or show-science) among the H. Instead, most people with such capacity were being funneled to quite a degree into service industries that do not ultimately make a nation a “great power”. You cannot build up scientific capacity to do hard stuff overnight and the results can be seen. The physicians and nurses on the ground are limited if there is no scientific capacity holding them up.

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Generating simple radially symmetric art

Many people experience beauty in structures with bilateral, radial and rotational symmetries with or without recursion. The recursive or nested structure are the foundation of the beauty in fractal form, the generation of which has become increasingly easy for the lay person with ever-improving computing power. One could generate beautiful fractal structures using a range of open source software; however, there is no substitute for writing ones own code and taking in some of the mathematics behind the beauty — truly fractal structures provide the clearest bridge between mathematics and beauty. While we have presented some discussion on such structures on these pages, that is not the topic of this note. Here, we shall talk about stuff that is mostly art for art’s sake (We fully understand that what constitutes art can have some subjectivity) that is generated based on simple repeats of certain motifs with an emphasis on radial and rotational symmetries.

For at least three generations, there has been a strand in our family with an interest in generating such art. While there certainly exist people with much greater skill than us (you can even see manifestations of genius in this regard), the driving force for us is the pleasure derived from process of generating such art. One experiences a climax, when the process of polishing the work culminates in a first person experience of beatific satisfaction. In the two previous generations, the main medium was the powder (rice flour, stone and other colored powders) used in traditional alaṃkāra. In our case it began with spending time in our youth with a kaleidoscope. That inspiration was then transferred to paper, pen and compass but eventually it transitioned to computer-aided in silico tools. Over the years we have used many tools each with its own advantages and disadvantages. The first programs we used were CorelDraw and Canvas. The latter, at that time, was available to only on a Mac. It was a decent program but expensive. Moreover, we never owned a Mac, and using it on a public or a borrowed Mac was hardly convenient. Hence, it fell to the way side. I continue to use CorelDraw for professional stuff, especially if the work needs freedom of the hand and has some complexity; however, it is expensive and a typical user might only be able access it via a funding agency. Then the open source Inkscape came along, which evolved to be a reasonable free substitute for CorelDraw. Although CorelDraw is “smoother” to use, the current version of Inkscape is not bad at all.

However, we wanted something more “programmable” where one could adjust various numerical parameters rather than going freehand — a language for graphics. The first such we looked at was MetaPost — it had, what to us were unfriendly aspects; however, the time we spent exploring it was not a total waste because in the second decade of the 2000s of CE we learnt of the existence of the PGF/TikZ (ironically named: “TikZ ist kein Zeichenprogramm) languages that greatly improved on MetaPost in our subjective opinion. Notably it could be used from within \LaTeX. Thus, we finally settled on TikZ as the language to write these pieces of art in. Following is an example of such with the compiled result appended below.

\usetikzlibrary{arrows, arrows.meta, patterns, shapes.geometric, decorations.shapes, shapes.misc, graphs, mindmap, calc, backgrounds}


\definecolor{col1}{RGB}{2, 35, 54}
\definecolor{col2}{RGB}{15, 184, 184}
\definecolor{col3}{RGB}{178, 209, 107}
\definecolor{col4}{RGB}{199, 186, 99}
\definecolor{col5}{RGB}{174, 137, 199}
\definecolor{col6}{RGB}{59, 148, 126}
\definecolor{col7}{RGB}{77, 148, 255}
\definecolor{col8}{RGB}{230, 229, 202}
\definecolor{col9}{RGB}{61, 69, 67}

\foreach \x in {0,36,72, ...,324}{
%wavy background
\draw[col1, fill=col1, rotate=\x, scale=.8] (0,0) -- (22.5:3.5) to [bend left=40] (-22.5:3.5) -- (0,0)--cycle;

%wavy dots
\draw[decorate, decoration={shape backgrounds, shape=circle, shape size=.8mm, shape sep=1.512mm}, col3, fill=col3, rotate=\x, scale=.77] (18:3.5) to [bend left=40] (-18:3.5);

\draw[col4, fill=col4, rotate=\x+18, yshift=2.2cm, scale=.2] (-1,1) ..
controls (-0.5,0.5) and (0.5,0.5) .. (1,1) .. controls (1.5,2) and (0,2) .. (0,2.5) .. controls (0,2) and (-1.5,2) .. (-1,1) --cycle;

\draw[col5, line width=1.5, rotate=\x, yshift=1.75cm, scale=.75] (0,1) .. controls (-0.5,0) and (0.5,0) .. (0,1) --cycle;

\begin{scope}[rotate=\x, xshift=2.1cm, scale=.3]
\draw[col7, fill=col7] (0,-1) .. controls (-0.5,-1) and (-0.5,1) .. (0,1) ..controls (-1,1) and (-1,-1) .. cycle;
\draw[col7, fill=col7] (0,0) circle (.25);
%pipal leaf
\draw[col8, line width=1, rotate=\x, xshift=1.35cm, scale=.15] (0,0) .. controls (-1,0.5) and (0,1.5) .. (1,1) .. controls (2,0.5) and (1.05,0.1) .. (2.1,0) .. controls (1.05,-0.1) and (2,-0.5) .. (1,-1) .. controls (0,-1.5) and (-1,-0.5) .. (0,0)--cycle;
\begin{scope}[col3, rotate=\x+18, xshift=1.3cm, scale=.5]
\draw[line width=\w] (0,0) to[bend right=\y] (1,\z);
\draw[line width=\w] (0,0) to (1,0);
\draw[line width=\w] (0,0) to[bend right=-\y] (1,-\z);
%dot in petal
\draw[col8, fill=col8, rotate=\x+18, xshift=2.1cm] (0,0) circle (.05);

%dot in pipal
\draw[col8, fill=col8, rotate=\x, xshift=1.46cm] (0,0) circle (.05);

\draw[col6, fill=col9, line width= 3, scale=.75] (-1.5,0) .. controls (-1.5,-2) and (2,1.5) .. (0,1.5) ..controls (-2,1.5) and (1.5,-2) .. (1.5,0) ..controls (1.5,2) and (-2,-1.5) .. (0,-1.5)..controls (2,-1.5) and (-1.5,2) .. cycle;

\foreach \x in {0,90,180,270}{
%releaux triangle
\begin{scope}[rotate=\x, xshift=.75cm, scale=.25]
\draw[col3, fill=col3] (-.5, -0.8660254) to[bend right=\y] (1,0) to[bend right=\y] (-.5,0.8660254) to [bend right=\y] (-.5, -0.8660254) -- cycle;
\draw[col1, fill=col1, xshift=.07cm] (0,0) circle(.3);

%central circles
\draw[col2, fill=col2] (0,0) circle (.45);
\shade[inner color=col8,outer color=black] (0,0) circle (.25);

Figure 1.

This example uses decadal symmetry with central tetrad element. In our subjective experience tetrad symmetry can be paired with other even symmetries as long as they central or the exterior most elements.

Figure 2.

Ideally all repeated motifs should have at least bilateral symmetry. However, one can get away with a layer or two of elements with just rotational symmetry, like the “S” element in Figure 2. The choice of color is another very important element — we like a degree of contrast in all the piece. Appended below are a range of productions illustrating different color choices.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

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The phantoms of the bone-pipe-2

Vidrum had been introduced to a synesthetic patient by a neurologist colleague. The patient’s manifestation of synesthesia left a rather profound impact on him; thus, when he had a break of an hour in his duties, wanting to explore the issue more, he decided to do some quiet reading on his computer in the library. He took his favorite seat beside the window looking out into a sylvan patch, and thought to himself: “This display of synesthesia would have interested Vrishchika a lot.” Even as he said so to himself, to his utter shock, he saw someone looking just like Lootika or Vrishchika go past him and take a seat another a little ahead. He almost exclaimed aloud: “that cannot be true! they are away in a faraway land enjoying the pleasures of conjunction with their puruṣa-s.” He looked at the girl again and realized it had to be Jhilleeka. “What is she doing here? This is not her kṣetra.” He walked up to her: “Hey Jhilli, what are you doing here. For a moment, I thought it was one of your sisters who had manifested themselves using their ghostly powers.” Jhilleeka smiled but seemed to be a bit at a loss to say anything. Vidrum went on: “I’ve not seen you in a while, but you have become a light-eyed version of your sisters.” Jh: “I’m taking that to be a compliment, but it must be just some homozygosity that found its way into me. Hope you are alright and fully recovered from the tumultuous events that are now past us.” V: “Let the past lie. But are you alright?” Vidrum pointed to her bandaged foot.

Jh: “Sort of. But that wound is the result of what one could call an adventure of sorts.” V: “What happened?” Jh: “If you run into my parents, do not tell them the whole thing. But if you want to hear the story, we can go outside.” V: “Sure.” By some fancy, I decided to take the shortcut through Viṇmārga back home from the university, which, as you know, passes through the rougher part of the city. From there, I took the bylane that leads to Mahiṣamūtra-mārga, and I quickly sighted that a knot of roughs led by Sphicmukha and Chāgalaṇḍa, who I believe were your classmates at school, had laid a predatory trap. They had set large branches of Vachellia bramble on the road to funnel drivers onto the side they had littered with nails and other metal objects. I avoided them from puncturing my bike but, in a flash of indiscretion, decided to kick one of the pegs out of the way while still riding — that did not go well, and I ended up with a deep cut on my leg. That’s why I came here to get a tetanus shot. I’m just waiting for my father to get back home.” V: “Some of the characteristic traits of your elder sisters are very much visible in you too. In any case, is this not your last month in college? I’m sure you are pursuing graduate school like your elder sisters.”

Jh: “That is correct. Prachetas and I did contemplate whether to start a company of our own and give grad school a skip given the deteriorating environment in educational institutions abroad. Thankfully, we got over that delusion after some further thought as we realized we were v1s to our core and not v3s by nature. Nor were we inclined to take the job offers we had because we have that independent academic streak. Moreover, we got into the same graduate school; so, we would not be facing the separation and the disjunction that Lootika and Somakhya faced when in graduate school.” V: “But those two told me that their disjunction was a key part of their character-building experience where they proved their individual worth by themselves.” Jh: “That may be so, but you can hardly deny that Lootika and Somakhya had the luck of being together all the way from school to the end of college — something the rest of us did not have the pleasure of. So that disjunction is not something they needed to cry about. Moreover, for all the ice in that period, the rest of us knew how Lootika kept thinking about Somakhya. We thought grad school is a good idea because, apart from the fact that we are fundamentally academically oriented, it gives an opportunity to prove ourselves under hostile fire. If luck, for some reason, doesn’t favor us, we can always quit it and get down to more real things like furthering our genes. If everything fails, in the least, I can teach my nephews and nieces mathematics and computation as they come of age.”

V: “Well, good luck. But even if you don’t start a company, I hope someday you and maybe Prachetas will build for me a robot that is like the late lamented Meghana.” Jhilleeka chuckled: “What purpose would such a robot serve even if we were to build it? Talk to Lootika and Somakhya — they will teach you a mantra of the Southern Path. Meditating thus on Śiva and Bhairavī, you will attain your desire more satisfyingly.” V: “Young lady, they are far away, and we are not in frequent touch. It is hard to find a time that matches to talk to them — let alone them imparting me a mantra. Moreover, do you even believe such mantra-s work?” Jh: “Well, you will learn if they work or not. Never mind if you don’t want to ask them about this, there is always the musical bone-pipe my sister Vrishchika gave you.” V: “Jhilli, do you think Meghana could be summoned that way?” Jh: “She may visit you that way, but it is not something you want to look forward to. It will not be pleasant. But the other phantoms who come to you via the bone-pipe might help you reach the equilibrium you seek.” Just then, Jhilleeka got her father’s call, and Vidrum had to return to his duties.


Lootika’s medallion

The following Saturday, Vidrum’s duties had ended by the afternoon; early that evening, he went with Sharvamanyu and Abhirosha for dinner. A: “So Vidrum, when will the construction of your new house begin?” V: “Sadly, it is not happening! If anything, my luck seems to remain the same” S: “But why — were you not set to hire the contractors?” V: “It is a strange story. But I decided it is better safe than sorry.” S: “I don’t get it! But if you don’t want to share it with us, fine.” V: “Oh no! I would gladly do so, but you may think I am crazily superstitious.” A: “Now that makes us even more curious.”

V: “Have you ever been gifted something interesting by the four sisters?” A: “Now, why do you ask that of all things? But yes.” Abhirosha pulled out an inlay-work medallion from her bag and showed it to Vidrum. “This is some art Lootika made for me. I realized that it was more than just art because she said that there was no need to display it but to just keep it with me, somewhere close.” Saying so, Abhirosha handed it to Vidrum. Looking at it closely, he passed it to Sharvamanyu: “It has a nice feel to it. Our friend has some eye for symmetry. I saw Vrishchika making something similar for her husband Indrasena shortly after the tumultuous events. What I received was something more sinister — a bone-pipe made from a human femur. Vrishchika gave it to me. But they all seem to know of it, for the other day, the youngest Jhilleeka asked me to ply it.”

A: “Now, how did you run into Jhilli?” V: “Well, that is a story of its own.” S: “Fine, but what does all this have to do with your abandoning the construction of your new house.” V: “Listen, it is a long and crazy story. If you blow into that bone-pipe you can get nice and haunting tunes. The haunting part is very real — it is not at all uncommon for a phantom to manifest thereafter and tell you something. The short story is that I was rather depressed with my luck that day and vocalized that matter to Jhilli. She reminded me of the bone-pipe her sister had given me and asked me to ply it.”

S: “OK, that sounds like an interesting object — a blast from the past — you never showed it to us?” V: “Well, I’ll show it to you guys the next time you’re home. I had put it aside, given all the trauma from the last visitation. But I realized paying attention to those visitations can actually be helpful. The encounter I had was somewhat dramatic.” A: “Ah! This sounds like the old times. Tell us the story.” V: “Sure. I blew into the pipe a song I heard in a movie — I’m sure it was one I had watched with you guys. The phantom came on very fast. I had hardly blown out a couple of lines, when I heard a gruff voice with a south Indian accent. I did not see anything, but I could feel an obvious presence. He asked to be seated on the couch across from my desk, saying that he needs a proper seat to ease his distress. I took a dictation of his story that I’ll read out once we are done with dinner.” It went thus:

“My name is Gunottaman (Guṇottaman), but most of the people who knew me called me Kāttutĕran, a moniker I acquired from my capacity to drive my father’s car at incredible speeds even as a ten-year-old. My family hails from the Dravidian country but had moved to the Karnāṭa country. While we came from a brahminical background, my father was the last in our lineage to have a slung a thread on his shoulder. He was a man of vision and modernity. He told us there was nothing to be gained by studying supernatural śloka-s and songs with which the brahmins earned a living by fooling gullible people. Instead, he said we should choose the Buddha, the Christ and Mahatma Gandhi as role models for leading a good and ethical life. In my teens, I read a little information pamphlet and added a new figure to that pantheon. He was the great biochemist Yerrapragada Subbarow. I was inspired by him to discover new drugs. Accordingly, I studied for a B.Sc. in chemistry and a further degree in chemical engineering from a reputed college.

Shortly after that, I became acquainted with a biologist known as Ayyangār. He had identified an amoeba-killing compound from an actinobacterium but did not know what it was. He saw the potential for making it into a treatment for amoebiasis, which raging in some villages. Having obtained a grant, he teamed up with me, and I showed that it was a peptaibol. Eventually, I even synthesized the peptaibol, which earned me a thesis and many accolades, including an invitation to work at a Japanese university. Having cleared that hurdle, I was now renowned as a double Ph.D. and was offered a professorship at a college. A couple of years into that, I realized that the humdrum teaching of dullards was not for me. I wanted to emulate my heroes and do good to humanity. I wished to make pharmaceuticals, but that path was not easy for a man with a modest income. By then, I was married and already had two children, and a third was on the way. But some luck came my way. I had a friend from college, Adhyankar (Āḍhyaṃkara), hailing from the merchant community. He had started a paint business that was flourishing due to the housing boom. One day, over lunch, he asked me if I could synthesize anti-fungal compounds for his paints. He had been importing these compounds and said that any breakthrough would result in a significant profit of which I would receive a share. While it was not the pharmaceutical work I wanted to do, I saw it as a break and took a one-year voluntary suspension from my college job to set up a lab funded by Adhyankar. My hard work paid off, and I synthesized a siloxane halamine derivative that could work well as an anti-fungal. With my team, we soon set up an industrial manufacturing process for producing and incorporating it into Adhyankar’s paints.

A major problem in our country is the discoloration of walls by cyanobacteria. Hence, I wondered if we could augment our paints with anti-cyanobacterials. During my Ph.D. in Japan, I had made acquaintance with a fellow graduate student who had identified and determined the structure of anti-cyanobacterial compound which had the sequence: Me_3R-V-V-OHMeR-MeR. I synthesized a truncated brominated derivative thereof that had 100-fold higher anti-cyanobacterial activity. When we brought this into production, I was able to negotiate a fuller partnership in the company of Adhyankar. The profits helped me to dabble with my true interests. I realized that the antiviral field was a wide-open opportunity, and Adhyankar was willing to again partner with me, thereby giving me a long rope to explore exciting possibilities.

By then, I had a flourishing family with three sons and a daughter. What is misery to some can be a gain for others. It was around that time the Great Dhori Virus Outbreak fell upon us. Building on my anti-cyanobacterial work, I had synthesized a bacterial cyanoalkaloid, whose halogenated derivatives had an excellent antiviral capacity that played a decisive role in flattening that outbreak. From the profits materialized during this time, I wanted to build a new lab and plant. I bribed a derelict temple’s management to procure some good land bypassing the usual bureaucratic strictures. During the building of the new lab, we unearthed a religious image that the Hiṇḍū-s worship under the name of some god, I think he is called Śiva. While I cared little for such superstition, I did not want it to be destroyed because it might be an object of veneration for people who believe in such things. Hence, I handed it over to some pundits at a temple. With a new lab in place, I often took my children there to intern and develop a scientific temper.

However, my fortune seems to have peaked there, and it was all downhill thereafter. My daughter acquired an undiagnosed neurological illness and committed suicide by jumping off the balcony in a fit of delirium. Then my youngest son developed a mysterious idiopathic anemia and died despite all our attempts to treat him. My next son, like me, was a great car enthusiast, but this proved to be the tragedy of our lives. He too enjoyed the thrill of speeding but sadly lost control of the car and expired on hitting a flyover pillar. Perhaps due to this stress or maybe due to her nature, my wife upbraided and slapped our eldest son one day in front of all his friends for not doing as well as we had expected in one of his exams. He was angered by that and ran away from home, and we never saw him again despite filing many a missing person report. Then when my turn came, it almost seemed like relief from all the suffering I was going through. I was altered by the alarm system regarding a problem in the lab. I was initially informed by the staff that there was nothing to fear. I thought it was just a false alarm and casually went in a little later to check things. At that point, there was a big phosgene leak, and I died from the exposure.

What happened thereafter was remarkable. I could see my corpse being donated by my wife to the hospital for study. With much horror, I watched it being cut up and my tissue being examined microscopically and analyzed. After what was left of my corpse was consigned to oxidation at the incinerator by Adhyankar, something even more striking happened. I found myself sitting in a ghostly corpus on a large boulder that lay outside my laboratory building. Marching in front of me was a vast horde of other ghostly beings. Some looked like skeletons, others had strange animal heads, yet others had a misty, shape-shifting nature. Far behind, I saw the leaders of that horde of ghosts — they were emitting a radiance and appeared more real than anything I had seen in life. I think they were gods, as I remember seeing images in the likenesses of them being taken out during Hiṇḍū festivals. One of them had an ape-face, another had six heads, yet another a proboscis, and still another was a dark bluish-black hue. A ghost from that immense horde came up to me and said: `You are appointed as the regent of this land that you once purchased through underhand means from the temple. You shall sit here and keep others away from it after your lab has been demolished. I spent a while wandering in my lab as though doing experiments but finally, one day, a government crew appeared and demolished it. I sat on the stone and made it my routine to haunt anyone who trespassed it grimly. When you bought this land, you came with a brahmin and his wife to take possession. They seemed to have some spells to those very same gods I saw when I was appointed as the guardian of the land. Hence, I was rendered powerless to do anything to you or them then. But now that it is just you, I can knead you like dough. If you were to build on this land, I shall reduce you to a fate that is not very different from mine. If you do not, and let my stone remain, then I will even use my ghostly powers to aid your quest for a new vehicle, a woman and a house.”

Abhirosha: “Vidrum, even I would have acted the same if I’d had encountered a phantasmagoria as this.” V: “Even if this were just an illusion, spurred by the visitation, I did some investigations that led me to a clear decision. I dug up old reports that the city auction had hidden from me. Those showed that indeed a chemical laboratory had stood on that plot. It was demolished after being decommissioned following an accident, and the plant nearby had been shut down for safety issues. I reasoned that the mysterious deaths of our visitor’s children were probably again from the poor safety leading to their affliction by toxic compounds. Who knows, some toxic stuff might still be lingering therein. Hence, I thought it prudent to abandon the plan of building my house on that site and let the agents of Mahādeva reclaim it.”


Vrishchika’s medallion

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Pandemic days: Galtonism hits India

At some point last year, we stopped writing any further dispatches regarding the pandemic catastrophe from the \omega \upsilon o \nu disease because everything was playing out more or less as laid out in the earlier notes. There was the whole public drama around the “lab-leak” hypothesis that was widely disseminated by the Jewish American intellectual Weinstein and his wife Heying. While they and their cohort made some good points, there were specific counterpoints that nobody in those academic circles was able to bring to the discussion. Having studied this virus closely and having discovered multiple new things about its evolution, we had laid those out in our earlier dispatches. So, we were not too disposed towards reiterating it.

The effects of the pandemic reached far beyond the human disaster — it played with the internal stability and politics of several nations across the world. It fueled the explosive growth of the American mental disease, navyonmāda, following incidents of police brutality, typical of the Mahāmleccha. As a result, among the Mahāmleccha, Vijaya-nāma-vyāpārī was overthrown by the navyonmatta activists in Big (primarily imaginary)-Tech and Media and replaced by their candidate Vṛddha-piṇḍaka and the sūtradhāriṇī Ardhā, who works the former like a putalikā on behalf of the operators in the deep-state and Big-Tech. Now in the driver’s seat, they moved quickly to impose the navyonmāda religion on the population of America — the full extant of the steps taken for its imposition are striking (supported by statistics and raw data) but cannot be narrated for now. Indeed, in retrospect, it now appears that the Nāriṅgapuruṣa was the last line of defense against dam-burst of navyonmāda. Its capture of a serious fraction of the Mahāmleccha elite is evidenced in nearly all the ejaculations of Piṇḍaka’s courtiers; now, they even intend flying the dhvaja of navyonmāda at their dūtya-s. In essence, the Piṇḍaka-śāsana is giving a taste of how it might have been for the heathens when the second and third unmāda-s were taking over West Asia and Europe.

If this had remained restricted to the Mahāmleccha, then the rest of the world might not be too bothered about it. In fact, rival powers would have rejoiced at it because it will eventually weaken the Mahāmleccha (at least temporarily). But like any physical infectious disease, this memetic one is also infectious. Just like the overt unmāda-s, navyonmāda too has a natural enmity towards the deva-dharma. Hence, it will act in similar ways to destabilize any political party or arrangement that might even marginally help H growth. It is brought into India via the first responders and mleccha-trained academics, and is also casually dispersed to the young urban population by the occidental media. During the Kangress era, the judiciary too was subverted by several crypto-proponents of sympathizers of this neo-religion. This is perhaps one of the most imminent dangers of navyonmāda given the power the court holds in the Indian political arrangement. It has also taken a deep root in centers of higher science and technology education such as the IITs, IISERs, and TIFRs, where several academics mimic their left-liberal counterparts in the west and engage in anti-Hindu action guided by navyonmāda. Thus, when navyonmatta-s from a scientific tabloid, like the Nature magazine, interview people for something regarding India, they goes to their co-religionists (i.e., navyonmatta-s) in Indian academia. These will invariably paint an anti-H picture. To give a concrete example, I know a senior academic who had served at the IISERs and Ashoka University (a navyonmāda nest) who wanted to give his students “a balanced view” that the great rājan Śivajī was a bandit. This means a whole crop of young H, especially those in the crucial research and technology fields, are being indoctrinated into the neo-religion. Given this situation, the rise of Piṇḍaka meant that this wing of internal navyonmatta-s would be strengthened in the deśa.

This is how it indeed played out. An uṣnīṣin rebellion passing off as a kīnāśakopa was fomented in the Pāñcanada, that too during an ongoing pandemic. The discerning clearly saw the pattern, as it followed along the lines of the earlier CAA riots, the Bhim riots, and the Patel riots. However, in this case, they were conclusively outed as a Swedish front-end for navyonmatta, who is on the Asperger’s spectrum, spilled the “toolkit” of the first responders leading to the quick arrest of some of their agents. Their subsequent interrogation revealed even more of their intent. The Indian government was rather mild with and gave a long rope to these rioters. This puzzled many nationalistic observers who were hoping for firm action that the śādhu was reputedly capable of. Our reading is that the H are relatively weak in their ground state and the security apparatus knows that. Moreover, they also feared opening of a border front with the Cīna-s while tackling internal rebellions. Hence, for good optics with the mleccha-s the H leadership did not act firmly. Thus, unlike the Cīna-s dealing with their rebels, the H have to go soft as they cannot take on the Mahāmleccha under Piṇḍaka, who will back the rioters, unlike the overthrown Nāriṅgapuruṣa. Our prediction from a little over a month back was that the Mahāmleccha would continue such action until at least the end of the year or till the possible event of their king Piṇḍaka falling prey to jara resulting in some internal turmoil.

Returning to the \omega \upsilon o \nu disease, the first wave in India was bad, but the nation as a whole fared much better than most other hard-hit countries. By the end of the civil year 2020 CE, it was coming down even as it was exploding in the USA. By the first two months of 2021, it looked as though India was on top of the pandemic and the vaccination program was initiated. It was going well by early March even though a lot of eligible people were not taking it. The corresponding program was doing much worse at that point in the US — some people were driving a long way to neighboring towns and cities to get their shots. A Pakistan physician was arrested by the mleccha-s for giving the vaccine that would be otherwise wasted to “people with Indian-sounding names!” because of their kṛṣṇa-rudhira sham policy. Unfortunately, on the H side a basic lesson of epidemiology was forgotten. Infectious respiratory disease epidemics show wave dynamics. The classic precedent of the Spanish flu of 1918 CE shows that the second wave can be worse than the first because more infectious and/or more virulent mutants can emerge, especially with the effective population size of the virus being large along with a large as yet uninfected population. The same dynamic was seen with the milder H1N1 epidemic in the USA. Even the limited SARS outbreak showed waves. Several countries had already experienced two waves of SARS-CoV-2, with the second being worse than the first. In several instances, like in UK, Brazil and Iran, we have seen SARS-CoV-2 variants with greater infectivity or virulence emerge and drive a new wave. This meant that India had to be ready for wave 2, potentially driven by a more infectious or virulent strain. It is in this regard the Health Ministry largely failed in it is messaging and warning of the public. However, nations do not fail or succeed based on their leadership alone. A much bigger part is played by the people’s social responsibility, deep state, and institutions. Modern H have been strikingly weak on each of these counts, especially when compared to the East Asians. At the times of non-crisis, the accumulated civilizational capital of the H nation could take them through, but any discerning observer knows that this could break in bad times.

Those bad times came with the entirely expected wave 2. While wave 2 was expected, the above H weaknesses poured more ghee into it, making it a conflagration of sa Devaḥ (The god). In our opinion, there were several deep failures beyond the Health Ministry’s negligence regarding the imminence of wave 2: 1) The people acted as though it was back to normal. There was no masking or social distancing, crowding at indoor entertainment spaces. 2) Wearing flimsy cloth or fashion masks in the name of comfort, frequent removal of the mask to speak and interact at close quarters, and improper use of the mask. 3) Poorly governed states like Maharashtra and Kerala let the outbreak remain bad, offering opportunities for the emergence and selection of new mutants. We suspect both the B.1.351 and B1.617 variants are major drivers of wave 2. The latter seems more infectious and clearly appears to break past any natural immunity or that acquired during wave 1. There are reliable reports of reinfection in wave 2 and might be cases of B1.617. 4) Crowded election rallies with no or improper masking in certain states and religious assemblies. While the Anti-H constituency and eventually also the Dillīśvara tend to emphasize the last one (e.g., the Kumbha), the data shows that massive outbreaks were building up far from the centers of these open-air gatherings. Hence, while such crowds might have played a role in local transmission, we do not think they were by any means the primary factor in the explosive second wave in India. 5) Many eligible people simply failed to take the opportunity to get vaccinated. 6) The weak research culture (traditionally disparaged and neglected by the rising urban middle class and in part addled with navyonmāda) in the country meant that study of the mutants, the efficacy of treatment, search for new drugs, epidemiology, bio-, and chemical technology for were all not up to mark to face a crisis. 7) Unlike some countries like the USA, India is a densely populated country where people live in close proximity with extensive casual social interaction. While the latter can be advantageous in some situations, in this situation, it is a disaster, especially when people with the illness do not self-quarantine responsibly. In the end, from all we have seen, the biggest failure was the first point — people simply not taking the threat seriously. Looking at the exponential phase of its growth, we can say that the plot was probably lost between March 15-20th when the tangent to the curve moved past 45^\circ. The result is an unprecedented public health crisis that the already weak institutions cannot bear. Reports on the ground mention an unending stream of cremations with people simply dying before they can even get access to supplemental oxygen.

When a nation is in crisis, its enemies and haters will try to make the most of it. With the precision of a Dutch clock, the mleccha haters from Big Media and navyonmāda-addled occidental academia who are big Piṇḍakānuyāyin-s (e.g., a Harvard University physician of Chinese ancestry who is vociferous on SM) aided by their marūnmatta allies came out like beetle-grubs from the wood-works peddling “cremation sensationalism” (aimed at their fire-hating Abrahamistic audience who mostly don’t know that H cremate), blaming it on H religious assemblies. The foreign policy of the Mahāmleccha state has a simple sūtra: Weaken, destabilize or destroy any non-pañcanetra state. To add to this, the national religion of the current regime is navyonmāda, which has a svabhāva-vairam with dharma and any political assembly that might even indirectly support it. Then there is predatory American Big Pharma which has always sought to profit off human misery (incidentally of their own people, including the śvetatvak-s). Hence, they saw a golden opportunity for 1) Playing the anger in the Indian middle class from the high death toll to engineer a janakopa against the Lāṭeśvara and his court; 2) Send a clear message to the Lāṭeśvara for being pally with their internal arch-enemy, the Nāriṅgapuruṣa; 3) Use the opportunity to aid dvitīyonmatta-s and navyonmatta-s by NGO channels by providing direct “aid” bypassing the Indian government headed by the Lāṭeśvara.

They executed this program reason fairly well until today: 1) They embargoed key ingredients for vaccine production and withheld the AstraZeneca vaccines that they are not even using. 2) They amplified the noise about the Indian failure via their usual Big Media outlets aided by navyonmatta academics (e.g., the said Harvard University academic of Cīna ancestry) to facilitate the Indian public opinion turning against the Lāṭeśvara’s court. 3) Once the situation was desperate in the deśa, over the current weekend, they suddenly got active and presented themselves as saviors (much like the English during the piṇḍāri wars). Their NSA and Secretary of State, in thinly veiled messages, talked of “working with their friends and partners” in India. Every discerning H knows who their friends and partners are and how much they work for the downfall of the H. Now everyone from the above courtiers to Piṇḍaka claimed that they would help by allowing the supply of materials to India. However, one notable point in all this was how for most part they (Piṇḍaka included) avoided directly mentioning the Indian government or leaders. This shows that, like the English tyrants of the past, they are attempting to use the dire situation to present themselves as saviors while discrediting the Indian leadership.

But why did they relent at all, especially on a weekend when they are normally relaxing? In part, they saw that the H in India were not buying their first-responder messages and were seriously angry against their blockade of material flow. Not only that, they saw that there was widespread public opinion against their actions even among the śvetatvak-s (barring some like the evil queen of the Śūlapuruṣa-s). Americans of H origin were pressing on them to release the blockade. It also appears that there was some straight-talking by the Indian NSA with his counterpart (speculation). Consequently, Piṇḍaka changed his line and claimed that he would open channels for raw material flow. As soon as he announced it, his deśī sepoys in Big Tech, Big Media, tinpot think tanks started amplifying Piṇḍaka’s announcement as a noble action of the great savior.

We do not know what exactly happened, but definitely, something hurried happened behind the scenes. One can never trust the Mahāmleccha — everyone who has done so has paid dearly. Hence, one can only hope that in this desperate situation, the Lāṭeśvara did not promise them something that could come to bite the H and him in the rear in the near future. This is a once-in-century pandemic, so there are bound to be failures, but that is no excuse for being unprepared, given all the precedence — this is analogous to the unpreparedness of the H rulers to Shihab al-dīn over 800 years ago despite the precedence of Mahmud of Ghazna. Some political setups will be better prepared for situations like this, e.g., the Cīna-s, who, unlike H, need not fear “death by democracy or judiciary” allowing them to play the long game without the wastage of time and money on constant elections, let alone the significant risks from the rallies [note, we are not per say advocating a Cīna form of government as the solution but simply stating a fact]. One can only hope this wakes up the H to the need for a way more comprehensive reform of the crappy “jugāḍ” mentality that they pat themselves on the back for, along with serious and unbiased cultivation of a more robust basic research culture. So far, the H have only mimicked the broken western system — often importing just the navyonmāda and marrying it with vicious regionalistic politics rather than bringing actual research excellence from the west into their premier institutes. This cannot change overnight, but without doing so, it is not going to be easy to meet the challenges of the current order. An example germane to the current situation is the modified nucleotide mRNA vaccines deployed in the USA. This needs an extensive biochemical knowledge base that cannot be developed without serious basic research. While along with the journey to Mars, the mRNA vaccines could very well be among the last great American achievements in the penumbra of their power being eclipsed by navyonmāda, it still has shown the gulf between their technological prowess and that of the rest of the world. While this pandemic will resolve eventually, it will be at high human cost and also we do not know for sure what it will leave behind. In the worst case scenario, there will be long term health issues (e.g. neurological and respiratory) that could hamper the work force. If the Lāṭeśvara is overthrown by the Indian democracy, as the Piṇḍaka-śāsana wants, the country will essentially be overtaken by agents of the mleccha-s and marūnmatta-s and greatly decline. But then who can predict the future?

Posted in History, Politics | Tagged , , , , | Leave a comment

Making an illustrated Nakṣatra-sūkta and finding the constellation for a point in the sky

The illustrated Nakṣatra-sūkta

Towards the latter phase of the Vedic age, multiple traditions independently composed sūkta-s that invoked the pantheon in association with their home nakṣatra-s as part of the śrauta Nakṣatreṣṭi or related gṛhya homa-s. Of these oldest and the most elaborate is seen in the form of the Nakṣatra-sūkta of the Taittirīya brāhmaṇa. From the time we first learned this in our youth, it has been a meditative experience that compensates for the bane of urban existence — bad skies. Passing from nakṣatra to nakṣatra, we could bring to our mind the various glorious celestial bodies that we had been recording since the 10th year of our life. Thus, the desire arose in us to create an illustrated Nakṣatra-sūkta that would aid in bringing them to mind as we recited it in an indoor urban setting. We have been making our own star maps for a while, each with its advantages and downside. For nice vector graphics (PDF), we decided to use the TikZ package for \LaTeX. The TikZ picture itself is generated by a script we wrote in R. The datasets used for the astronomical bodies are:

  • Since we did not want it to be too cluttered nor stress the \LaTeX compilation with memory issues, we stuck to the Bright Stars Catalog with about 9096 stars for plotting.
  • The stars were colored discretely using their spectral type from the catalog. We only include the types W (very rare), O, B, A, F, G, K, M and C for our palette.
  • The double stars were obtained from the Washington Double Star catalog and mapped on the Bright Stars Catalog.
  • The variable stars were taken from the confirmed variability record in the Bright Stars Catalog and supplemented with information from the Hipparchos survey.
  • The deep sky objects were obtained from The NGC 2000.0 Catalog and corrected where necessary. For galaxies, the orientation angle was assigned as in Stellarium. We generally plotted only the brightest of these, which can be seen by small telescopes (e.g. 20 x 3in binoculars, 3-4in refractors, 6-10in Newtonian reflectors) that we have used in our observing career.
  • For the Milky Way, we used a file specifying different contours that used to be available from old planetarium software like HNS.
  • The constellations boundaries as specified by the International Astronomical Union were based on the corrected version of Davenhall and Leggett’s catalog available via Vizier.
  • The constellation figures are based on those drawn by Hans Augusto Rey(ersbach) in his 1952 book “The Stars: A New Way to See Them”.

We generated the star maps by IAU constellation and mapped the nakṣatra asterisms onto them as per the earliest Vedic traditions (when known) or the traditional identification widely accepted by Hindus (when the Vedic identity was unclear; see notes in PDF for details). At the end of the sūkta we provide brief notes on the Vedic tradition of the nakṣatra-s. One issue that came up in this process was the mapping of any given point in the sky onto a constellation. This takes us back to the history of the origin of modern constellations. While most of them in the northern hemisphere have their roots in ancient cultures, the precise boundaries are of recent vintage. The man behind that was Benjamin Apthorp Gould (1824-1896 CE). Born in the USA, he showed precocious mental ability and went on to become a doctoral student of Carl Gauss at the age of 20. While with Gauss, he did considerable work advancing our understanding of the asteroid belt. Inspired by the tradition of the creation of detailed star catalogs championed by Gauss’s colleague Carl Harding and student Johann Encke, Gould also went on to be one of the most outstanding star catalogers of the age. Going to Argentina to study the southern skies, he pioneered the use of photography in mapping the heavens. As part of this work, he defined the constellation boundaries for the southern constellations in 1877 CE. This was then extended by Eugène Delporte (a prolific asteroid discoverer) for the northern constellations under the IAU in 1930 CE. So the question is, given these boundaries, how do we say which constellation a point in the sky belongs to?

Nancy Roman designed a beautiful algorithm for this in 1987 CE. It goes thus: We first need to precess the coordinates of our current epoch to those of 1875 CE, which correspond to the epoch used by Gould when he first defined the boundaries. We briefly describe below the algorithm for the precession to a given epoch without going into the trigonometry and calculus involved in arriving at it (that can be found in a textbook on basic numerical procedures in astronomy, e.g., the freely available textbook, Celestial Mechanics, by Professor Tatum). For simplicity (sufficient for most purposes in terms of accuracy), we take a constant rate of precession of the equinoctial colure as p=50.290966''/y, i.e., per year. We take the inclination of the earth’s axis to be: I= 23^\circ 26' 21.406''. We then compute the parameters m, n in degrees thus:

m= \frac{p}{3600}\cos(I), \; n= \frac{p}{3600}\sin(I)

Let, \alpha be the Right Ascension (celestial longitude; here taken from (0^\circ,360^\circ))and \delta be Declination (celestial latitude; here taken from (-90^\circ,90^\circ)) of the point in the sky we wish to precess. We then compute the corrections:

a= m + n \sin(\alpha)\tan(\delta), \; d= n \cos(\alpha)

Let z be the signed difference in number of years between the epoch we wish to precess to and the current epoch. Then we get the precessed coordinates as:

\alpha_p=\alpha+az, \; \delta_p= \delta + dz

If \alpha_p<0, \alpha_p= \alpha_p + 360^\circ

Having precessed the coordinates to 1875 CE using the above, we look up the table created by Roman of just 357 rows which takes the below form:

\begin{tabular}{|r|r|r|l|} \hline RA low & RA up & DE low & Constellation \\ \hline 0.0000 & 360.0000 & 88.0000 & UMi\\  120.0000 &217.5000 &86.5000 &UMi\\ 315.0000 &345.0000& 86.1667 & UMi\\ 270.0000 &315.0000 &86.0000 & UMi\\ 0.0000 &120.0000 &85.0000 &Cep\\ 137.5005 &160.0005 &82.0000 & Cam\\ \hline \end{tabular}

The lookup procedure goes thus:
1) Read down the DE low column until you get a declination lower than or equal to the declination of your point.
2) Move to the corresponding RA up column and read down until you get a right ascension higher than that of your point.
3) Move to the corresponding RA low column and read down until you get a right ascension lower than or equal to that of your point.
4) Check the corresponding RA up column and see if it is higher than the right ascension of your point. If yes, the constellation column gives the constellation in which the point lies. If not, go back to the first step 1 and continue downward in the DE low column from the first DE you obtained lower than or equal to that of your point to find the next such value and repeat the following steps until the condition in the final step is met.
Thus, we can obtain the constellation of any celestial object given its coordinates.

Posted in art, Heathen thought, History, Scientific ramblings | Tagged , , , , , , , , , , , , | Leave a comment

Johannes Germanus Regiomontanus and his rod

Even before we had become acquainted with the trigonometric sum and difference formulae or calculus are father had pointed to us that there was an optimal point at which one should stand to observe or photograph features on vertical structures, like on a tall gopura of a temple or a tree. That point can be calculated precisely with a simple Euclidean construction. Hence, we were rather charmed when we encountered this question in a German book on historical problems in mathematics. It was posed in 1471 CE by Johannes Germanus Regiomontanus to a certain professor Roderus of Erfurt (Figure 1): At what point on the [flat ground] does a perpendicularly suspended rod appear the largest (i.e. subtends the largest angle)? Let the rod be of length a and it is suspended perpendicularly at height h from the ground. The question is then to find the point P at which \angle\theta would be the largest. This is also the kind of question that often repeated itself in some form in the lower calculus section of our university entrance exams. So it is not a difficult or unusual problem, but it has a degree of historical significance. Before we look into its solution, let us first talk a little about its proposer, who as an enormously important but not widely known figure in the history of science and mathematics in the neo-Occident.

Figure 1. The rod of Regiomontanus

Born in 1436 CE at Unfinden, in what is today Germany, Regiomontanus seems to have shown signs of early genius. Seeing this, his parents sent him at age 12 to Leipzig for formal studies and then he proceeded to Vienna to obtain a Bachelor’s degree at age 15. His genius came to the notice of Georg von Peurbach, a German astrologer, who wished to produce a corrected and updated translation of the Mathematike Syntaxis (Almagest via Arabic) by the great Greco-Roman astrologer and mathematician Klaudios Ptolemaios of Egypt. He hoped in the process to establish the geocentric theory on a firm footing and use the newly introduced Hindu decimal notation for the ease of calculations. However, von Peurbach’s Greek was not up to the mark to effectively translate the original but he transmitted his mathematical and astrological knowledge to Regiomontanus, whom he treated as his adopted son, before his death at age 38. On von Peurbach’s deathbed, Regiomontanus promised to continue his work on the Syntaxis and also create a synthesis of the mathematical knowledge that was present in it with the new knowledge of the Hindus and the Arabic neo-Platonic revolution that was entering Europe from the Mohammedan lands.

The Regiomontanus took up the task with great diligence by mastering the Greek language and started composing verse in it. He then took to traveling around Greece and Italy collecting Greek and Latin manuscripts collection to revive the lost knowledge of the ancients. In the process, he found a manuscript of the yavana Diophantus that he could now handle using the elements of Hindu bījagaṇita transmitted to Europe from the Mohammedans. He then became the court astrologer of the Hungarian lord Matthias Corvinus Hunyadi who staved off the further penetration of Europe by Mehmed-II, the conqueror of Constantinople, through several campaigns. As a ruler with literary interests, he had looted several manuscripts from Turkish collections in course of his successful raids. These offered additional opportunities for the studies of Regiomontanus. Having established an observatory in Hungary for Matthias, he returned to Germany and built an observatory equipped with some of the best instruments of the age and also adopted the newly introduced printing technology to start his own press. As a result, he published a widely used ephemerides with positions of all visible solar system bodies from 1475 to 1506 CE. He also published a remarkable geometric work titled “De Triangulis Omnimodis (On triangles of every kind)” wherein, among other things, he introduced the Hindu trigonometric tradition to Europe. To my knowledge, it also contains the first clear European presentation of the sine rule and a certain version of the cos rule for triangles. Regiomontanus also recovered and published the striking Latin work “Astronomica” of the nearly forgotten heathen Roman astrologer Marcus Manilius from the time of the Caesar Augustus. This beautifully poetic work would be of interest to a student of heathen religious traditions and Hindu belief systems because neo-Hindu astrology was after all seriously influenced in its belief structure of the Classical world. As a sample, we leave some lines of old Manilius here:

impensius ipsa
scire iuuat magni penitus praecordia mundi,
quaque regat generetque suis animalia signis
cernere et in numerum Phoebo modulante referre. (1.16–19)
It is more pleasing to know in depth the very heart of the universe and to see
how it governs and brings forth living beings by means of its signs and to speak
of it in verse, with Phoebus [Apollon] providing the tune.
-translated from the original Latin by Volk

Two years after the publication of his ephemerides, Regiomontanus was summoned to Rome to help the Vatican correct its calendar. He died mysteriously at the age of 40 while in Rome. His fellow astrologers believed it was prognosticated by a bright comet that appeared in the sky in 1476 CE. Others state that he was poisoned by the sons of the yavana Georgios Trapezuntios, whom he had met during his manuscriptological peregrinations. He had a kerfuffle with Trapezuntios after calling him a blabberer for his incorrect understanding of Ptolemaios and apparently the latter’s sons had their revenge when he was visiting Rome. Thus, like his friend von Peurbach, Regiomontanus died before he could see the published copy of his work on the Syntaxis. However, it was posthumously published as the “Epitome of the Almagest” in 1496 CE, 20 years after his demise in Rome. Looking at this book, one is struck by the quality of its production and the striking synergy of its text and lavish mathematical illustrations. Even today, with the modern computer languages like \LaTeX (TikZ included) and GeoGebra and our collection of digital fonts one would be hard-pressed to produce something nearing the quality of Regiomontanus’ masterpiece published at the dawn of the Gutenberg printing revolution.

Figure 2. A yavana and a śūlapuruṣa in anachronistic conversation: The frontispiece of Regiomontanus’ Epitome of the Almagest showing him questioning Ptolemaios under the celestial sphere.

Regiomontanus is said to have had a lot more material to write and publish that never saw the light due to his unexpected death. One of these was the possibility of the motion of the earth and heliocentricity. In this regard, we know that he criticized astrologers of the age for accepting the Ptolemaic model as a given without further analysis. Moreover, he demonstrated that his own astronomical observations contradicted predictions made by the geocentric models of the time. We are also left with tantalizing material reported by his successor Schöner that hint that he was converging on the movement of the Earth around the sun. After Regiomontanus had passed away, the young German mathematician Georg Joachim Rhäticus deeply studied the former’s works to become a leading exponent of trigonometry in Europe. He befriended the much older Polish astronomer Copernicus and taught the latter geometry using the “De Triangulis Omnimodis” of Regiomontanus, a copy of which with Copernicus’ marginal notes still survives. Rhäticus also urged Copernicus to publish the heliocentric theory. This raises the possibility that Rhäticus was aware of Regiomontanus’s ideas in this regard and it helped crystallize Copernicus’s own similar views. In the least, the geometric devices that both Copernicus and later Tycho Brahe needed for their work were derived from Regiomontanus, making him a pivotal figure in the emergence of science in the neo-Occident. [This sketch of his biography is based on: Leben und Wirken des Johannes Müller von Königsberg by E. Zinner]

Figure 3. Construction to solve the Regiomontanus problem.

Returning to his problem, we can game it thus (Figure 3): The rod of length a suspended perpendicularly at height h subtends the \angle\theta at the ground. This angle can be written as the difference of two angles: \angle\theta =\angle\alpha-\angle\beta. Let the distance of the point on the ground from the foot of the perpendicular suspension of the rod be x. We can write the tangent difference formula for the above angles using Figure 3 as:

\tan(\theta)=\tan(\alpha-\beta)= \dfrac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)}= \dfrac{\dfrac{a+h}{x}-\dfrac{h}{x}}{\dfrac{x^2+h(a+h)}{x^2}}=\dfrac{ax}{x^2+h(a+h)}

We can see from Figure 3 that as the point on the ground moves towards the foot of the suspension, both \angle\alpha, \angle\beta \to 90^\circ, thus \angle\theta \to 0^\circ. If the point on the ground moves away from the foot of the suspension, both \angle\alpha, \angle\beta \to 0^\circ and again \angle\theta \to 0^\circ. Thus, somewhere in between, we will have the maximum \theta and it will be in the interval [0^\circ,90^\circ]. In this interval, the tangent increases as the angle increases. Thus, it will reach a maximum when the function y=\tfrac{ax}{x^2+h(a+h)} reaches a maximum. We would find this maximum by differentiating this function and finding where \tfrac{dy}{dx}=0. This approach, using calculus, is how we would have answered this question in our university entrance exam. One will observe that this function has a rather flat maximum suggesting that, for the purposes of viewing a feature on a tall vertical object, a relatively approximate position would suffice. While this principle of extreme value determination by calculus was known in the Hindu mathematical tradition by at least the time of ācārya Bhāskara-II (1100s of CE), there is no evidence that any of this Hindu knowledge of calculus was transmitted to Regiomontanus. In Europe, a comparable extreme value principle was informally discovered much later by the French mathematician Michel Rolle in 1691 CE who actually rejected differential calculus. So how would Regiomontanus have solved in 1471 CE?

It is believed that he used the logic of the reciprocal. When y=\tfrac{ax}{x^2+h(a+h)} is maximum its reciprocal y=\tfrac{x}{a}+\tfrac{h(a+h)}{ax} would be minimum. We can see that if x becomes large, then \tfrac{x}{a} term would dominate and it would grow in size. Similarly, when x becomes small, the \tfrac{h(a+h)}{ax} will dominate and it would grow in size. The 2 opposing growths would balance when \tfrac{x}{a}=\tfrac{h(a+h)}{ax}. This yields x=\sqrt{h(a+h)}. With this in hand, we can easily use the geometric mean theorem in a construction to obtain the desired point P (Figure 3). This also yields another geometric relationship realized by the yavana-s of yore regarding the intersection of the tangent at point P on a circle and a line perpendicular to it that cuts a chord (here defined by the suspended rod) on that circle: The distance of the point of tangency P from its intersection with the line containing the said chord is the geometric mean of the distances of their intersection to the two ends of the chord.

We may conclude with some brief observations on the history of science. Regiomontanus is a rather striking example of how the founder of a scientific revolution can be quite forgotten by the casual student due to the dazzling success of his successors. In the process, the existence of scientific continuity between the Ptolemaic system and the heliocentric successor might also be missed by the casual student. His life also provides the link between the popularization of the Hindu decimal notation in the Occident by Fibonacci and the birth of science in those regions by the introduction of Greek and Hindu tradition via the Arabic intermediate. While Hindu astrology was influenced by the Classical astrological tradition there is no evidence that the Ptolemaic system ever reached India. The Hindus instead developed their own astronomical tradition that appears to have rather early on used a potentially heliocentric system of calculation culminating in the work of ācārya Āryabhaṭa-I, who also discovered a rather brilliant algebraic approximation for the sine function. However, soon there was a reversal to a geostationary, giant-earth model under Brahmagupta, the rival of the Āryabhaṭa school. In the realm of astronomy, the totality of these developments resulted in epicyclic systems or eccentric systems that paralleled the Occidental models in several ways. On the mathematical side, it spawned many high points, such as in trigonometry, ultimately resulting in the emergence of an early form of differential calculus by the time of Mañjula that was subsequently advanced by Bhāskara-II. This line of investigation culminated in the works of the Nambūtiri-s in the Cera country with the emergence of what could be termed full-fledged calculus. Remarkably, this was paralleled by the revisiting of Āryabhaṭa-I and the move towards heliocentric models by the great Nīlakaṇṭha Somayājin. Partial heliocentric models for at least the inner planets, along with the prediction of the Venereal transit of the sun was also achieved by Kamalākara, a Mahārāṣṭrī brāhmaṇa, in the 1600s. Notably, only the earlier phase of the Hindu trigonometric tradition was transmitted to the Occident at the time of Regiomontanus. None of the Hindu studies towards calculus found their way there and they appear to have been rediscovered in the Occident about 2 centuries after Regiomontanus. Despite possessing a mathematical and astronomical edge, in the centuries following Nīlakaṇṭha, the Hindu schools, facing a dilution from the chokehold of the Mohammedan incubus, did not spawn a scientific upheaval of the order that took place in Europe in the centuries following Regiomontanus.

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A great statistician, and biographical, numerical musings on ancient game

Recently my friend brought it to my attention that C. Radhakrishna Rao had scored a century. Born in 1920 CE to Doraswamy Nayadu and A. Laxmikanthamma from the Andhra country, he is one of the great mathematical thinkers and statisticians of our age. He came from a high-performing family but even against that background he was clearly an outlier showing early signs of mathematical genius and extraordinary memory beyond mathematics. An example of this was seen in his youth in an award he received for his anatomical knowledge, wherein he displayed his perfect recall of all bones and structures of the body. He might have been an outstanding mathematician but the lack of opportunities to pursue research in India or elsewhere during WW2 led him to going to ISI, Kolkata and becoming a statistician. By the age 20, he was doing and publishing his research by himself and eventually was awarded a PhD for his pioneering statistical work on biometrics at the Cambridge University with Ronald Fisher as his supervisor. By the age 28, he was a professor who had authored several works at the frontier of statistics. Over his 100 years he has been prolific and actively publishing into this advanced years — an outlier in every sense — a truly rare genetic configuration.

CR Rao wrote a very accessible book for a lay audience titled Statistics and truth: putting chance to work. This small book provides a great introduction to the utility and the consequences of well-founded numerical and probabilistic thinking with examples from diverse sciences. We found the book particularly attractive because, despite being a mathematical layman, we stumbled onto the probabilistic view of existence around the 15th year of our life. Rao’s book then lent proper shape to our thoughts that had been born from several experiments and explorations. To us, the probabilistic view is the fructification of an ancient strand of Hindu thought first articulated in a ṛk from the pathetic sūkta of Kavaṣa Ailūṣa (RV10.34.8):

tripañcāśaḥ krīḻati vrāta eṣāṃ
deva iva savitā satyadharmā ।
ugrasya cin manyave nā namante
rājā cid ebhyo nama it kṛṇoti ॥

Three times fifty plays the swarm of these,
like the god Savitṛ of true laws.
To the fury of even the fierce they bow not ;
even the king verily makes his bow to them.

The ṛk is referring to the game of chance, apparently one of the favorite games of the old Ārya-s played with vibhīdaka/vibhītaka nuts. Rao’s essays inspired us to explore the basic numerical aspects of this game at the end of junior college (Also the time we were studying the RV and AV). We present a freshly illustrated version of that here for other simple-minded folks. The game may be reconstructed thus: A hole was dug in the ground and 150 nuts were thrown into it. Then the player drew a handful of those to get out n nuts (probably there were some constraints against cheating by drawing just 4 nuts that are not entirely clear. A possible alternative formulation involves casting the 150 nuts towards the hole and only those n that fell into the hole were considered for the ensuing operation). If n\mod 4 \equiv 0 then it was a Kṛta (K) or the best result. The next 3 successively lower ranked results were n\mod 4 \equiv 3, a Treta (T); n\mod 4 \equiv 2 a Dvāpara (D); n\mod 4 \equiv 1, a Kali (L). It is unclear if the results were named for the 4 yuga-s or vice versa. In our childhood, our grandmother played this game with us albeit with tamarind seeds she had saved after peeling off the fruit. We manually worked out the number of different combinations (hence, order does not matter) formed from the 4 types of results (K, T, D, L) that can be seen in 1, 2, 3… successive draws: in 1 draw you can have K, T, D or L \to 4 possible combinations. In 2 draws you can have: KK, KT, KL, KD, TT, TD, TL, DD, DL or LL \to 10 possible combinations. So on. The sequence of the number of possible combinations goes as: 4, 10, 20, 35… This gave us an introduction to some the principles of combinatorics that only later in life we learned to be governed by the multinomial theorem:
Kṛta, Treta, Dvāpara, Kali \mapsto m=4; n=1, 2, 3... successive draws; hence, the total number of possible combinations in n successive draws is:

N={{n+m-1} \choose {m-1}}

We wondered about the precise chance of getting a combination in consecutive set of draws. We finally understood this only upon apprehending the multinomial theorem. This allowed us to compute say, the chance of getting 4 kṛta-s in 4 consecutive draws as \tfrac{4!}{4!\cdot 0! \cdot 0! \cdot 0!}\cdot \tfrac{1}{256}=0.00390625, which is pretty low. On the other end the chance of get all the 4 results in 4 consecutive draws, i.e. KTDL, is much higher: \tfrac{4!}{1!\cdot 1! \cdot 1! \cdot 1!}\cdot \tfrac{1}{256}=\tfrac{3}{32}=0.09375. Since the vibhīdaka game was for gambling, we can assign the scores from 4 for K to 1 for L and measure ones cumulative gains over multiple draws. We asked, for example, in 4 successive draws what will be distribution of scores (Figure 1) — what score will one have the highest chance of obtaining. We can see that the scores will be distributed between between 4 (LLLL) to 16 (KKKK). We had intuitively realized in our childhood that one had the greatest chance of of having the midpoint score of 10. With the multinomial distribution we could calculate the precise probability of getting the score 10 as 0.171875. This gave us a good feel for the multinomial distribution and how we could get a central tendency in terms of the most probable consequence (score) even multiple scores had the same number of generating combinations (first vs second panel).

vibhIdaka_4drawsFigure 1. The number of distinct combinations and probabilities of getting a given score in 4 draws.

Thus, we can reach any integer by the sum of the scores in a certain number of draws (order does not matter as only the sum matters). The draws resulting in scores adding to the first few integers are shown in Table 1.
Table 1

Integer Draws Number
1 L 1
2 D, LL 2
3 T, DL, LLL 3
4 K, TL, DD, DLL, LLLL 5

Inspired by Hofstadter, after some trial and error, we were able to formulate an alternating recursion formula to obtain this sequence of the total number of ways of reaching an integer as a sum of integers from 1..4. We first manually compute the first 4 entries as above. Then the odd terms are given by the recursion:
The even terms are given by:
f[n]=f[n-3]+f[n-1]-f[n-4]+\left \lfloor\tfrac{n}{4}-1\right\rfloor+2
\lfloor x \rfloor in the floor function or first integer \le x
Thus, we have \mathbf{f: 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108 \cdots}

We also devised an alternative algorithm that is well suited for a computer to extract this sequence. This algorithm revealed a close relationship between this sequence and geometry of triangles. Effectively, the above sequence f gives the total number of integer triangles that have perimeter P \le n for n \in 4, 5, 6 \cdots. Thus, for n=4 we can have only 1 integer triangle, 1-1-1, that has P \le 4. For P \le 5 we have 2 triangles (1-1-1, 1-2-2) and so on (Figure 1, Table 2). Since the smallest integer triangle has P=3 we can get the 0th term of f[0]=1. Then we can state that f[P-3] provides the number of integer triangles with P \le n; n=3, 4, 5 \cdots \infty.

vibhIdaka_tri01Figure 2. First 18 integer triangles

Figure 1 shows the first 18 integer triangles, i.e. those with P \le 12. One immediately notices that in this set the isosceles triangles dominate (Table 2). Of these every P divisible by 3 will yield one equilateral triangle; thus equilateral triangles are the most common repeating type of triangle. There are only 3 scalene triangles in the first 18 integer triangles of which one is the famous 3-4-5 right triangle, which is also the first Brahmagupta triangle (integer triangles with successive sides differing by 1 and integer area). We first computed the the number of triangles with P \le n that are isosceles. This sequence goes as:

\mathbf{f_i: 1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78 \cdots}

Strikingly, every alternate term in this sequence from the second term onward is a triangular number, i.e. the sum of integers from 1\cdots n. The terms between them are the integer midpoints between successive triangular numbers. This understanding helps us derive a formula for this sequence:

f[n]=\left \lceil \frac{n^2}{8} \right\rfloor

Here the \left \lceil x \right\rfloor function is the rounding up function, wherein if k is an integer \left \lceil k+ \tfrac{1}{2} \right\rfloor =k+1 and the rest are rounded to the nearest integer.
Table 2

P ≤ n # triangles # isosceles # scalene
3 1 1 0
4 1 1 0
5 2 2 0
6 3 3 0
7 5 5 0
8 6 6 0
9 9 8 1
10 11 10 1
11 15 13 2
12 18 15 3
13 23 18 5
14 27 21 6
15 34 25 9
16 39 28 11
17 47 32 15
18 54 36 18
19 64 41 23
20 72 45 27
21 84 50 34
22 94 55 39
23 108 61 47
24 120 66 54
25 136 72 64
26 150 78 72
27 169 85 84
28 185 91 94
29 206 98 108
30 225 105 120
31 249 113 136
32 270 120 150
33 297 128 169
34 321 136 185

Remarkably, we find that the first scalene triangle appears at P=9 and then scales exactly as f but with an offset of 9. Thus, the number of scalene triangle with P \le n= f[P-9]. The sequence f scales approximately as a polynomial with positive cubic and square terms, whereas the number of isosceles triangles with P \le n scales as \left \lceil \tfrac{n^2}{8} \right\rfloor. Hence, even though the isosceles triangles are dominant at low n they will become increasingly rare (Table 2) and their fraction of the total number of triangles will tend to 0.

We can also look at the largest angle of the integer triangles (Figure 2). These are plotted along the arc of the unit circle defined by them and scaled and colored as per their frequency of occurrence. As noted above, every third perimeter will define an equilateral triangle. This will result in the smallest of these angles \arccos\left(\tfrac{1}{2}\right) = 60^\circ being the most common. The zone exclusion in its vicinity shows that one needs large sides to approximate the equilateral triangles (e.g. the bigger Brahmagupta triangles). Beyond these, other major angles that are repeatedly observed are: \arccos\left(\tfrac{1}{4}\right) = 75.52^\circ; \arccos\left(\tfrac{1}{6}\right) =80.406^\circ; \arccos\left(\tfrac{1}{8}\right) = 82.83^\circ; \arccos\left(\tfrac{1}{3}\right) = 70.53^\circ. For example, the common version of the \arccos\left(\tfrac{1}{4}\right) triangle arises whenever the perimeter P= 5k; k=1,2,3 \cdots. Thus, these are all versions of the 1-2-2 triangle. However, rare scalene versions can arise, for example, in the form of the 6-7-8 triangle and its higher homologs. Apart from the trivial equilateral triangles, 2 other integer rational sector triangles, the right or 90^\circ (bhujā-koṭi-karṇa triples) and the 120^\circ triangles (e.g. 3-5-7, P=15) appear repeatedly with a lower frequency defined by their triple-generating equations.

vibhIdaka_cos01Figure 3. The plots of the largest angles for integer triangles with P \le 34

Finally, this search of integer triangles also provides a mean to construct triangles, one of whose angles are approximately a radian (Figure 3). In first 511 triangles, (P\le 40), the 5-13-15 triangle provides an angle that approaches 1 radian the closest: 1.0003^c.

vibhIdaka_tri02Figure 4. Triangles with an angle approximating a radian.

The above observations gave us useful introductory lesson on the path to statistical mechanics. Let us consider the isosceles triangles as representing great order (because the is less freedom in their sides) and the scalene triangles as representing greater disorder (more freedom in their sides). A simple multinomial derived score results in the proportion of the order configurations decreasing over time (more draws), i.e. disorder dominates, resembling entropy in the physical world. Among the more “ordered” states the dominant one tends to be that which is in the most “central” configuration, i.e. the equilateral triangle. Finally, certain peculiar configurations can repeatedly emerge if they happen to have special generating equations like the 90^\circ or 120^\circ triangles.

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Modulo rugs of 3D functions

Consider a 3D function z=f(x,y). Now evaluate it at each point of a n \times n integer lattice grid. Compute z \mod n corresponding to each point and plot it as a color defined by some palette that suits your aesthetic. The consequence is a what we term the “modulo rug”.
For example, below is a plot of z=x^2+y^2.

matrixmod01_318Figure 1: z=x^2+y^2, n=318

We get a pattern of circles around a central circular system reminiscent of ogdoadic arrangements in various Hindu maṇḍala-s. From the aesthetic viewpoint, the best modulo rugs are obtained with symmetric functions higher even powers — this translates into some pleasing symmetry in the rug. Several examples of such are shown below.

matrixmod06_318Figure 2: z=x^4-x^2-y^2+y^4, n=318

matrixmod08_315Figure 3: z=x^4-x^2-y^2+y^4, n=315

matrixmod13_309Figure 4: z= x^6-x^4-y^4+y^6, n=309

matrixmod12_318Figure 5: z=x^6-x^2-y^2+y^6, n=318

matrixmod07_312Figure 6: z=x^4-x^2+y^2-y^4, n=310

All the above n are composite numbers. Accordingly, there is some repetitiveness in the structure. However, if n is a prime then we have the greatest complexity in the rug. One example of such is plotted below.

matrixmod14_311Figure 7: z=x^6-x^4+x^2+y^2-y^4+y^6, n=311

See also: 1) Sine rugs; 2) Creating patterns through matrix expansion.

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A guilloche-like trigonometric tangle

Coprimality, i.e., the situation where the GCD of 2 integers is 1 is one of the fundamental expressions of complexity. In that situation, two numbers can never contain the other within themselves or in multiples of them by numbers smaller than the other. In other words, their LCM is the product of the 2 numbers. There are numerous geometric expressions of this complexity inherent in coprime numbers. One way to illustrate it is by the below class of parametric curves defined by trigonometric functions:

x=a_1\cos(c_1t+k_1)+a_2\cos(c_2t+k_2)\\[5pt] y=b_1\sin(c_3t+k_3)+b_2\sin(c_4t+k_4)

The human mind perceives symmetry and certain optimal complexity as the hallmarks of aesthetics. Hence, we adopt the following conditions:
1) a_1, a_2, b_1, b_2 are in the range \tfrac{3}{14}..1 for purely aesthetic considerations.
2) k_1, k_2, k_3, k_4 are orthogonal rotation angles that are in the range [0, 2\pi]
3) c_1, for aesthetic purposes relating to optimal complexity, is an integer in the range [5,60]
4) c_2 captures the role of coprimality in complexity. It coprime with c_1 and is in the range [40,141]
5) c_3 = |c_1-c_2|.
6) c_4=c_1+c_2-c_3
The last two conditions are for making the curve bilaterally symmetric — an important aesthetic consideration.

The result is curves with a guilloche-like form. For the actual rendering, they are run thrice with different colors and slightly different scales to give a reasonable aesthetic. Our program randomly samples through the above conditions and plots the corresponding curves. Below are 25 of them.


Figure 1.

Here is another run of the same.

tritangs04Figure 2.

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Huntington and the clash: 21 years later

This note is part biographical and part survey of the major geopolitical abstractions that may be gleaned from the events in the past 21 years. Perhaps, there is nothing much of substance in this note but an uninformed Hindu might find a sketch of key concepts required for his analysis of geopolitics as it current stands. The biggest players in geopolitics are necessarily dangerous entities; hence, things will be in part stated in parokṣa — this goes well with the observation in our tradition that the gods like parokṣa.

In closing days of 1999 CE, we had our first intersection with Samuel Huntington and his hypothesis of the clash of civilizations. We found the presentation very absorbing because it lent a shape to several inferences, we had accumulated over the years both in Bhārata and on the shores of the Mahāmleccha land. The firsthand experience on shores of the Mahāmlecchadeśa was very important for there is no substitute to fieldwork in anthropology — it enabled a direct interaction with the various denizens of the land and allowed us, for the first time, to extract precise knowledge of their ways and attitudes. A key concept articulated Huntington was “the clash of civilizations”, the title of his book. This is a central concept on which all geopolitical analysis rests. However, we parsed it as a network wherein the nodes are civilizations and not all edges have the same nature or valence (Figure 1). Since the closing days of the Neolithic, the core of the civilizational network (at least in the Old World to start with) has been rather dense. Further, the civilizational network is dynamic both in terms of its nodes and edges. Some civilizational nodes decline or become extinct over time taking away the edges that were connected to them (dashed lines in Figure 1). The edges themselves might change from agonistic (shown as light cadet blue arrows in Figure 1) to antagonistic (shown as red inhibitory edges) ones or vice versa over time. Some edges might be complex and cannot be easily characterized (with no heads, e.g. the “Galtonian” edges in green linking the Anglospheric powers to China). The characterization of the edges might also vary from the viewpoint of the pakṣa of the characterizer. Regarding that last point, the characterization presented here is with limits reasonably predictive and useful from the Hindu standpoint.

So, how does basic clash of civilization articulated by Huntington play out in the framework of this civilizational network? The simplest thing is to look at the flux at a given node. This is a sum of the “weights” of the edges coming into that node. Thus, it is easy is to perceive that the Hindu civilization is currently a node with notable negative flux — this immediately indicates that it is node at the adverse receiving end of the clash of civilizations.

geopol_netFigure 1. A simplified and partial view of the civilizational network.

Some literature
Since Huntington’s publication several disparate works have been published or translated that have a bearing on the Hindu construction of a geopolitical world view. We just outline a few below:
• Amy Chua, an academic of Chinese ancestry, published a work illustrating the role of strongly coherent minorities with high human capital relative to their host populations in civilizational clashes, especially the destruction of states and in some cases civilizations from within. One dynamic she highlighted relates to the Occidental itch to bring “democracy” to states containing such minorities. We may add that sometimes what comes in the name of “democracy” is in reality a “gift-wrapped” strain of the Marxian doctrine. This democratization or Marxian liberalization allows the under-performing “masses”, full of resentment against the over-performing minorities, to get back at them often resulting in intra-national civilizational clashes. If the over-performing minority was the cause for holding the nation together and/or its productivity, it results in national collapse upon their defeat or expulsion. In other cases, is festers as a long-term conflict following the Huntingtonian dynamics. The objective of the enemies of the Hindus is to make this dynamic play out on the brāhmaṇa-s.

• The translation and the publication of the English works of the German academic Jan Assmann helped introduce important terms such as “counter-religions”. While he originally introduced it to understand the rise of the ekarākṣasonmāda-s of West Asia, it also serves as an excellent framework to describe the emergence of subversive religious movements in the Indo-Iranian sphere. The first such, which seems to have marked a schism within the Indo-Iranian tradition, was the counter-religion promulgated by Zarathustra. On the Indo-Aryan side, a cluster of such movements occurred nearly 2500 years ago culminating in the counter-religions of the Tathāgatha, the Nirgrantha and the Maskarin of the cowshed. Subsequently, we had a near counter-religious movement in the form of the Mahānubhāva upheaval, which contributed to the weakening of the Hindu resistance to the Army of Islam. Few centuries later, similar memes and the half-digested ekarākṣasonmāda eventually resulted in the subversion of the pāñcanadīya saṃpradāya into the uśnīśamoha. The other term Assmann introduced was the “Mosaic distinction” that helps explain the vidharma tendencies in counter-religions, especially ekarākṣasonmāda.

• The mūlavātūla indologist Sheldon Pollock published a work on the history of Indian tradition. While recognizing the positive and enormous influence of the Sanskrit cosmopolis, Pollock tried to subtly sneak in the navyonmāda framework into Hindu studies. Along with this, he provided the foundation for the powerful American indological school to present a late date for the rise of Sanskrit as a medium of Hindu expression. This helped create the idea of a non-existent Sanskritic Hindu civilization before the common era (Sanskrit was just some hidden language used in the sacred texts of brāhmaṇa-s), thus, making it younger than the mūlavātūla and probably even the pretonmāda tradition. Further, as per this theory, the transformation did not arise from with the H but was probably fostered by the Iranians, perhaps with some Greek influence. More insidiously, it opened the door for other indologists of this school to link the dharma with their pet boogeyman, the śūlapuruṣa movement of the 1930-40s. The importance of this sleight of the hand will become apparent with the next item. Unfortunately, the positive side of Pollock’s work studying the knowledge systems of the Hindu cosmopolis should have been done by Sanskritists from our pakṣa within a proper H framework — instead the H pakṣa took off on flogging dead Germans and producing little positive work.

• The recent volume by Lindsay and Pluckrose probed deep into the proct of the navyonmāda tradition that arose in the śūlapuruṣīya lands and grew into a viṣāla-viṣa-vṛkṣa nurtured by the Phiraṅga and Mahāmleccha. This work helps understand the roots of its arborizations in the form of both the śākhā-s (i.e. the Freudian, e.g. Wendy Doniger, and philological, e.g. Richard Davis) of new American indology that subverted the tradition of the old Daniel Ingalls. Given its origins in the conflict within the śūlapuruṣīya lands, and being a pracchannonmāda itself, it is not surprising that one of its projects in the indological domain is fleshing out the above-stated point of connecting dharma to the movement of the ghātaka-netā śūlapuruṣaṇām. Additionally, it has received nourishment from the founding lords of the Soviet Rus empire and served to cover up their genocidal activities. It also has been active in furthering the Maoistic strain of Galtonism (see below). It attained ascendancy in mleccha-lands by precipitating the overthrow of Vijaya-nāma vyāpārin and placing Piṇḍaka as the puppet mleccheśa from behind whom their supporter, the ardhakṛṣṇā, operates. Aided by their longstanding backer the duṣṭa Sora, they have now taken aim at the Hindus having presented them as a movement comparable to their archenemies of yore, the śūlapuruṣa-s.

The conquest of the internet
In 1999 CE, the internet was still young and a mostly free place for expression. It was seen as heralding a new mode of expression for individuals who had no voice until then. But in the coming decade this gradually declined as the principle of freedom of expression slowly eroded. The mleccha deep-state has long sought to spy on its citizens and the opportunity to do this came with the marūnmatta attack on the mahāmleccha on September 11, 2001. The mleccha powers could now institute sweeping curbs on the people in the name of protecting them. However, this was only a bīja for the total destruction of the freedom of expression that was to come with the takeover of the internet by the guggulu-mukhagiri-jāka-bejhādi- duṣṭāḥ and the viṣāmbhonidhi Wikipedia. This take over aligned with the subversion of these vyāpāra-s by the navyonmāda. The prelude to this was seen when a servant of guggulu was expelled for voicing his opinion. While people thought it was just an internal company matter, it was clear that the navyonmāda was moving to end to freedom expression. A feedback loop developed between new social media and another major development, i.e. the ubiquity of the smart phones. The latter made every man perpetually visible to the operatives of the mleccha deep-state. The dangers of the reach of these duṣṭa-s along with the mleccha deep-state was exposed by their rogue spaś Himaguha who escaped to the khaganate of Putin. The real action was seen in the past year in collusion with the conventional media to overthrow the mleccharāṭ prajalpaka Vijaya and replace him with their favored man Piṇḍaka, now provided with a court of navyonmatta-s. With that the internet became a weapon for the navyonmatta-s who are directing its full force at their longstanding foe, the Hindus.

Some basic principles for the vigraha of the loka-saṃgrāma
• The foundations of Hindu polity lie in the actions of the deva-s in the śruti by which they overthrew the ditija-s in battle after battle by ūrja, māya and astrāṇi. This was translated for the human sphere by pouring the heroes into the divine bottles in the Itihāsa-s. Finally, it was codified by the clever Viṣṇugupta who aided the Mauryan to overthrow the evil Nanda-s and the yavana-s. It was presented for bāla-s by the wise Viṣṇuśarman, an acute observer and pioneer in the study of biological conflicts. He laid out the forms of vairam. Among those is svabhāva-vairam.

• Being ekarākṣasonmāda-s and vidharma-s (counter-religions), the unmāda-s and dharma are locked in svabhāva-vairam — a conflict that ends only in the extinction of one of the parties in the long run. The ekarākṣasonmāda-s have destroyed many of our sister religions and we remain the only remaining bulwark against them. Some object that the Cīna-s and Uṣāputra-s are also there — so why claim that we are the bulwark. We argue that the Cīna-s are seized by their own sādhana of legalism (see below) that has rendered them quite weak in terms of religion. The Uṣāputra-s, while doing well for themselves, are not a force that can restore heathenism in the world, especially given their currently aging and declining population. The graph in Figure 1 and history shows that there is some truth this “viśvaguru” quality of the H, even if it has declined over the last millennium.

rogād rogaḥ | iti roga-paramparā | ko .ayam rogaḥ? mānasikaḥ | kutra rogasya janma? marakatānām uttare .asmadīyānām mitanni-nāma-rāṣṭrasya paścime .abrahmaś ca mūṣaś ca joṣaś cetyādīnām rākṣasa-graheṇa grasta-manaḥsu | tasmāt pretaḥ śūla-kīlitaḥ | tadanantaraṃ mahāmadaḥ | navajo rudhironmādo dāḍhikamukhasya | tasmād idānīṃtano navyonmādaḥ | parasparaṃ yudhyante kiṃ tu dharma-prati teṣām virodhaḥ saṃyuktaḥ | kasmāt? | vidhārma-bhāvād viparīta-buddhyā roga-tulyaikarākṣasa-viśvāsād deva-mūrti-dveṣāc ca | tasmād ucyate mleccha-marūnmattābhisaṃdhiḥ | tasya bṛhadrūpaṃ sarvonmāda-samāyoga-rākṣasa-jāla-śambaram | idaṃ hindūkānām paramam vairam ||

• The understanding of the Chinese state in most Occidental and Indian presentations ranges from misguided to deeply flawed. Two key concepts are required to understand its behavior and threat potential. The first, the doctrine of “legalism” or fa jia, whose early practitioner Lord Shang played a notable role in the rise of the Chin — in many ways he can be seen as the Viṣṇugupta of the Cīna-s who laid path for their unification under Chin Shi Huang, who played the role comparable to our Mauryan Candragupta. This doctrine, while often denied, has dominated Cīna imperial action since. While it is a rather sophisticated system, which is outside the scope of this note, a key feature is mutual spying that helps keep society in check — a convergent feature with other totalitarian systems. In it the ruler might keep the people busy with a benign “outer coat” that keeps the imperial designs out of their sight, or to paraphrase the neo-emperor Deng Xiaoping, they will adjust to follow the wind blowing from the rulers. Over the ages, the Cīna imperium has used Confucianism, Bauddham, Turkism, socialism and westernism as the outer coats to conceal their imperial actions. This legalism makes the Cīna-s ruthless and dangerous adversaries who are difficult to read. Even if they might not be rākṣasonmatta-s, the imperial focus of the system makes them hungry for land and ādhipatyam. For this they might play a long game, slowly encroaching on land, millimeter by millimeter and playing victim when their land-grab is noticed. Using that confusion, they would try to settle the situation in their favor. However, their aging population is the biggest road block to their total victory.

• The second concept that we have laid out in these pages in some detail is Galtonism. It describes a certain type of sinophilia that permeates the West in a form first articulated by the English intellectual Galton. In it, the Occidental center sees a great power in China and is almost in awe of it from the cracking of their psychometric yardsticks such IQ, and finds them to be of a “identifiable” fair complexion (at least the more northern subset) and a very orderly people. Thus, in contrast to the Hindu, they are willing to concede a global role for the Cīna-s, despite they being heathens. Conversely, they see in the Hindu simultaneously a defiant “other”, “an ugly people” and an idiot incapable of playing any great global role. In fact any attempt on their part to do so is seen as a dangerous challenge to their ekarākṣasam undergirding that should be squelched right away. A distinct strain of Galtonism is that seen in the navyonmatta-s (e.g. starting with their boosterist, the naked Needham, down to duṣta-Sora): for them the Cīna state is a culmination of their own utopian doctrine — of course they would ignore the fact that their own implementations fail and try to claim the genius of the Cīna-s for themselves. Thus, they play a potent role as ready apologists for the Cīna imperium.

In retrospect
Looking back, late Huntington was right in terms of the great clash between the marūnmatta-s and mleccha-s that was to play out in his own last years. He was also right in that the Cīna-s would ally with the marūnmatta-s to get back at their foes. However, this did not develop globally as the Cīna-s had their own marūnmatta terrorism, which they recognized as an unmāda and treated as such. Hence, the Cīna-s limited its use to India, since there was the ever-willing TSP available as a bhṛtya who would not blow back. In the end, despite the rise of the Khilafat under Dr. Abu Bakr al Baghdadi, the mahāmleccha triumphed in this round of the conflict though their cousins in Europe might be eventually conquered.

What Huntington did not foresee was that the battle would be brought to the world by a new force, the navyonmāda, backed by the sora-jāka-mukhagiryādi-duṣṭāḥ. Like Constantine seizing the Roman empire for the pretasādhaka-s, these have placed a pliable man Piṇḍaka at the helm surrounded by navyonmatta-s. This war has already reached the Hindus. It will ally with the Cīna-s and the marūnmatta-s against their common foes. In an extreme scenario it might provide the final bridgehead the marūnmatta-s need for their conquest of mleccha lands.

The Cīna-s and marūnmatta-s have a degree of immunity to the navyonmatta-s. That is in part because the former have sealed off their internet and created their own parallel world like that of Viśvāmitra for Triśaṅku. The marūnmatta doctrine is a superior, fecundity-supporting version of the navyonmāda; hence, it is going to be hard to breach. In the long run the dynamics of navyonmāda are unclear due its contra-reproductive strategies. However, in the short run it could wreak havoc on the Hindus, especially their elite, who seem to be particularly susceptible to this disease. Going forward, at least for the next several years, models of all the older conflicts in geopolitics have to be updated to account for the role navyonmāda will play. Whatever the case, as far as H go, it will ally with the other unmāda-s against them. It will also split the mūlavātūla-s into pro- and anti- camps, a dynamic that might cause some instability to it.

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The phantoms of the bone-pipe

As Vidrum was leafing through some recent case studies to gather the literature for his own production, he received a call from his chauffeur. He had fetched Vidrum’s new car. Vidrum went out to take a look at it. As he saw it gleaming in the mellow light of the parkway lamp he thought of his old friends for some reason: “Clever Lootika or Vrishchika would have said that it looks like a work of the Ṛbhu-s. That triplet of deities meant a lot to the four sisters, but I had never heard of them before I came to know them. May be after all there is a reason why they say the brāḥmaṇa-s are the conduit for communicating with the gods. No wonder this new car looks good but for some reason I experience no thrill of the kind I experienced when I got my first bicycle or for that matter my lamented old car.” He was snapped out of his musing by his chauffeur who asked him if he would want to go out on a test drive. Vidrum: “Sure. Let us drive till the foothills of the temple of Durgā past the pond of lotuses and then go over to the hotel Kūrmahrada and buy some dinner to take home.”

Back home from the test drive, Vidrum rang in his butler and informed him that he had obtained dinner from outside and offered the butler and his wife a packet of the same too. He then asked the butler to prepare cold turmeric and almond milk for the night and dismissed him. As he was enjoying his mouth-tingling dinner he lapsed into a train of thought: “I wish Somakhya was around to cast a spell of protection on my car. Time and again my mind goes back to my lamented first car. I remember that day clearly.” As his mind drifted there, his joy from the tasty dinner flattened quite a bit. Having concluded his meal, he went over to the little shrine Somakhya had installed for him to worship the 16-handed Vīryakālī, whose original was enshrined at the edge of the cemetery in his ancestral village. When Somakhya and Indrasena had learned of that shrine they were exci