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.

Figure 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 \textit{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.

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:

$y=\pm\dfrac{B}{2}\sqrt{\dfrac{L^{2}-4x^{2}}{L^{2}+8wx+4w^{2}}}$

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 device 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 yield 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.

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

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$_4$s, 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$_4$s 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.”

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 |
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.

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 |
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.

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.”

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
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.

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.

Game 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 $o=0.6$.

Game 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.

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 wife birthed a boy.
The boy became a man
<repeat>

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.

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 ways, 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.

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 $j$:

$p_{ij}=u^{D_{ij}}(1-u)^{L-D_{ij}}$

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:

$(1-u)^L=0.97$

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.

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:

Figure 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 Said 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.

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.

Figure 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,
You have healing powers in your thumb,
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.

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:
1) https://www.indictoday.com/videos/manasataramgini-civilization-counter-religion-continuity-collapse-i/
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.

2) https://www.indictoday.com/conversations/manasataramgini-civilization-counter-religion-continuity-collapse-ii/
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.

3) https://www.indictoday.com/conversations/manasataramgini-civilization-counter-religion-continuity-collapse-iii/
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.

4) https://www.indictoday.com/conversations/manasataramgini-civilization-counter-religion-continuity-collapse-iv/
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.

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.

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.

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:
śrauṣaḍ
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.

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 ॥
OṂ BHŪR BHUVAḤ SVAR JANAD VṚDHAT KARAD RUHAN MAHAT TAC CHAM OṂ viṣṇave svāhā + idaṃ viṣṇave na mama ॥

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.

śrauṣaḍ
pātho ghṛtasya guhyasya nāma ।
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 ॥
OṂ BHŪR BHUVAḤ SVAR JANAD VṚDHAT KARAD RUHAN MAHAT TAC CHAM OṂ viṣṇave svāhā + idaṃ viṣṇave na mama ॥

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 ॥
OṂ BHŪR BHUVAḤ SVAR JANAD VṚDHAT KARAD RUHAN MAHAT TAC CHAM OM indravantaḥ svāhā ॥

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 ।
jyeṣṭha indriyāya ṛṣibhyo
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?”

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
yato jajña ugras tveṣanṛmṇo RU
patiṃ vo aghnyānāṃ |
anu yaṃ viśve madanty ūmāḥ ṢO
dhenūnām iṣudhyaso3m ॥

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?”

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.”

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 $n$th 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 $n$th triangular number.

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.

Figure 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.

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.

\documentclass[margin=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows, arrows.meta, patterns, shapes.geometric, decorations.shapes, shapes.misc, graphs, mindmap, calc, backgrounds}

\begin{document}
\begin{tikzpicture}
\pgfdeclarelayer{background}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}

\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
\begin{pgfonlayer}{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;
\end{pgfonlayer}

%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);

%onion
\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;

%petal
\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;

%chandrabindu
\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);
\end{scope}
%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]
\def\y{20}
\def\z{sin(30)}
\def\w{1}
\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);
\end{scope}
%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);
}

%tetrafolium
\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]
\def\y{30}
\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);
\end{scope}
}

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


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.

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.

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.

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$_3$R-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

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?

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.

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.

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
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).

Figure 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
5 KL, TD, TLL, DDL, DLLL, LLLLL 6
6 KD, KLL, TT, TDL, TLLL, DDD, DDLL, DLLLL, LLLLLL 9
7 KT, KDL, KLLL, TTL, TDD, TDLL, TLLLL, DDDL, DDLLL, DLLLLL, LLLLLLL 11

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:
$f[n]=f[n-3]+f[n-1]-f[n-4]$
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 $P \le n; n=3, 4, 5 \cdots \infty$.

Figure 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.

Figure 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: $1.0003^c$.

Figure 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.

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 $z=x^2+y^2$.

Figure 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.

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

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

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

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

Figure 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.

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

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.

Figure 2.

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.

Figure 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.

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 excited. They told him of the great significance of that goddess as per some ancient text whose name he had forgotten. He had promised to take them along with Lootika and Vrishchika to his village someday. He felt his meditation gave him some focus to put a few words to paper reporting a fatal case involving the infection of a sanitation worker by Burkholderia mallei. As he was doing so, he remembered that Somakhya had given him a monologue on the bacterium when they were in college but Vidrum was not yet a suitable vessel to have imbibed any of it. He kicked himself for the same because he knew it was something important for his current investigation but he just did not recall anything. But Somakhya and his gang were far away and in limited contact; hence, he had to contend with whatever he had.

His mind went back to his old car and the events around it careened across his mental screen: “I still remember those days. I had a fun ride and drove by the shuttle-stop at the college to park the car and collect some stuff from my cubicle. On my way out I saw Vrishchika and her mother waiting for the shuttle. I offered to give them a ride home. They accepted it with much gratitude but, as was typical of them, the two remained silent through much of the ride. Unfazed by the bustle of the city passing by us, Vrishchika’s mom was grading exam papers. That brought back memories of her class. Being busy with her daughters, she only taught part time, but was perhaps the only one of all our female and most male instructors who had the command and the ability to make everyone understand — some of that she had passed on to her daughters. But her exams were always hard. Nevertheless, she was kind unlike many of our sadistic lecturers, and passed everyone in the final exam. As we neared their home, Vrishchika’s mom looked up and said: I guess you don’t have anything at hand for dinner — let me pack you some.’ I graciously acquiesced. When we got to their home, I was seated in the hall as Vrishchika and her mom went into the kitchen. Vrishchika came out with a small spoon and asked me to try a pickle and see if I liked it. It was great. Her mother then responded from the kitchen that she would aliquot a bottle of the pickle for me. As she was doing so, Vrishchika asked me if I was finding my new apartment boring without the friends’ from the cemetery. I had to confess that I missed a bit of all that drama though I certainly found the quiet rather beneficial. Vrishchika darted into her room and brought out a curious object. It was a musical bone-pipe made from a human femur. Vrishchika waved it in the air it made some haunting music, like the wooden pipe made by the tribesmen from the northern Marahaṭṭa country or southern Mālava. She said I could blow into it and I might get a visit from interesting phantoms if they happened to be pleased with the music. I was apprehensive of any such gift but she told me it will do me good in life. I wished Vrishchika good luck because she was leaving abroad and I was not sure if I would ever see her again.

With my dinner in hand, I drove back home with the bone-pipe. That weekend I blew out of it the tune of a film song. To my surprise I felt a presence as I used to feel in my old house. There was no one in my room but I could still feel someone seated next to me. Just as we would do when plying the planchette, I asked if somebody was around. I mysteriously fell asleep at that instant and saw a vision of my new car being destroyed and me dying in the crash. I woke in fear and wished I could talk to Vrishchika, Somakhya or Lootika about it but they were all gone. I remembered a strange statement from Lootika when my first bike was stolen, which she did not elaborate on: Wheel after wheel would be destroyed but your wheel would keep turning.’ The day that prophesy came true is still fresh in my mind. By some terrible coincidence my car fell into a ditch the very place Meghana had died. I was indeed lucky not just to evade the appointment with Citragupta but to escape unscathed from such a tremendous crash. I thought to myself — may be, l still have great acts to do in life.”

Vidrum shaken out of his reverie by his butler’s knock on the door to give him his beverage. He felt a sudden urge to play the bone-pipe. He lit a bundle of sage and as it was smoldering he took out the pipe from a box where he carefully stored it and blew out a tune of a folk song to the 1000-eyed goddess that his grandmother had taught him as a youth. For a while nothing happened and Vidrum relaxed into the wafting odor of the sage along with his beverage. With his mind crowded by various events of the past, he almost forgot that he had plied the pipe when he jolted by the presence of a strong fellow with some East Asian ancestry, albeit felt only vaguely. The intensity of the presence soon increased as it grabbed him by his legs and thrust into his chair. There he felt another presence seize his very personage and launch him into a bout of frenzied writing spanning a few pages.

Somakhya and Lootika were sipping their tea as the sun’s declining rays streamed into their room. Somakhya passed his tablet to Lootika with the scan of a manuscript on it: “varārohe, what do you make of this?” L:“Why dear, though eminently legible, this is a very strange handwriting with a form reminding one of sparklers on a Dīpāvalī night. What is this strange manuscript?” S: “That metaphor for the writing is indeed very apt. It was something Vidrum sent me. He apparently took it down some time ago in a frenzied ghost-dictation induced by the bone-pipe your first sister had found in our youth. He prefaces it with the comment that it would be of greatest interest to us. Why don’t you read it out aloud?” L: “Ah, the musical bone-pipe. I had nearly forgotten about that one. Should I call the kids; may be they would like the story?” S: “I’ve not read it yet. So, let us examine it first to make sure it might be of interest or even appropriate for them. So, let them continue practicing the workout of the conics that my father has sent them.”

Lootika read it out: “I’m glad I’ve found someone to tell my story as also a bit of that of my friend. It was my friend who thrust you into your chair so that I could tell my story. I was a brāhmaṇa, Bāẓ Nayan by name. My ancestors had come all the way from Jammu and settled in the hamlet of Indargaon. We followed the Mādhyaṃdina school of the Śukla-yajurveda. Seeing my precocious capacity in absorbing the Veda after my upanayana, I was sent far from home to the gurukula at Rishikesh. There, I acquired the śruti to completion and also become a scholar of vyākaraṇa mastering the ins and outs of sage Pāṇini and his commentator Patañjali. But tragedy struck at that point as my family was wiped out in an earthquake. Left with no one, I went and sought refuge at the feet of svāmin Ātmānanda giri, a great advaita yati. He employed me as a teacher for his brahmacāri-s. Later he acquiesced to my intention to acquire an English education and study linguistics. I did so in Shimla for 5 years and translated the pariśiṣṭa-s of Kātyāyana into English. During my stay there, I befriended a Gorkha, Jang Bahādur, who had to retire from the army after sustaining injuries in a battle with the Cīna-s. Unfortunately, svāmin Ātmānanda giri’s āśrama was washed away in a great flood and I was again left with no one in the world other than my friend Jang Bahādur.

One day he showed me an advertisement in a paper for a Sanskrit professor in a great peninsular city and suggested that we go there. He said he might find a security job there. I thought it was a great idea and after a long journey by train with barely any money we made it to the city. Thankfully, they spoke and understood some Hindi there and we could make our way to a rat-and bug-ridden lodge to stay while we found a job. The job was at the College of Antiquities which was one of premier research centers in the country. The clerks there asked me for a domicile certificate and a nationality certificate. As I had neither, they rudely shooed me away. I had to make a living initially as a cook and then as an arcaka at a temple of the terrifying Vināyaka. Jang Bahādur found a job as the nightwatchman for a street.

I still believed I was Sanskrit professor material — I was confident that few people knew the intricacies of the vyākaraṇa, as it applied to the śruti, the sāmānya language or the vulgar Prākṛta-s, as I did. Hence, I went back to the college in the hope of meeting some Sanskritists who would see my true worth. I found my way into the campus by somehow convincing the guard to let me in by claiming I was paṇḍita who had been called for a meeting. I searched around to reach to office of a brāhmaṇa from the peninsula, Somaśiva Śarman. He was a learned scholar of both the pure āryavāk as well as its vulgar vikṛti-s, who was engaged in the project of a great compendium of pre-modern knowledge. He looked at me quizzically, wondering if a brāhmaṇa could ever have a name as mine. He asked my gotra and śākhā and then asked me recite sections from the Vājasaneyi śruti. I noticed he was beginning to believe me as I did so. I took the opportunity and pulled out my precious typeset manuscript of the edition and the translation of the Kātiya pariśiṣṭa-s from my bag and handed it over to him. He studied it intensely for a while and looking up remarked that he needed to hire me right away. He moved the bureaucracy with much effort to get me the post of a staff-paṇḍit at the college.

Lootika paused and interjected: “O Bhārgava, this is most interesting. The phantom’s tale is directly intersecting with our lives. Śilpikā is none other than our learned language teacher, whom we gave much grief as children, and the young lady he mentions is undoubtedly your own mother — perhaps from the days just before your birth.” S: “It has to be so. I really hope Śilpikā did not cast a spell on us that our children regress to the mean. Yet, this phantom seems unfamiliar to me. Pray continue dear.”

L: “That lady’s paper suggested a topic for my dissertation, which was always as at the bottom of my mind. It was a detailed comparative study of all Yajurveda texts. I worked hard and wrote an over 1000-page monograph of the subject. It featured many new translations, detailed analysis of the śrauta practices and the like. Jayasvāmin and Somaśiva presided as my preceptors and I was awarded the coveted title of Professor. I was at the height of my powers and wanted to publish my dissertation as a two-volume work. But the gods apparently had other plans. Two disasters struck our college and me personally. That summer, during the vacations, a band of uśnīśin terrorists broke into our museum. Jang Bahādur bravely defended the premises but was killed in the process and the terrorists made away with the Gupta gold coins and melted them down to finance their operations. A few months later, left-liberal activists, claiming to be righting the wrongs done to the depressed classes, demolished a wall of our college and fired our archives. As a result, my typeset dissertation was burnt down and lost.

Sometime before that, Cidānanda yati, a survivor of the advaitāśrama that was washed away, had come to city and was conducting classes on Śaṅkarācārya’s tradition. He called me to meet him and help him with the translations of the texts of Appayya Dikṣita and Girvāṇendra Dikṣita that he was preparing. In course of those discussions, he introduced me to a song composed by one of the Śaṅkaramaṭha-s and told me that no amount of yaci bham and phak were going to take me anywhere if I did not awaken from the dream of phenomenal existence into satchidānanda. The former would only take me on the path of rayi to the realm of the Moon, where the forefathers dwell. Then I would return to be born again, he said. Instead, if I followed the path of austerity, celibacy, faith and Brahmavidyā, I would have the great awakening into the sole reality that is Brahman. Soon thereafter a great comet appeared in the sky. A little later, the Japanese man who was the first man to observe that comet died.

Those events got me thinking about the arrival of the Pitṛrāṭ. I was sad on one hand about the loss of my dissertation on the other the possibility of never transcending the realms to know that there is only satchidānanda. I rationalized that there was no reason to fear death at all — after all it is something no one experiences. When one is alive there is no obviously no experience of death. When one is dead there is no experience of death; so, why fear something one never experiences. However, for many death can come with suffering. There was no way to prevent that even if one did not fear death for the suffering before death was after all a real experience. But then if one experienced that which is satchidānanda then one would realize that the suffering was just like in a dream. But if one does not experience satchidānanda then what are the experiences after death, if they exist at all, I wondered.

I was to get the answer for all this soon. Sundara Somayājin was a Soma ritualist from the Drāviḍa country. He was performing a Somayāga in the city. I had long wanted to witness that kind of a Yāga and went to meet him. We had a philosophical discussion on the Vedic religion. He said that what was important was the correct svara of recitation and precise execution of ritual actions as per the ritual treatises. It really did not matter if the gods like Indra or the Aśvin-s existed. They were anyhow not gods to be worshiped in the same sense as the ‘real̍ gods’ like Śiva or Gaṇeśa or Viṣṇu, he said. I became very afraid when I heard this. I reminded him that such words were uttered by the ignorant to the great ṛṣi Nemo Bhārgava:

nendro astīti nema u tva āha ka īṃ dadarśa kam abhi ṣṭavāma ||
“Indra does not exist, o Nema” So indeed he says.
“Who ever has seen him?” “Whom shall we praise forth?”

Thus the ignorant questioned the existence of the great god. But he made his presence felt:
ayam asmi jaritaḥ paśya meha viśvā jātāny abhy asmi mahnā |
ṛtasya mā pradiśo vardhayanty ādardiro bhuvanā dardarīmi ||

Here I am, o chanter: see me here.
I’m at fore in all the species by my greatness.
The directives of the natural laws magnify me.
As the smasher, I keep smashing the worlds.

The Somayājin dismissed me by saying that it was all arthavāda and after all no Indra appeared at some given time to some Nema because the Veda was coeval with the beginning of time. In one of the talks in the college, I had heard another śrauta ritualist talk about the Seat of Vivasvān. He had mentioned that its knowledge was very important to evade the arrow of Ugra Deva when one is performing a yāga. Unfortunately, our Somayājin did not seem very aware of it and was to learn the reality of Indra very soon. Perhaps, due to some pāpa I had committed in a past janman, I too was to bound in karman with him. After the Ṣoḍaśin-graha was taken, the sky blackened with a mass of clouds and a great streak of lightning followed by a thunderous peal struck the pandal that had been erected for the yāga. An electrical explosion and fire followed and the learned Sundara Somayājin was borne to the abode of Vivasvān’s son, like Meghanāda struck by the Indrāstra discharged by the Saumitri or like Arṇa and Citraratha being felled by Maghavan beyond the Sindhu or like the Sāmavedin-s of Vaṅga or Aṅga being washed away by a blow from his vajra. I too was consumed by the fierce Kravyāda on that day.

A month or so after my expiration, a band of socialists paid by a mleccha instigator, claiming to be acting on behalf of the depressed classes, attacked my college campus again. My house and belongings were among the things consumed in their arson. In my heydays, among my many foreign visitors was a Gaulish woman, Laetitia Vernon by name, who sought my help to read Sanskrit legal texts. Despite my many stern warnings, my only son got infatuated with her and having married her left for the shores of a mleccha land. My son having adopted the mlecchānusāra did not perform any kriyā-s for me. He instead wrote an article in my memory saying that the secular India was coming of age with progress and equity even as the brahminical superstition was becoming a thing of the past. Consequently, upon my death I wander as a brahmarakṣas. At least me and my friend Jang Bahādur are united in death and we lead a mostly quiet incorporeal existence haunting the little hill that lies between my college and the river.”

Some notes on the Brahmayajña brāhmaṇa and Uttama-paṭala of the Atharvaṇ tradition

The Brahmayajña brāhmaṇa (1.1.29 of the Gopatha-brāhmaṇa) of the Atharvaveda provides a glimpse of the Vedic saṃhitā canon as known to the brāhmaṇa authors of the AV tradition. The Brahmayajña might be done as part of the basic rite as done by dvija-s of other śākhā-s or as part of the more elaborate AV tradition of the annual Veda-vrata. The annual vrata-s of the Atharvaṇ brāhmaṇa-s include the Sāvitrī-vrata, Veda-vrata, Kalpa-vrata, Mitra-vrata, Yama-vrata and Mṛgāra-vrata. The kṣatriya-s and vaiśya-s should do at least 3 and 2 of them respectively, with the first 2 being obligatory. During these vrata-s the ritualist follows certain strictures like not consuming butter milk nor eating kidney beans, common millets, or the masura lentils at the evening meal, bathing thrice a day and wearing woolen clothing. Before performing Brahmayajña, he performs the ācamana as per the vidhi which states:
sa ācamanaṃ karoti |
He performs the ritual sipping of water.

This calls for the special Atharvaṇic ācamana described in the final section of the ācamana-brāhmaṇa of the AV tradition (GB 1.1.39):
It has also been thus stated in the ṛk:

“āpo bhṛgvaṅgiro rūpam āpo bhṛgvaṅgiromayaṃ |
sarvam āpomayaṃ bhūtaṃ sarvaṃ bhṛgvaṅgiromayam ||”
The waters are of the form Bhṛgu-Aṅgiras incantations. The waters are imbued with the Bhṛgu-Aṅgiras incantations.
All being is imbued with the waters; [thus,] all [being] is imbued by the Bhṛgu-Aṅgiras incantations.

•The Atharvaṇ-s justify the above ṛk is by noting that the Paippalāda Atharvaveda begins with the ṛk “śaṃ no devīḥ…” to the waters (see below).

antaraite trayo vedā bhṛgūn aṅgiraso ‘nugāḥ ||
Within these [waters] the three [other] Veda-s follow the Bhṛgu-Aṅgiras incantations.

“apāṃ puṣpaṃ mūrtir ākāśaṃ pavitram uttamam” iti ācamyābhyukṣy ātmānam anumantrayata | [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”. Thus, he sips the water and having sprinkled water (practically mārjanam) he recites the incantation indra jīva etc: Enliven, o Indra; Enliven o Sūrya. Enliven, o gods. May I live. May I complete my term of life.

•The flower of the waters in the above incantation is an allusion to the ṛk describing the the ancient action of the Atharvaṇ-s in kindling the fire in waters [from a lotus]: tvām agne puṣkarād adhy atharvā nir amanthata |”
•He does the ācamana by taking three sips each with two successive words from the mantra apām puṣpam…

iti brāhmaṇam ||
Thus is the brāhmaṇa.

Now for the Brahmayajña:
kiṃ devatam iti ? ṛcām agnir devatam | tad eva jyotiḥ | gāyatraṃ chandaḥ | pṛthivī sthānam |
“agnim īḷe purohitaṃ yajñasya devam ṛtvijaṃ | hotāraṃ ratnadhātamam ||”
ity evam ādiṃ kṛtvā ṛgvedam adhīyate ||
Who is the deity? Agni is the deity of the ṛk-s. That is indeed light. Gāyatrī is its meter. The earth is its station.
“I praise Agni, the officiant of the ritual, the god and ritualist; the hotṛ and the foremost giver of gems.”
Thus, having placed it at the beginning the Ṛgveda is studied.

yajuṣāṃ vāyur devatam | tad eva jyotis traiṣṭubhaṃ chandaḥ | antarikṣaṃ sthānam |
iṣe tvorje tvā vāyava stha devo vaḥ savitā prārpayatu śreṣṭhatamāya karmaṇe ||
ity evam ādiṃ kṛtvā yajurvedam adhīyate ||
Vāyu is the deity of the Yajuṣ-es. That is verily light; Triṣṭubh is its meter. The atmosphere is its station.
“To you for nourishment, to you for strength. You are the Vāyu-s. May Savitṛ impel you the most excellent ritual.”
Thus, having placed it at the beginning the Yajurveda is studied.

sāmnām ādityo devatam | tad eva jyotiḥ | jāgataṃ chandaḥ | dyauḥ sthānam |
“agna ā yāhi vītaye gṛṇāno havyadātaye | ni hotā satsi barhiṣi ||”
ity evam ādiṃ kṛtvā samāvedam adhīyate ||
The Āditya is the deity of the Sāman-s. That is indeed light. Jagati is its meter. The heaven is its station.
O Agni, come to the oblations, praised with songs to the ritual offering. Sit as the hotṛ on the ritual grass.
Thus, having placed it at the beginning the Sāmaveda is studied.

atharvaṇāṃ candramā devatam | tad eva jyotiḥ | sarvāṇi chandāṃsi | āpaḥ sthānam | <śaṃ no devīr abhiṣṭaya> ity evam ādiṃ kṛtvātharvavedam adhīyate ||
The moon is the deity of the Atharvaṇ incantations. That is indeed light. All are its meters. The waters are its station. “May the divine [waters] be auspicious for us…” Thus, having placed it at the beginning the Atharvaveda is studied.

adbhyaḥ sthāvara-jaṅgamo bhūta-grāmaḥ saṃbhavati | tasmāt sarvam āpomayaṃ bhūtaṃ sarvaṃ bhṛgvaṅgiromayam | antaraite trayo vedā bhṛgūn aṅgirasaḥ śritā ity ab iti prakṛtir apām oṃkāreṇa ca | etasmād vyāsaḥ purovāca:
“bhṛgvaṅgirovidā saṃskṛto ‘nyān vedān adhīyīta |
nānyatra saṃskṛto bhṛgvaṅgiraso ‘dhīyīta ||”
From the waters the families of immotile and motile organisms have come into being. Hence, all being is imbued with water; [thus] all is imbued with the Bhṛgu-Aṅgiras incantations. The three other Veda-s are situated within these Bhṛgu-Aṅgiras incantations. Therefore, indeed it is water and the origin of water is by the Oṃkāra. In this regard Vyāsa had formerly said:
“He who is sanctified by the Bhṛgu-Aṅgiras incantations may study the other Veda-s.
The one sanctified elsewhere should not study the Veda of the Bhṛgu-Aṅgiras-es.”

•Regarding the origin of all beings from water: this is articulated early on in the ṛk: yo apsv ā śucinā daivyena… (RV 2.35.8) of Gṛtsamada Śaunahotra.

sāmavede ‘tha khilaśrutir brahmacaryeṇa caitasmād atharvāṅgiraso ha yo veda sa veda sarvam |
iti brāhmaṇam ||
Now there is also the khila of the Sāmaveda: “Therefore, he who as a celibate student knows the Veda of Atharvāṅgiras-es knows all this.”

Thus is the brāhmaṇa.

•The statement from the Sāmaveda-khila is also taken to justify the punarupanayana that is performed in order for those of other traditions to study the Atharvaveda.

Notes
Several notable points are raised by the Brahmayajña brāhmaṇa of the AV, not just regarding the AV tradition but also regarding its interaction with the other Vedic schools and their own evolution. It is quite obvious that the Brahmayajña brāhmaṇa represents a relatively late brāhmaṇa composition with a specific aim of justifying the primacy of the AV, probably in the context of the intra-brahminical competition for the position of the brahman in the śrauta ritual. This is explicitly supported by the fact that it cites Vyāsa [Pārāśarya] who appears in late Vedic texts and is remembered by tradition as the redactor of the 4 fold form of the śruti. In a similar vein, the citation of the Sāmaveda-khila suggests that it was composed after the terminal sections of the Sāmavedic tradition had been completed.

The opening ṛk of the RV is compatible with any of the śakha-s of the Ṛgveda. The Yajurveda that it refers to is clearly the Vājasaneyi saṃhitā (either Mādhyaṃdina or the Kāṇva śākhā-s). The Samaveda could again be any of the Samavedic saṃhitā-s. The Atharvaveda is probably the Paippalāda saṃhitā because the vulgate and the Śaunakīya begin with “ye triśaptāḥ…” However, we must note that we do not know the beginning of the lost AV śākhā-s.

Why is this notable? The AV-pariśiṣṭa 46 (Uttama-paṭala) gives the beginning and end verses of the four Veda saṃhitā-s along with several AV verses to be used in the annual Veda-vrata. Notably, these are partly different from those of the Brahmayajña brāhmaṇa. Interestingly, according to the Uttama-paṭala, the RV ends with the famous ṛk: “tac chamyor āvṛṇimahe…”. This is not present in the Śākala-pāṭha which instead ends with the short Saṃjñā-sūkta. The former ṛk was claimed by Michael Witzel to be the last ṛk of the Bāṣkala RV. However, as Vishal Agrawal correctly noted its is stated to be the last ṛk by even the Śāñkhāyana and Kauśītaki traditions. Thus, the Uttama-paṭala is referring to some RV śākhā other than Śākala, though we cannot be sure of its identity.

The Uttama-paṭala gives the Sāmaveda’s first verse as “agna ā yāhi…”, which is known to be the first ṛk of all surviving śākhā-s of the SV. However, the last ṛk is given as:
“eṣa sya te dhārayā suto ‘vyo vārebhir havane maditavyam | krīḍan raśmir apārthivaḥ ||”
This is different from the ṛk “svasti na indro vṛddhaśravāḥ…” with which the surviving SV śākhā-s conclude. It is a divergent variant of the ṛk RV 9.108.5 not attested elsewhere. In fact the extant SV saṃhitā-s contain a version that follows the RV cognate. Thus, evidently the Uttama-paṭala is referring to a now lost SV śākhā.

The situation with the YV is the most interesting. The cited starting mantra goes thus:
“iṣe tvorje tvā vāyava sthopāyava stha devo vaḥ savitā prārpayatu śreṣṭhatamāya karmaṇa āpyāyadhvam aghnyā indrāya bhāgam ūrjasvatīḥ payasvatīḥ prajāvatīr anamīvā ayakṣmā mā va stena īśata māghaśaṃso rudrasya hetiḥ pari vo vṛṇaktu dhruvā asmin gopatau syāta bahvīr yajamānasya paśūn pāhi ||”

Remarkably, this mantra is not found in any of the extant YV saṃhitā-s. However, the “indrāya bhāgam” is reminiscent of the “indrāya deva-bhāgam” found in the Āpastamba-śrautasūtra and the Bhāradvāja-śrautasūtra or the “devebhya indrāya” found in the Maitrāyaṇīya saṃhitā. Moreover, the last mantra of the Yajurveda is given as “dadhikrāvṇo akāriṣam…”. In other YV saṃhitā-s, this mantra occurs in the Aśvamedha section and is used among other thing by the ritualists to purify their mouths after the obscene sexual dialog. However, it is not the last mantra of the Aśvamedha section in any of the extant saṃhitā-s. This indicates two things: first, the Uttama-paṭala is recording a now lost YV śākha of the Kṛṣṇayajurveda (KYV). Second, while today Āpastamba and Bhāradvāja are attached to the Taittirīya-śākhā, they were once the sūtra-s of a lost KYV śākhā. This loss likely happened relatively early. It was probably associated with the southward movement of the Āpastamba-s and Bhāradvāja-s, who then shifted to the related Taittirīya-saṃhitā (TS). The text of the Baudhāyana-śrautasūtra precisely follows the TS; hence, it was definitely one of the original sūtra-s of the Taittirīya-śākhā.

Finally, the AV of the Uttama-paṭala begins with “ye triśaptāḥ…” indicating that it was recording the original śākhā behind the vulgate or the Śaunakīya.

Thus, we see a striking difference between the two AV traditions of the Gopatha-brāhmaṇa and the Uttama-paṭala. While the tendency is to see the AV-pariśiṣṭa-s as late and post-dating the brāhmaṇa, we have to be more cautious. First, the AV-pariśiṣṭa-s are a rather composite mass recording a range of traditions that with a wide temporal span. Some material like the Nakṣatra-kalpa-sūkta could closer to the late brāḥmaṇa material in age whereas, at the other end, the tortoise-soothsaying (Kurmavibhāga) is likely a late text. We posit that the Uttama-paṭala belongs to the an early layer of the AV-pariśiṣṭa-s — this provides a reasonable hypothesis for the divergence between it and the brāhmaṇa. First, it should be noted that the AV-Paippalāda-AV-Śaunakīya/vulgate divergence is rather deep — mirroring the deep divergence of the Kṛṣṇa and Śukla branches of the Yajurveda. This split might have gone along with some geographical separation in the initial phase of their divergence. This geographical separation model would suggest that the Brahmayajña brāhmaṇa tradition was associated with the AV-Paippalāda or a related lost AV school that was in the vicinity of the old Vājasaneyin-s. This is also supported by certain parallels seen between the Gopatha-brāhmaṇa and the Śatapatha-brāhmaṇa in the śrauta sections. In contrast, the Uttama-paṭala as associated with the Śaunakīya or a related school that developed in the vicinity of a lost KYV śākhā.

We have evidence that the interactions between the KYV and AV traditions might go back even deeper in time: for example, this is clearly supported by the AV-related Bhavā-Śarvā-sūkta of the Kaṭha-s that was likely present in the lost Kaṭha-brahmaṇa and the various shared sūkta-s and upaniṣat material between the Taittirīya and the AV. Finally, we have evidence from what is today Gujarat that at a later period there was a certain equilibriation of the AV schools with the combination of the Paippalāda and Śaunakīya material. This parallels a similar acquisition of some Kaṭha material by the Taittirīya. Thus, there appears to have been a relatively complex web of fission and fusion interactions between the śākha-s over a protracted period.

Some notes on the Henon-Heiles Hamiltonian system

Anyone familiar with dynamical systems knows of the Henon-Heiles (HH) system. What we are presenting here is well-known stuff about which reams of material have been written. However, we offer certain tricks for visualizing this system that make it easy for lay readers with just a high school knowledge of mathematics to play with. The HH system was discovered by the French astronomer Henon and his colleague Heiles when they were studying the motion of stars in the galaxy under the influence of the gravity of the total matter in the galaxy. The true astronomical significance of these equations outside the scope of the current discussion. Our own original interest in this problem was primarily from the perspective experimental mathematics (“play physics”), starting as an extension to our interest in defining ovals by means of ordinary differential equations (ODEs). The system defined by Henon and Heiles considers the motion of a body in 2 dimensional Euclidean space, i.e. a fixed plane. The phase space describing the motion is thus defined by the variables $(x, y, p_x, p_y)$, where $x, y$ describe the position in two dimensions and $p_x, p_y$ describes the momentum in the two directions. Given that the momenta are $p_x= mx'; p_y=my'$, for a body of unit mass the momenta become the derivative of the position variables with respect to time $(x'=\tfrac{dx}{dt}, y'=\tfrac{dy}{dt})$. Henon and Heiles considered a potential described by the equation:

$V(x,y) = \dfrac{x^2+y^2}{2}+x^2y-\dfrac{y^3}{3} \kern 3em \cdots \S 1$

The potential energy of a simple harmonic oscillator in the $x$ direction is $V(x)=\tfrac{kx^2}{2}$. By taking a unit force constant $k$ we see that the terms $\tfrac{x^2+y^2}{2}$ in $\S 1$ represent two orthogonal simple harmonic oscillators. The further nonlinear term, $x^2y-\tfrac{y^3}{3}$, in $\S 1$ is a perturbation that couples these oscillators. This potential takes the form of a cubic hyperboloid-paraboloid and is visualized in Figure 1.

Figure 1.

The kinetic energy of the body is given by $T=\tfrac{mv^2}{2}$; where $v$ is the velocity of the body. Thus, for the above-defined HH system we get $T=\tfrac{x'^2+y'^2}{2}$. The Hamiltonian of a system, which represents its total energy, is given by $H=T+V$. Since this is an energy conserving system, its total energy is equal to a scalar constant $E$, i.e. the energy level of the system. Thus, for the HH system we get:

$H=\dfrac{x'^2+y'^2+x^2+y^2}{2}+x^2y-\dfrac{y^3}{3}= E \kern 3em \cdots \S 2$

If we section the 3D curve $V(x,y)$ by planes corresponding to different energy levels $z=E$, we get the equipotential curves within which the $x,y$ would lie for a given energy level (Figure 2). We observe that if $E=\tfrac{1}{6}$, the equipotential curve becomes 3 intersecting lines that form an equilateral triangle defined by the equilibrium points $(-\tfrac{\sqrt{3}}{2}, -\tfrac{1}{2}); (0,1); (\tfrac{\sqrt{3}}{2}, -\tfrac{1}{2}))$. Within this equilateral triangle, the body exhibits bounded motion. Thus, for all $E<\tfrac{1}{6}$ we get bounded trajectories in the $x-y$ plane. As $E$ becomes smaller the central equipotential boundary tends towards a circle and degenerates to a point at 0. However, for $E>\tfrac{1}{6}$ we get curves that are open; hence, at these energy levels the body can escape to infinity via the open lanes. Thus, there is a clearly defined escape energy level for this system, $E=\tfrac{1}{6}$.

Figure 2. The energy levels correspond to $E=\tfrac{1}{32}, \tfrac{1}{16}, \tfrac{1}{12},\tfrac{1}{8},\tfrac{1}{6}, \tfrac{1}{4}, \tfrac{1}{3}, 1, 2, 3, 4$

To study the trajectories under this system we first obtain the equations for the force acting on a body of unit mass (acceleration) in each direction from the above potential by taking the negative partial derivative with respect to each positional variable:

$x''= -\dfrac{\partial V(x,y)}{\partial x} = -x-2xy$

$y''= -\dfrac{\partial V(x,y)}{\partial y} = -y -x^2+y^2$

From the above can now get a system of ODEs thus:

$x'=p_x$
$y'=p_y$
$p_x'=x''= -x-2xy$
$p_y'=y''=-y -x^2+y^2 \kern 3em \cdots \S 3$

The solutions to this system $\S 3$ yield a curve in the 4-dimensional phase-space $(x,y, p_x, p_y)$. To solve $\S 3$ we first need to obtain some initial conditions for a given energy level $E$ using the Hamiltonian $\S 2$. We do that by setting $x_0=0$. We then choose some values of $y_0, p_{y0}=y_0'$. From those we can calculate $p_{x0}=x_0'$ thus:

$p_{x0}= \sqrt{2E-p_y^2-y^2+\dfrac{2y^3}{3}}$

One can see that this places a constraint on the allowed $y_0, p_{y0}$ — they have to be chosen such that $p_{x0}$ is real. Once we have these initial conditions we can solve the above ODEs with efficient LSODA solver written by Alan Hindmarsh and Linda Petzold or you can write your own solver by the method of Runge and Kutta as we did in our youth. Initial results below are shown using the LSODA solver. However, we will see below that we can also obtain solutions without using traditional ODE solutions. Figure 4 shows an example of solution for the energy level $E=\tfrac{1}{8}$ and initial conditions $x_0=0; y_0= 0.1, y_0'= 0.14$ in the 3D space defined by $x, y, y'$

Figure 3.

To get a better understanding of its behavior, we can visualize the solution in several other ways Figure 4. First, we can simply look at the way $x, y$ change with time (first 2 top left panels of Figure 4). As expected, $x(t), y(t)$ would be oscillatory functions that cannot be defined using any elementary functions. We can also examine the positional trajectory of the body in its plane of motion by plotting $x, y$ (top right panel of Figure 4). From the equipotential curves defined above from $\S 1$, we can see that this trajectory would be bounded by the closed loop of the curve defined by the equation (shown in blue):

$\dfrac{x^2+y^2}{2}+x^2y-\dfrac{y^3}{3}=E$

We can also plot $y, y'$ (bottom left panel of Figure 4) which shows how momentum changes with the position in the $y$ direction. This curve will be bounded by a special oval (shown in blue) that is determined by letting $x=0; x'=0$ in the Hamiltonian $\S 2$. This gives us a cubic curve defined by the equation (in standard $x-y$ coordinates, not the $(x, y)$ of the phase space of the solutions of $\S 3$):

$\dfrac{x^2+y^2}{2}-\dfrac{x^3}{3}=E \kern 3em \cdots \S 4$

The closed loop of the cubic $\S 4$ is the bounding oval, which was what got us first interested in the HH system in the 16th year of our life.

Figure 4.

Finally, the bottom right panel of Figure 4 shows the Poincare section that records the points where the curve shown in Figure 3 pierces the plane $x=0$ (See below for further discussions). It is obvious that these are a subset of the $y, y'$ plot and will thus be bounded by the same oval $\S 4$.

The way to compute the Poincare section is to search the $x$ values of solution for cases where the sign of $x_n$ and $x_{n+1}$ are different. Such successive points will bound the segments of the curve that pierce the plane $x=0$. Given that our steps for numerical integration are small, we can calculate the corresponding values of $y, y'$ using linear interpolation: $y=\tfrac{y_n+y_{n+1}}{2}; \; y'=\tfrac{y_n+y_{n+1}}{2}$. Plotting the thus calculated $y, y'$ will give us the Poincare sections for a given starting point. Now, we can also calculate the solutions for above system $\S 3$ without solving the ODEs by converting it into a discrete difference equation. These difference equations have a step parameter $\epsilon$, which if kept small can yield solutions equivalent to that obtained by solving the ODEs. The system of difference equations goes thus:

$p_{xn+1}= p_{xn}+\epsilon (-x_n-2x_ny_n)$
$p_{yn+1}= p_{yn} + \epsilon (-y_n+y_n^2-x_n^2)$
$x_{n+1}= x_n+\epsilon p_{xn+1}$
$y_{n+1}= y_n+\epsilon p_{yn+1}$

Figure 5.

We empirically determined that by setting $\epsilon =0.02$ we can get results similar to the solution of the ODEs with time steps of 0.01. This provides us an easy mechanism, with somewhat higher speed than the ODE solver, to obtain equivalent solutions for the HH system. This in turn allows us to explore the Poincare sections for different initial values at a much higher density. Figure 5 shows one such exploration of Poincare sections for the energy level $E=0.128$ with 100 different initial conditions, each plotted in a different color. The result is a beautiful oval with an inner decoration by a strange attractor reminiscent of one of the ovoids produced for the Russian royalty. The attractor shows clear preferred regions for the intersections of certain orbits and regions where the intersections are chaotically distributed. To better understand the relationship between the structure of the Poincare sections and the form of the orbits on the $x-y$ plane we take the case of $E=\tfrac{1}{8}$ and examine 12 initial points chosen from different regions of the Poincare sections, i.e. defined $y_0, y_0'$, with $x_0=0$ (Figure 6).

Figure 6.

The trajectories of these initial points on the $x-y$ plane are plotted in Figure 7. Towards the narrow end of the bounding oval we have an oval exclusion zone and the towards the broad end of the oval we have a candra-bindu (crescent and dot) clearing zone. The initial point 1 lies at the center of the narrow end oval clearing. This initial point and the center of the crescent clearing at the broad end (not shown) yield a trajectory with a single loop with 3 apexes (top row, leftmost plot of Figure 7). The next trajectory (top row, next plot moving left to right) is a straight line at a $45^\circ$ incline and corresponds to the center of the two “eyes” of the Poincare section (point 2 in figure 6 is one of these eyes).

Figure 7. The trajectories of the points corresponding to Figure 6 in left to right in 3 rows from top to bottom.

The basis of these trajectories can be understood from the plots of the functions $x(t); y(t)$ (Figure 8; for every point in Figure 6 and trajectory in Figure 7 the corresponding $x(t); y(t)$ are shown one below the other from top to bottom in 2 columns). The first two trajectories result from oscillations where both $x(t)$ and $y(t)$ have period 1 — they show the same repeating pattern after one oscillation. Thus, these two cases can be said to be in a 1:1 resonance. In the second case, they are additionally in phase, i.e. the crest and trough at the same time.

Point 3 samples the center one of the four “islands” which surround the above-mentioned “eyes” of the Poincare section. Each of the island-centers results in a trajectory like the 3rd plot (Figure 7, top row). Point 4 samples one of the small crescents in the vicinity of the oval exclusion zone around point 1 and results in the trajectory seen in plot 4 of Figure 7. These two trajectories result from $x(t); y(t)$ where both have a periodicity of 4, i.e. a 4:4 resonance. Of the two the trajectory 3 arises from a case where in addition to 4:4 resonance the two oscillators are also in phase.

Point 5 (Figure 6) and its corresponding trajectory (Figure 7) corresponds to two period 5 oscillators in a 5:5 resonance (Figure 8). Such 5:5 resonance oscillators are a pervasive feature of the HH system at this energy level and correspond to the 5 islands of exclusion around the oval exclusion around point 1, the center of the bindu and the two exclusion zones flanking either tip of the crescent.

Figure 8

Point 6 corresponds to a trajectory arising from a 8:9 resonance; point 7 evolves into a more complex 5:25 resonance; the trajectory of point 8 simulates a 3D ribbon and arises from the even more complex 11:37 resonance.

The trajectories arising from points 9, 10 and 11 exhibit what might be termed quasiperiodic behavior. In the case of the evolution of point 9, $x(t)$ has a quasiperiod of 4, i.e., it has a similar pattern repetition after every 4 oscillation but the successive repeats are not identical but change slightly over time. $y(t)$, on the contrary, has a strict period of 1. In the evolution of point 10, $x(t)$ has a quasiperiod of 5 which is overlayed on a nearly regular higher period of 15. These two points are representative of the evolution of the points in the zones close the bounding oval on its narrow side. One may note that the evolution of point 11 is like a “broadband” version of the 5:5 resonance points. Keeping with this, $x(t)$ has a strict period of 5, whereas $y(t)$ has a quasiperiod of 5 with higher-order repeat patterns of multiples of 5.

Finally, the evolution of point 12 is chaotic, i.e. the oscillations have no discernible period. The irregularity is marked in $y(t)$ but is $x(t)$ it manifests more slowly over time. These chaotic trajectories form the bulk of the central uniform distribution of points in the Poincare section. The appearance of chaos can be seen as the limit of the trajectories with increasingly complex or longer period resonances. The quasiperiodic orbits with a nearly regular short period internal repeat structure might be seen as lying at the edge of long periods and true periodicity. In terms of energy levels, chaos starts appearing in the central regions close to $E=\tfrac{1}{10}$ and by $E=\tfrac{1}{8}$ constitutes the bulk of the internal structure of the Poincare section with internal islands of periodicity and quasiperiodicity in the anterior periphery of the oval. By the limiting $E=\tfrac{1}{6}$ nearly all of the trajectories become chaotic.

In conclusion, the HH system qualitatively shows all the typical forms of oscillatory behaviors observed in natural systems (e.g. variable star light curves, weather patterns, population dynamics and far-from equilibrium oscillatory chemical reactions): periodicity with different resonances, quasiperiodicity and chaos. It thus provides a good example how any system whose phase space is defined by even simple ODEs with non-linear terms can exhibit the behavioral diversity characteristic of natural systems.

Yajus incantations for the worship of Rudra from the Kāṭhaka ritual manuals

The loss of the northern and northwestern Kṛṣnayajurveda traditions due to the Mohammedan depredations of Northern India (aided an abetted by the predatory Anglospheric regimes) has been one the great tragedies faced by Hindudom. Hence, it is rather important to collate and restore whatever remains of these traditions, namely those belonging to the Kaṭha and Kapiṣṭhala schools, which were once dominant in the greater Panjab and Kashmir. In the 1940s, vidyābhāskara, vedāntaratna Sūryakānta, saṃskṛtācārya of the Pañjāba-viśvavidyālaya, Lavapura (modern Lahore) had collated several Kaṭha fragments that came from lost texts outside of the relatively well-preserved saṃhitā. These came from the lost brāhmaṇa and the surviving āraṇyaka, as well as the lost mantrapāṭha of the Kaṭha-s that went with the sūtra-s of Laugākṣi. Notable in this regard, were the following manuscripts that Sūryakānta found in what is today the terrorist state: 1) A Śāradā manuscript which was written in 1033 Vikrama-saṃvat, bright āṣāḍha aṣṭami (approximately June of 1111 CE) in Gilgit. Ironically, this manuscript was found in the possession of a mulla named Hafiz ar Rahman of the Panjab [footnote 1] and contained 340 folios. This was an extensive paddhati with several Kaṭha mantra-s and brāhmaṇa sections used in their late gṛhya rituals. Another Śāradā manuscript, found in the possession of the same mulla, of 180 folios contains overlapping content from brāhmaṇa and mantra material used in Kaṭha rituals. Finally, there was the D.A.V. college manuscript with two parts of 189 and 169 folios respectively that was again an extensive paddhati with overlapping material. The above Rudra-mantra-s come in the sections labeled Rudra-mantrāḥ or Śatādhyāya(Rudra)mantrāḥ and comprise their second division, coming after the Śatarudrīya. The fate of these manuscripts after the vivisection of India in 1947 CE remains unclear. In the past year, the eGangotri trust has made freely available two independent texts which span the mantra-s in question from the Raghunātha Mandira Sanskrit collection, Jammu. One is a Śāradā manuscript of the Śatādhyāya-dīkṣa and another is a print version of the Śatādhyāya produced in the 1920s by the Kashmirian brāhmaṇa-s, Tārachanda Kaulā and Keśava Bhaṭṭa. These have helped correct some problematic parts of the Sūryakānta texts.

The first part of this fragment is a rather important because is the only occurrence of a variant version of this famous incantation to Rudra found outside the Atharvaveda saṃhitā-s. The said incantation occurs as sūkta 11.2 in the AV-vulgate (often taken to be the Śaunaka saṃhita) and as sūkta 16.104 in the Paippalāda saṃhitā. In totality, the two AV versions resemble each other more closely and have a more extensive set of mantra-s. This clearly establishes that it was not a late acquisition of the Kaṭha-s from the neighboring Paippalāda-s, who were also prominent in the same region (e.g. the Kashmirian intellectual bhaṭṭa Jayanta). Two further points are notable. This text is entirely rhotacizing (e.g. arikravebhyaḥ) relative the fully or partial lambdacizing AV saṃhitā-s (AV-vul: aliklavebhyaḥ; AV-P ariklavebhyaḥ). On the other hand, it has mṛḷatam, mimicking the Ṛgveda dialect, instead of the AV mṛḍatam. Similarly, this text shows the archaism of using the RV-type dual form Bhavā-śarvā as opposed to the AV Bhavāśarvau. This was likely originally part of the Kaṭha-mantrapāṭha which went the sūtra-s of Laugākṣi.

It shares with the AV and Śāṅkhāyana-RV traditions, the conception of Rudra in his twin form — Bhava and Śarva. In the Śāṅkhāyana-śrautasūtra (4.20.1-2), Bhava and Śarva are called the sons of Rudra Mahādeva, thus presenting them as ectypes of the Aśvin-s, who are the sons of Rudra in the RV [footnote 2] and mirror the para-Vedic Skanda-Viśākha dyad who are coupled with Rudra (e.g. in gṛhya-pariśiṣṭa-1 of the Kauthuma Samaveda: oṃ rudraṃ skanda-viśākhayos tarpayāmi ।). In contrast, while Bhava and Śarva are used as epithets of Rudra in other Yajurveda traditions (e.g. Taittirīya), they are not presented as twins. This suggests that the the Kaṭha tradition developed in proximity to the locale where AV traditions original diversified in which the cult of the twins Bhava and Śarva, like that of the Greek Dioscouri, was dominant.

The second part is homologous to the equivalent section of the Aruṇa-praśna of the Taittirīya āraṇyaka, which is used in the Āruṇaketukacayana ritual, where the bricks of the citi are replaced by water-filled pots. It might have been part of an equivalent lost section of the Kaṭha brāhmaṇa. It is largely equivalent to the TA version with a few variants that we have retained due to consistency across Kaṭha manuscripts. Variants of the final mantra are found as AV-vulgate 7.87.1; AV-P 20.33.7 and Taittirīya saṃhitā 5.5.9.3; Kaṭha saṃhitā 40.5.33. The Kaṭha version is oddly formed and unmetrical both in the saṃhitā and across the prayoga manuals. Hence, we retain it as is without emendation or metrical restoration based on the other saṃhitā-s.

footnote 1: He could have descended from converted brāhmaṇa-s
footnote 2: https://manasataramgini.wordpress.com/2020/01/12/the-asvin-s-and-rudra/

RV 10.5

The Ṛgveda is replete with obscure sūkta-s but RV 10.5 might easily take a place in the top tier of those. One might even ask why even attempt to write a commentary on this. We admit we could be plainly wrong in reading the words of our ancestors but the allure of attempting to understand the recondite cannot be passed up. We know nothing of the true composer of this sūkta for the anukramaṇi attributes it to the god Trita Āptya, a watery deity of old IE provenance with cognates Thrita and Thraetona Athvya in the Iranian world and Triton in the Greek world. The sūkta itself is directed towards Agni.

ekaḥ samudro dharuṇo rayīṇām
asmad dhṛdo bhūri-janmā vi caṣṭe |
siṣakty ūdhar niṇyor upastha

The one sea, the receptacle of all riches;
he of many births from our heart looks on.
He clings to the udder in the lap of the two hidden ones.
In the midst of the fountain, the bird’s path is set down.

The opening ṛk is already fairly obscure. We believe the sea here is as literal as it gets. In later Hindu tradition, the sea is seen as the receptacle of riches and the same metaphor finds an early expression here. Now, that sea is juxtaposed with one of many births. The deity of the sūkta is given as Agni and there is no reason at all to doubt that — although he is explicitly mentioned only in the final ṛk of the sūkta, many epithets throughout the sūkta confirm him as the deity . Sāyaṇa informs us that the many births of Agni correspond to this multiple kindlings in the ritual altars of such as the Āhavanīya, the Gārhapatya and the Dakṣiṇāgni in diverse yajña-s. This just one of the ways in which Agni may be seen as having many births. Alternatively, in mythological time he is again said to have many births — a possible allegory for the precession of the equinoctial colure. However, the twist in this sūkta is the “internalization” of the yajña, as Agni is said to be in the heart of the ritualists. This takes us to the next foot where he is said to “cling to the udder in the lap of the two hidden ones”. Agni in the lap of the two parents can be a metaphor for the two pieces of the ritual fire-drill or Dyaus and Pṛthivi in a cosmic context. However, neither of them are hidden and this comes in the context of the internalized yajña implied in the earlier foot. Hence, we take hidden to mean something internal, probably the mind and the intellect (which are not visible entities), whose action composes the sūkta like Agni being generated by the fire-drill. Then in the final foot, we come to Agni being identified as a bird and his path being set down in the midst of the fountain. Sāyaṇa interprets this as Agni as the lightning in the midst of the cloud. This is appears to be version of the famous representation of Agni as Apām Napāt. However, we believe that two distinct metaphors, one physical and one mental or internal, are being intertwined here. The sea and the fountain are physical — they are allusions to the famous fire in water, with the fountain as the underwater plume associated with these fires. These sites in the Black Sea-Caspian Sea region could have been accessed by the early Indo-Europeans and those sightings left an impression on their tradition.

samānaṃ nīḷaṃ vṛṣaṇo vasānāḥ
saṃ jagmire mahiṣā arvatībhiḥ |
guhā nāmāni dadhire parāṇi || 2

The virile ones dwell in the same nest,
the buffaloes have come together with the mares,
The kavi-s guard the seat of the natural law (ṛta),
they have placed the highest names in concealment.

This may be interpreted as a metaphor for the feeding of the ritual fire with oblations. The oblations are likened to the virile buffaloes, while the mares are the tongues of Agni (seen as female Kālī, Karālī, etc). This brings us to the famous imagery of the fire within water as the Vaḍavāgni or the equine fire (something Sāyaṇa seems to intuitively grasp), whose flames might be seen as the mares. The kavi-s here might be seen as tending to Agni who is seen as the seat of the ṛta. Sāyaṇa mentions the secret names to be Jātavedas, Vaiśvānara, etc which have secret meanings.

mitvā śiśuṃ jajñatur vardhayantī |
viśvasya nābhiṃ carato dhruvasya
kaveś cit tantum manasā viyantaḥ || 3

The two imbued with the truth and illusion conjoin,
having measured [him] out, the two birthed the child, making him grow,
[who is the] nave of all that moves and stands still.
Indeed they [the beings] with their mind seek the connection (lit: thread) of the kavi [Agni].

Here we agree with Sāyaṇa’s interpretation that it refers to the birth of the cosmic manifestation of Agni as the sun from his parents, the two world-hemispheres. This is mirrored in the ritual by the generation of Agni by the two parts of the fire-drill. In this solar form, he is seen as a nave of all that moves and moves not and connection to him is mentally sought by all beings.

ṛtasya hi vartanayaḥ sujātam
ghṛtair annair vāvṛdhāte madhūnām || 4

For the wheel-tracks of the law, the well-born one,
refreshing offerings, for booty, serve from the days of yore,
the world-hemispheres having worn the mantle,
with ghee and honeyed food augment [the child Agni].

Here the world halves are explicitly mentioned; this clarifies the reference to the cosmic Agni, i.e. sun. The ṛta’s wheel-tracks, i.e. movements of celestial bodies like the sun further build this connection and support the rendering of ṛta as the “natural law” which is manifest in celestial movements that have continued since the ancient days. Them following the cosmic Agni is intertwined with the metaphor of the ritualists seeking booty serving Agni with refreshing offerings. The mantle of the world-halves is a likely allusion to the days and nights.

sapta svasṝr aruṣīr vāvaśāno
vidvān madhva uj jabhārā dṛśe kam |
antar yeme antarikṣe purājā
icchan vavrim avidat pūṣaṇasya || 5

Desirous [of them], the seven shining sisters,
the knower (Agni), held up from the honey to be seen,
He held [them] up within the mid-region, the earlier born one,
seeking a mantle, he found that of the earth.

This ṛk is rather obscure. Sāyaṇa explains the seven sisters as the seven tongues of Agni (Kālī, Karālī, etc) that he has held up within the mid-region for all to see. However, the celestial connection hinted at by the sisters being held up in the sky (?) suggests that it could be an allusion to the Kṛttikā-s (Pleiades) the asterism associated with Agni. . However, this interpretation will not hold if we strictly take antarikṣa to be the atmosphere. We follow Sāyaṇa to take the adjective aruṣīḥ to mean shining rather than red (which ironically would fit his tongues interpretation better) Further, we also follow Sāyaṇa in interpreting the obscure word Puṣaṇa as the Earth.

sapta maryādāḥ kavayas tatakṣus
tāsām ekām id abhy aṃhuro gāt |
āyor ha skambha upamasya nīḷe
pathāṃ visarge dharuṇeṣu tasthau || 6

The kavi-s have fashioned the seven boundaries,
just to one of those the troubled one has gone,
in the nest of the highest Āyu, the pillar
stands in foundations [situated] where the paths diverge.

Sāyaṇa takes the seven maryādā-s to be ethical strictures: sins like killing a brāhmaṇa or bedding ones teacher’s wife, beer, etc lie outside the boundaries of proper conduct. Indeed, this moral sense appears to be in play when the same ṛk is deployed in the Atharvan marriage ceremony: Kauśika-sutra 10.2.21: sapta maryādāḥ [AV-vulgate 5.1.6] ity uttarato .agneḥ sapta lekhā likhati prācyaḥ | To the north of the marital ritual fire 7 lines are drawn towards the east. Then while reciting this ṛk, the couple places a step on these lines to signify the ethical strictures that accompany marriage. While this implication might be the secondary sense of the first foot, we believe that its primary sense is distinct. In the ritual sphere, it is an allusion to the seven paridhi-s, the firesticks which enclose the fire. These in turn appear to be a symbolic representation of celestial “boundaries” for the purpose of the yajñ. This implied by the yajuṣ incantation that is recited as paridhi-s are laid down (e.g. in Taittirīya Śruti): viśvāyur asi pṛthivīṃ dṛṃ̐ha dhruvakṣid asy antarikṣaṃ dṛṃ̐hācyutakṣid asi divaṃ dṛṃ̐ha agner bhasmāsy agneḥ purīṣam asi || This incantation is for the rite with three paridhi-s (madhyma, uttara and dakṣiṇa). They are respectively associated with the earth, the atmosphere and the sky. The seven-paridhi ritual might have likewise symbolized the six realms and the central plane of one version of vaidika cosmography (speculation). This ṛk returns to some of the themes found in the first and second ṛk-s. The nest and the “dharuṇa”, here meaning the foundation, are mentioned again. Āyu, in the general sense, may be understood as the ancestor of the pañcajana-s, the son of Pururavas and Urvaśi. However, when Agni is seen as the fire of Āyu-s, he is called the best of the Āyu-s. This is made explicit in the ritual context in the Yajurveda by the incantation (e.g. in the Taittirīya Śruti): vider agnir nabho nāmāgne aṅgiro yo .asyāṃ pṛthivyām asyāsuṣā nāmnehi … Thus, the pillar of Agni by the name Āyu, is the skambha referred to in this ṛk. It is said to stand in the foundation where the paths diverge. Thus, the pillar should be understood as the axial pillar with the divergent paths being that of the gods (the northern path) and that of Yama with the dead (the southern path). The point of divergence is of course the equinoctial colure which intersects the plane on which the axis stands.

asac ca sac ca parame vyoman
agnir ha naḥ prathamajā ṛtasya
pūrva āyuni vṛṣabhaś ca dhenuḥ ||

Both the unmanifest and the manifest are in the primal sky,
Dakṣa’s birth is in the womb of Aditi,
Agni, indeed, for us is the first borne of the law,
in his former life both bull and cow.

This last ṛk talks of the role of Agni back in time during the cosmogonic period by giving a summary of what is covered in the famous cosmogonic sūkta-s, like RV 10.72 and RV 10.129. Here the unmanifest (literally the non-existent) and the manifest all that came into being are said to exist in that primal sky (parame vyoman) just as in RV 10.129 (the famous Nāsadīya sūkta). The generation of beings is seen as occurring with the Āditya Dakṣa being born from Aditi (and vice versa as per RV 10.72). This posits initial cyclical reproductions of male from female and vice versa. But it results in an apparent paradox of who came first, the male or the female. The ṛṣi of this sūkta tries to break the paradox by stating that it was Agni who was the first-born entity of the ṛta, who in that former state was androgynous. Thus, the author invokes hermaphroditic reproduction as the ancestral state. This was also the position of Vaiśvāmitra-s of maṇḍala-3, who present a comparable set of cosmogonic constructs in the context of Indra and Varuṇa (RV 3.38), emerging from the god Tvaṣṭṛ, who in some ways is like the yavana Kronos. They are said to have partitioned the hermaphroditic ancestral bovine into male and female, similar to Zeus and Apollo cleaving the hermaphrodites into separate sexes in the yavana world.

Bṛhaspati-śanaiścarayor yuddham-2020 ityādi

The below is only for information. Parts of it should not be construed as any kind of prognostication on our part.

The great Hindu naturalist Varāhamihira describes various kinds of planetary conjunctions or grahayuddha-s in his Brihatsaṃhitā (chapter 17) thus:

yuddhaṃ yathā yadā vā bhaviṣyam ādiśyate trikālajñaiḥ |
tad vijñānaṃ karaṇe mayā kṛtaṃ sūrya-siddhānte ||
The time and nature of planetary conjunctions (graha-yuddha) can be predicted by astronomers. That science has been [taught] in astronomical work composed by me [based on] the Sūrya Siddhānta. [Here he is referring to his Pañcasiddhānta]

viyati caratāṃ grahāṇām uparyupary ātma-mārga-saṃsthānām |
ati-dūrād dṛg-viṣaye samatām iva samprayātānām ||
The planets revolve in space in their respective orbits that are positioned one above the other. [However,] due to their great distance, when observed they appear as if revolving on the same surface (i.e. the sky). [Here, Varāhamihira clarifies that even though it is called a yuddha how it must be understood in the scientific sense.]

āsanna-krama-yogād bheda+ullekha+aṃśu+mardana+asavyaiḥ |
yuddhaṃ catuṣprakāraṃ parāśara ādyair munibhir uktam ||
In the order of the proximity of the conjunct planets: 1. bheda (occultation); 2. ullekha (near tangential contact); 3. aṃśumardana (the grazing of rays); 4. asavya (apart) are the four types of conjunctions described by Parāśara and other sages.

bhede vṛṣṭi-vināśo bhedaḥ suhṛdāṃ mahākulānāṃ ca |
ullekhe śastra-bhayaṃ mantrivirodhaḥ priyānnatvam ||
In the bheda conjunction, there is drought and friends and persons of great families become enemies; in the ullekha conjunction there is fear of weapons, a rebellion of ministers, but there is [abundance of] good food.

amśu-virodhe yuddhāni bhūbhṛtāṃ śastra-ruk-kṣud-avamardāḥ |
yuddhe ca+apy apasavye bhavanti yuddhāni bhūpānām ||
In the aṃśumardana conjunction, kings go to war and people are afflicted by weapons, disease or famine. In apasavya (asavya) conjunction, rulers go to war.

ravir ākrando madhye pauraḥ pūrve +apare sthito yāyī |
paurā budha-guru-ravijā nityaṃ śītāṃśur ākrandaḥ ||
The Sun in mid-heaven is [called] ākranda; paura in the east and when stationed in the west a yāyin. Mercury, Jupiter, and Saturn are always paura. The Moon is always ākranda.

ketu-kuja-rāhu-śukrā yāyina ete hatā ghnanti |
Ketu, Mars, Rāhu and Venus are always yāyin-s. These are either struck (defeated) or strike (win). Depending on whether the ākranda, yāyin or paura losses or wins the objects associated with of their respective categories [suffer or prosper].

paure paureṇa hate paurāḥ paurān nṛpān vinighnanti |
evaṃ yāyy ākrandā nāgara-yāyi-grahāś ca+eva ||
A paura defeated by another paura, results in city-dwellers and kings being smitten. Similarly, if a yāyin or an ākranda is defeated by another or respectively by a paura or yāyin [the objects associated with them are affected accordingly].

dakṣiṇa-diksthaḥ paruṣo vepathur aprāpya sannivṛtto +aṇuḥ |
adhirūḍho vikṛto niṣprabho vivarṇaś ca yaḥ sa jitaḥ ||
That [planet] which is positioned to the south, at a cusp, showing rapid variability of brightness, goes retrograde immediately after conjunction, with the smaller disc, occulted, gets dimmer or changes color is said to be defeated.

ukta-viparīta-lakṣaṇa-sampanno jayagato vinirdeśyaḥ |
vipulaḥ snigdho dyutimān dakṣiṇadikstho +api jayayuktaḥ ||
If the planet appears endowed with the appearance opposite of the above-described it is deemed the victor. So also that which appears bigger, smooth in the motion or brighter is considered the winner even if stationed to the south.

dvāv api mayūkha-yuktau vipulau snigdhau samāgame bhavataḥ |
tatra +anyonyaṃ prītir viparītāv ātmapakṣaghnau ||
If both are endowed with bright rays, growing larger and smooth moving it becomes a samāgama conjunction. There is [consequentially] a conciliation between the objects associated with the two planets; if it is the opposite (i.e. both are small, growing dim, etc) both the associated objects will be destroyed.

yuddhaṃ samāgamo vā yady avyaktau svalakṣaṇair bhavataḥ |
bhuvi bhūbhṛtām api tathā phalam avyaktaṃ vinirdeśyam ||
In cases where it the characteristics are not clear as to whether the conjunction of two planets is in a yuddha or a samāgama conjunction, it is likewise unclear as to what the fruits will be for the rulers.

Thus, Varāhamihira describes the general omenology of planetary conjunctions as per hoary Hindu tradition. This year is marked by a remarkable conjunction of Jupiter and Saturn that will peak as per geocentric coordinates on the day of the winter solstice (December 21, 2020; Figure 1).

Figure 1

This has caused tremendous fear and excitement among those with a belief in such omenology. Even as we were examining the conjunction for purely astronomical reasons, we received one such call from a believer. Hence, we looked up Varāhamihira to see what he has to say. Notably, this conjunction on Dec 21 is the closest since the famous Keplerian conjunction of 1623 CE. Thus, it is definitely an aṃśumardana and nearly an ullekha. From the above, we can see that, as per tradition, it is a grahayuddha of the aṃśumardana type between two paura planets, which prognosticates death from weapons, disease, or famine. Further, it is evident that Jupiter is to the south and it reduces in magnitude as it emerges from the conjunction. So as per the Hindu typology of conjunctions, it is defeated by Saturn (Figure 2).

Figure 2

Varāhamihira further provides specific omenology for the defeat of Jupiter by Saturn (BS 17.19):

bhaumena hate jīve madhyo deśo nareśvarā gāvaḥ |
saureṇa ca+arjunāyana-vasāti-yaudheya-śibi-viprāḥ ||
If Jupiter is beaten by Mars, Madhyadeśa region, kings and cows suffer. [When beaten by] Saturn, the Arjunāyana, Vasāti, Yaudheya, Śibi [peoples] and the Brāhmaṇa-s suffer. [It is noteworthy that one of the used words for Jupiter in the last 2000 years is Jiva. This word is not encountered in the earlier layers of the Sanskrit tradition. It is a loan from Greek, the vocative declension of Zeus, and is one of the marks of the influence of Greek astrology (e.g. the Paulīśa-siddhānta and Yavana-jātakam) on its Indian counterpart.]

It was that latter prognostication that caused the fear in our interlocutor. These conjunctions of Jupiter and Saturn have an interesting geometric feature: the great trigon. For superior planets one can approximately calculate the frequency of conjunctions thus: Let $p_1, p_2$ be the periods of revolution of the two planets and $p_2 > p_1$. Then $\tfrac{360^\circ}{p_1}, \tfrac{360^\circ}{p_2}$ will be the mean angular speeds of the two planets respectively. The difference in their speeds would be:

$\dfrac{360^\circ(p_2-p_1)}{p_1p_2}$

Hence, the time the faster planet will take relative to the slower planet to complete one revolution (i.e. catch up with it again) will be:

$p_c=\dfrac{360^\circ}{\dfrac{360^\circ(p_2-p_1)}{p_1p_2}} = \dfrac{p_1 p_2}{p_2-p_1}$

This $p_c$ will be the duration between successive conjunctions. The period of Jupiter is 4331 days of Saturn is 10747 days. Hence, $p_c= 7254.56 \; \mathrm{days} = 19.86245\; \mathrm{years}$. Thus, the Jupiter-Saturn conjunctions will repeat approximately every 20 years. One can see that the next two conjunctions will occur with respect to the original one at:

$\left (p_c \dfrac{360^\circ}{4331} \right ) \mod 360 \equiv 243.0112^\circ; \left (2p_c \dfrac{360^\circ}{4331} \right ) \mod 360 \equiv 126.0224^\circ$

Thus, the three successive conjunctions will trace out an approximate equilateral triangle on the ecliptic circle — the great trigon. This trigon caught the attention of the great German astrologer Johannes Kepler, the father of the modern planetary theory in the Occident. He described this in his work the “De Stella Nova in Pede Serpentarii” that announced the discovery of his famous supernova. Since the successive trigons are not exactly aligned, they will progress along the ecliptic (Figure 3). We see that after 40 successive conjunctions it occurs very close to the original conjunction $\approx 0.4488778^\circ$ (Figure 3). This amounts to about 290182.4 days (794.4979 years). This is the roughly 800 cycle that Kepler was excited about and thought that he was in the 8th such cycle since the creation of the world (being conditioned by one of the West Asian diseases of the mind).

Figure 3. Progression of the trigons in the cycle of 40

The above formula can also be used to calculate the successive oppositions (when the configuration is Sun–Earth–superior planet) or inferior conjunctions (Earth–inferior planet–Sun) or superior conjunctions (Earth-Sun-inferior planet). We those obtain other interesting patterns. One such is successive oppositions of Jupiter which happen every 398.878 days. Thus, they nearly inscribe a hendecagon on the ecliptic circle (Figure 4).

Figure 4. Successive oppositions of Jupiter

Now the successive inferior or superior conjunctions of Venus happen every 583.9578 days. Thus, these successive events trace out a near pentagonal star (Figure 5). This comes from the fact the ratio of the orbital periods of Earth to Venus is nearly $\tfrac{13}{8}$ which is a convergent of the Golden Ratio.

Figure 5. Successive conjunctions of Venus

Beyond this, we may also note that the successive oppositions of Saturn nearly inscribe a 28-sided polygon (roughly corresponding to one per nakṣatra) whereas those of Mars nearly inscribe a polygon of half that number (14 sides). The minor planet Ceres nearly inscribes an 18-sided star in the ecliptic circle (Figure 6).

Figure 5. Successive oppositions of Ceres

The cosine principle, radial effect and entropy in the generalized Lozi map

The generalized Lozi map is a good way to illustrate the cosine principle and the radial effects (in lay circles to which I belong in this regard, as opposed to mathematicians). The generalized Lozi map is a 2-dimensional map defined thus:

$x_{n+1}= 1 + y_n + a|x_n|$
$y_{n+1}= -x_n$

The map is area-preserving and yields “aesthetic” images for $a \in [-0.6,1.1]$. Additionally, values $a=-1; a=\sqrt{2}$ are also somewhat aesthetic and interesting. We have previously described the cosine principle for various dynamical systems, but we reiterate it here for the generalized Lozi map as it is one of the easiest ones to explain to a layperson. First, a few words on how we visualize this map. We start with the vertices of a 60-sided polygon circumscribed by a circle of radius $r$, centered at $(0,0)$, and record the evolution of each vertex for a 1000 iterations under the map. Since the map has an absolute value term, it will be bilaterally symmetric along the line $y=-x$. Hence, we rotate the iterates by an angle of $-\tfrac{\pi}{4}$, then scale and center the points, and plot the evolutes of each vertex (orbit of the vertex) in a different color. The examples of 9 such mappings starting with the said polygon in a circle of radius $r=0.2$ are shown in Figure 1.

Figure 1

The values of of the parameter are chosen such that $a=2\cos\left(\tfrac{2\pi}{p/q}\right)$, where $p,q$ are integers. We observe that the value of $p$ determines a key aspect of the shape of the map, i.e. in each map there is a central, largely excluded area that takes the form of a polygon with $p$-sides. This is the cosine principle. More generally, the shape of the central region of the map is determined by the $p$ corresponding to the $2\cos(\theta)$ closest to $a$. Note that for the case $\tfrac{\pi}{2}$, we take a number relatively close to 0, for at 0 the map is degenerate. Outside of the polygonal exclusion zone, we may find chaotic behavior but it is still bounded within a unique external shape. The chaos is particularly apparent in the cases when $a=-1; 1; \sqrt{2}$ when the map respectively yields the headless gingerbread man, the classical gingerbread man and the tripodal gingerbread man strange attractors. At the other values of $a$, we see bands of chaos interspersed with rings of closed loops that resemble the period-doubling phenomenon in other strange attractors prior to the outbreak of full-fledged chaos.

In Figure 2 we produce the same plot by changing the radius of the circumscribing circle of the initial polygon to $r=0.45$. We can see that at this radius, for the low $p$ the cosine principle remains dominant, but for large $p$ the polygonal zone gets “smoothened” out (e.g. for $p=9..11$). This indicates the radial principle, i.e. the effect of the starting radius on the degree of expression of the cosine principle in the map.

Figure 2

The degree of chaos can be seen as the measure of entropy of the map. By following the colors, one can see that when $a=-1; 1; \sqrt{2}$ the orbits of a given starting vertex under the map are all over the place within the attractor boundary. In contrast, for the other values of $a$, the evolutes are mostly limited to particular bands. When $a \approx 0$ then the evolute of each vertex is limited to a certain concentric curve. Thus, the former lie at the high end of the entropy spectrum and the latter at the low end. A proxy for the entropy distribution of the attractor can be obtained by computing the coefficient of variation, $c$, i.e. the ratio of the standard deviation to the mean of the distances of the evolutes of a particular vertex from the center of the map:

$c=\dfrac{\sigma_d}{\mu_d}$, where $\sigma_d$ is the standard deviation and $\mu_d$ the mean distance from the center

We plot $c$ for the maps with $r=0.2$ (Figure 1) and $r=0.45$ (Figure 2) for each vertex at 60 angles from $0..2\pi$ respectively in Figures 3 and 4. The mean $c$ is shown as $\mu$ for each plot.

Figure 3

Figure 4

We note that $\mu$ for $a=-1; 1; \sqrt{2}$ is significantly (an order of magnitude) greater than the $\mu$ those for the other $a$. Further, the radial effect can also be seen affecting the entropy of a map. While it remains roughly the same or is lower for the high entropy triangular, hexagonal and octagonal $a$, for the remaining polygonal $a$ the entropy rises at $r=0.45$ relative $r=0.2$. In the pentagonal case, it is mostly across the board while we see specific peaks in the decagonal and heptagonal case.

We next examine the radial effect and entropy more systematically for a fixed value of $a$ by choosing the hendecagonal value $a=2\cos\left(\tfrac{2\pi}{11/3}\right)$. The map is shown in Figure 5 and the entropy proxy $c$ in Figure 6.

Figure 5

Figure 6

Here we see two disconnected effects of the radius. First, at certain values the inner hendecagon is lost (e.g. $r=0.1$) or becomes smoothened out (e.g. $r=0.5; 0.6$). Second, the entropy of the orbits of certain vertices dramatically rises for some values (e.g. $r= 0.5..0.8$). The radial effect on neither the entropy nor the expression of the polygonal inner zone is the same across different $a$ values. However, more generally, the lower the number of polygon sides, stronger is the polygonal expression across $r$.

Finally, we touch upon a general philosophical point that can be realized from such chaotic systems. While it is not specific to this generalized Lozi attractor, we take this opportunity to articulate it because we have presented the entropy concept. Most people agree that the attractors with neither too much entropy nor too little entropy are aesthetically most pleasing. This also has a counterpart in biology. Selection tends to prefer systems with an optimal entropy. Too much entropy in a structure (say a protein) and it is too disordered to be useful for most functions. Too little entropy and it is again too rigid to be useful for much. Moreover, from an evolvability viewpoint, too rigid a structure offers too little option for exploring multiple functions in biochemical function space. Too much disorder again means that it explores too much space to perform any function well enough to be selected. Hence, structures with some entropy optimum tend to be selected rather than those with minimum or maximum entropy. Selection can be conceived as maximizing a certain function, say $f(x)$ for simplicity, in a given entity under selection. This $f(x)$ will then be the fitness function. We can see from the above that $f(x)$ cannot directly or inversely track mean entropy because that will not maximize fitness which is at some optimal entropy. It has to hence track something else. This would depend on the optimal band of entropy that is selected by the given constraints. For example, one field of constraints could select for an optimal band of mean $c$, like $\mu \in [0.03, 0.1]$. Such a field will select $a$ corresponding to the pentagon, heptagon, nonagon and decagon while avoiding the triangle, hexagon and octagon for too high entropy and the square and hendecagon for too low entropy (Figure 1, 3). This constraint field will also select for other values (e.g. Figures 7 and 8) that have $\mu$ is in this interval (Panels 1, 2, 6). Thus, the $f(x)$ will be a function with local peaks that is very different from the underlying reality of a continuous entropy distribution from low to high. Thus, selection translates the underlying reality into a sensed structure very different from it. The philosophical corollary to this is that a sensed structure will be different from and unlikely to reflect the underlying reality.

Figure 7

Figure 8

Prakīrṇaviṣayāḥ: Life, brains, warfare and society

1 On big brains
An occidentally conditioned person remarked that “we were making bad use of the great brains we have evolved. Instead of using it for human betterment, we were expending it on killing each other with sophisticated weapons.” I could not but help smiling for we have long held that the recent explosive growth of brain size in humans is a likely signal of evolution due to biological conflict. Thus, we posit (like others who have independently done so) that intraspecific and interspecific (e.g. with australopithecines, Homo naledi, Neanderthals, Denisovans and the like over time) conflict led to the escalation of brain growth in human lineages. After we emerged as victors against our related species and eventually settled down as farmers, we began a transition to domestication along with the animals we had allowed to survive as domesticates for our needs. In course of this domestication, it looks as though our brain size came down a notch. Paralleling this, domestication in other animals also appears to have caused a reduction in their brain sizes. In some cases, we see strong evidence that this arose from the reduction in conflict. This possibility was noticed early on by Charles Darwin himself: “ …no animal is more difficult to tame than the young of the wild rabbit; scarcely any animal is tamer than the young of the tame rabbit…” This has since been confirmed by a modern study, which showed that the domestication of the rabbit resulted in: 1) reduction in brain size relative to body size; 2) a reduction in the amygdala and an enlargement of the medial prefrontal cortex; 3) reduction in white matter throughout the brain [footnote 1]. These changes have been proposed to result in a decreased flight response in the domestic rabbit. Similarly, the domestic pig and probably also the domestic cat and dog have smaller brains than their wild counterparts. We saw a poignant illustration of this in the form of a domesticated white lab mouse that had escaped from the lab was savoring its newfound freedom. However, its lack of smarts for life in the wild quickly made it a victim for a crow couple. Thus, if the brain of an organism is an instrument in an arms-race, the brain-size and the level of ambient biological conflict have a positively correlated relationship. More generally, “losing the martial edge” from domestication has also been seen on a civilizational scale — for example, among the steppe peoples who transitioned to a sedentary existence.

We do not as yet fully understand all the reasons why large-brained organisms arose repeatedly among those with smaller brains. But once it is in place, biological conflict can keep it growing in size. Several birds on islands are renowned for their intelligence and might have even bigger brains than their mainland counterparts [footnote 2]. At the face of it, it might look paradoxical — an island usually has less danger from predation and related conflicts than the mainland: think of the flightlessness of the dodo or the solitaire. One hypothesis that explains this is the opening of new niches on the island to the colonizer, which increases intra-specific competition as its population expands. Given the potential habitat and resource diversity, or difficulty in accessing the latter on the island, the plasticity of behavior and therefore a larger brain can be decisive in the intra-specific conflict. An example of the use of a larger brain in exploiting difficult resources is seen in the case of the cane toad introduced to Australia. The toxic toad kills species like the Varanus lizard that eat it resulting in a major decline in their populations. On the other hand, the big-brained Torresian crow has learned to rip it apart and eat it from the ventral side and thus avoid its poison glands. Thus, the rise of smarter birds on islands via intra-specific conflict could be related to the phenomenon that drove the expansion of the human brain.

Recent studies providing constraints on the distribution of earth-like exo-planets suggest that there must be $\le 6 \times 10^9$ stars with Earth-like planets in the Milky Way. This is a large enough number that it brings home the reality of Fermi’s paradox: “If there are extra-solar system civilizations why have we not heard from them yet?” One noted astronomer suggested that this might mean that human-type intelligence is likely to be exceptional across the Milky Way. We take a slightly more nuanced view informed by biology with regards to the reality of Fermi’s paradox. It is clear that most organisms that profoundly modify their host planet might not do so with any intention of signaling to life forms on other planets. For example, cyanobacteria altered the earth and made its atmosphere oxidizing resulting in a whole lot of new dioxygen chemistry that made organisms like ourselves possible. Cyanobacterial metabolites might signal the presence of life to an observer on another planet, but this is hardly intentional. Similarly, William Hamilton, just before his death, proposed that the bacteria might have caused’ the emergence of atmospheric clouds to disperse themselves or their spores. While this might seem far-fetched at first sight, since the work by Sands we have known that bacteria can nucleate clouds and ice (e.g. Pseudomonas syringae). More recently, the role of marine bacteria in seeding Arctic clouds has been demonstrated [Footnote 3]. Thus, there might be an “agency” on part of the bacteria in visibly modifying the planetary atmosphere to facilitate their spread. However, while it is likely that bacteria-like forms can effectively signal the presence of life on a planet through more than one means, they do not seem to be doing so with the intent of informing aliens. This kind of signaling seems to need a large centralized brain of the kind we have. Such brains are only present in animals among earthly life and have evolved only a few times in the past 700-1000 million years animals have been on this earth: cephalopods, some lineages of avian dinosaurs, some placental mammals. While we might not fully understand why a lineage evolves a bigger brain than its sister groups in the first place, a major driver of escalating growth appears to be biological conflicts. Thus, on other planets too we expect brain-like structures to evolve if they were to provide an edge in the arms race rather than for signaling to aliens. Its growth will be driven by the arms race and not the need for space exploration. Further, if auto-domestication happens as a consequence it might eventually decline in size. Thus, auto-domestication might be seen like how yogin-s described siddhi-s — they come as byproducts but focusing on them can take you down from the goal of yoga.

We have had the unique distinction of being born in the age of space exploration. Some people, inspired by the excitement of it, have remarked that space exploration might provide selective advantages by allowing the colonization of new planets; hence, intelligent life should eventually turn to such an endeavor. We take a dimmer view. First, we believe such colonization might be an option for the basal prokaryote-like life forms that are likely to widely populate the universe. It might not be necessarily intentional but, like the seeding of clouds and ice on earth by bacteria, certain adaptations might have facilitated such escape and transmission especially in the earlier days of the solar star cluster, and its cognates throughout the Milky Way. However, for larger big-brained organisms like ourselves both the physics and the biology make such prospects of such a gain fairly unlikely. For the most part, space exploration is a byproduct of the development of weapon delivery and surveillance systems, like rockets and satellites, which actually mean something for the conflicts (i.e. dual-use technology). Once the utility of space exploration for the main product declines, the interest in space exploration for its own sake will also be limited. In the best case, we could have many intelligent civilizations that are “mining” nearby planetary bodies for various resources that give them an edge. Thus, we would say that Fermi’s paradox should be taken as the null hypothesis because theory predicts that the primary driver of big brain-like structures would be biological conflicts on the host planet, and space exploration would merely be its rare sideshow.

Finally, we should note that a big brain is also a big memetic ecosystem where viral pathological memes can take root. This probably goes hand-in-hand with domestication, which releases some of the strong survival pressure that an organism faces in a natural environment. For instance, on the wild steppe one has to ensure that food is available to tide through the harsh winter months. This cuts out a lot of the avenue for slacking. In contrast, in a city with a well-provisioned supermarket supplying soft syrupy viands at an arm’s reach and a low price takes the mind away from survival and allows for slacking. Against this background, the emergence of diseases, like American Naxalism, which would otherwise reduce survivorship, can take root and thrive. These new diseases of the mind along with their ancestral versions, i.e. the unmāda-s from West Asia can eventually recycle the civilizational state back to a more basic condition. Thus, the civilizational state we are in would cycle up and down without for the most part reaching out to life on other planets. The same would hold for them too.

2 The parasite within
Small genomes, like those of small RNA or DNA viruses (e.g. the SV40 virus), are lean and mean. They code for little else beyond a minimal apparatus to replicate their own nucleic acid and the bare essential apparatus to take hold of the host systems for producing more of themselves. In contrast, large viruses, like say a poxvirus, a mimivirus, a pithovirus or a Bacillus virus G have giant genomes. In addition to the replication apparatus, they code for a transcription system, and have a degree of self-sufficiency and independence from the host systems. They code for several elaborate means to more subtly hijack and control the host in several ways. While the former class primarily depends on fast replication of their little genomes to overwhelm the host or at least get some copies of themselves made before the host immunity overwhelms them, the latter is a different type of player. They do not replicate as fast but compete hard with the host while taking their own time to replicate their relatively large genomes accurately. This means that they code for a diverse array of capacities to keep the host immunity at bay even as they assemble their elaborate copy-making machinery inside the host cell. A curious thing about such larger viral genomes is that they invariable carry parasites within their own genomes. These may take the form of introns, inteins, and other mobile parasitic genetic elements that invade the genome and the genes of the bigger viruses. The inteins and introns have been selected to mediate their own splicing either at the level of the protein or the RNA transcript. Thus, they do not fatally cripple their host. However, they lodge themselves in genes like the DNA polymerase; hence, the host-virus simply cannot get rid of these genomic parasites. Thus, it is given that as genome sizes grow beyond a certain point, where they shift to the paradigm of slower replication and harder competition, parasitic elements make their home in them.

The relationship between the selfish elements and the cellular genomes is even more complicated: we have shown that transcription factors that regulate gene expression and transcription regulatory elements to which they bind repeatedly evolve from the freeloading mobile elements. So over evolution, they offer raw material for innovation. Sometimes they provide new weaponry for pathogenic organisms and new defensive strategies for cells against other invaders. On the other side, they might breakout to give rise to new viruses. Thus, viruses like retroviruses share an ultimate common ancestry with freeloaders like introns. Other mobile genomic parasites have similarly given rise to viruses such as adenoviruses. Thus, over evolutionary time these freeloaders come with both downsides and upsides. In an environment where the system robustness allows them to be accommodated within the bandwidth allowed by selection, they will persist and end up conferring some advantages to the host genomes that maintain them as opposed to those that do not.

We wondered if the social analogs of genetic systems might have a similar two-sided relationship with respect to freeloaders. One might protest that memes are fine but how can social systems be analogized? We say that even if the mapping is not exact, these can be usefully brought into the orbit of generalized genetic systems (much like the proposed replication of clays). While we are not sufficiently motivated to describe this in full here, we try to illustrate it by example. In essence, it may be seen as a meme or its variant. Take a social system like a government. Various positions interact in a network like a network of genes in different functional ensembles: the Department of Defense, the Department of Biotechnology, the Judiciary, etc. which house within them various positions. Now, the individuals occupying a given position can be seen as fungible, e.g. a judge position can be filled by another person, but the position remains. So, it can be seen as copying itself on that fungible substrate. It can expand too: the same organization can be reproduced recursively from state to state. In the least, the comparison made us realize that certain things that people get very worked up by, such as apparently non-functional and corrupt positions in a bureaucracy, are likely to arise organically and will not go away easily. They might possess “addiction modules” like toxin-antitoxin systems that bring the system down if an eviction is attempted. Further, if there is strong selection that forces them to become extinct it might also bring down other aspects of the organization that are considered useful. We cannot rule out that over evolutionary time some of the social freeloading positions (e.g. positions relating to some branches of the humanities academia) offer selective advantages to the system in certain environments. On the other hand, such social positions can also provide the raw material for the emergence of destructive elements that are more like viruses (e.g. certain policing, religious and academic positions in society).

3 War and innovation
There is a fractal structure to the organization of space. As a result, we have few blue whales and elephants and lots and lots of bacteria. Consequently, many more biological conflicts play out among bacteria — there is a non-stop warfare between different bacteria and also between bacteria and their viruses. These battles are life-and-death struggles — as an old English tyrant gleefully remarked about how no quarter was offered to the Hindus in the war of 1857 CE, so it is in these battles fought by bacteria — “kill or get killed’’ is the name of the game. As a result, natural selection has produced an extraordinary repertoire of weaponry. We have shown that these bacterial conflicts are at the root of all innovation in biology. The origin of eukaryotes was marked by some revolutionary structural adaptations that rendered them immune to some of the weaponry used in these conflicts. For example, the dominance of the tailed bacteriophage was passé in eukaryotes. Eukaryotes mostly do without weaponry like restriction-modification and CRISPR systems. Thus, the armaments of the old-world suddenly came to a stop in the eukaryotic realm. Notable, the eukaryotic reorganization also meant that they were going to less innovative not just in terms of weaponry but more generally in terms of inventing new stuff. Yet, eukaryotes show remarkable systems innovations: where does this come from? What we found was that they get most of their innovation from the “weapons systems” of bacteria through lateral gene transfer and reuse them mostly as “peacetime technology” for various cellular systems like their chromatin structure, RNA-processing, signaling inside and between cells, multicellularity, etc. and also their own defense needs. This brought home an important point that without the pressure of warfare there will be no fancy peacetime technology.

In human endeavor, we see this in the form of various technologies, including space exploration and medicine, being driven by the military needs as the engine of innovation. Hence, we suggest that the utopian society of peaceniks would cease to innovate meaningful technology. However, it is conceivable it turns its mind towards making technology that is primarily of the form of addictions that might eventually render it supine before a more robust culture from within or without.

4 The eternal struggle
One major difference between the Abrahamistic counter-religions of the Messianic variety and the Indo-European religions is the single endpoint utopianism preached by the former. This is the driving force behind its secular mutations, including that in whose grip the modern Occident is currently convulsing. In contrast, at least since the breakup between us and our Iranian cousins (of the Zoroastrian flavor), both sides had ingrained in them the concept of the eternal or repeated episodic struggle of the deva-s and asura-s. It probably was already present in the ancestral matrix of the religion. On the Iranian side, Zarathustra caricatures it in the form of the lands the āirya-s being repeatedly invaded by Angra Mainyu. In fact, this dualism is very important to the philosophy of the Zoroastrian branch of the Iranians. On our side, the Mahābhārata and the purāṇa-s emphasize the repeated cycles of the devāsura-saṃgrāma with neither side gaining total victory. Individual victories might be achieved: Namuci, Vṛtra, Naraka, Prahlāda, Andhaka, and so on might be eliminated but new ones arise. Most importantly, right from the brāhmaṇa texts, we have the emphasis on different upāya-s being used by either side to gain victory; in each new round, a new upāya is needed for victory. Through the teachings of the diverse upāya-s, the brāhmaṇa texts lay out important teachings for humans in daily life. We hold that the devāsura-saṃgrāma is one of the most important teachings of our tradition and a mythic codification of one of the highest realizations of the Indo-Iranians. It essentially tells us the truth of nature — the eternal struggle — like between prey and predator or virus and host or producer and consumer. For example, T4-like bacteriophages and bacteria have been fighting it out for more than 2 billion years, which from long before the Pleiades existed in the sky. Thus, this battle is eternal, and each round can be won by a new upāya which becomes part of the genetic record, much the upāya-s recorded in the brāhmaṇa passages. New ūpāya-s may be discovered which supersede old ones, much like the brāhmaṇa telling us of how the old performance of a ritual might be replaced by a new brahmavāda. The Hindu-s need to pay heed to this teaching. Just as cellular life and the viruses are locked in eternal conflict, so also are we with counter-religious viruses of the mind. They will mutate and new forms will arise, and we have to keep trying new upāya-s. There will be losses, but the end goal is not to become extinct — one cannot avoid losses entirely. Thus, rather than hoping for a utopia to be ushered in, like that wished for by our enemies upon our total destruction under a leader like the āmir al momīn, we have to be prepared for round after round of saṁgrāma.

Footnote 1 https://www.pnas.org/content/115/28/7380
Footnote 2 https://www.nature.com/articles/s41467-018-05280-8
Footnote 3 10.1029/2019GL083039

The old teacher

The summer of the year after the tumultuous events, Lootika and Somakhya were traveling to visit their parents. They were supposed to attend the marriage of Somakhya’s cousin Saumanasa but they found social engagements with a subset of the clan quite wearisome. Thankfully, a perfect excuse appeared for them to give it the slip, and they returned to their parents’ city after Somakhya’s parents had left for Kshayadrajanagara for the marriage. Thus, Somakhya was staying with his in-laws till his parents returned. The morning after they had arrived Lootika had risen early and left to give a talk at the university and then engage in some sartorial explorations with her old friend Kalakausha and her friend’s sister Kallolini. Somakhya after finishing his morning rituals sauntered into the kitchen to chat with Lootika’s mother who was busy at her cooking.

LM: “May Savitṛ grant success to your ritual. I’m sort of envious of your mother. She is lighting fast in the kitchen like that legendary English surgeon Liston but unlike him rather infallibly consistent.”
S: “You must tell her that…”
LM: “I did but I had to clarify that it was a compliment as I went on to narrate to her the enthusiastic lopaharṣaṇa of the English surgeon.”
S: “I’d be happy to help if I could speed things up for you in any way.”
LM: “Don’t be stupid and just stand there just outside the kitchen and chat with me while I wrap this up. I am substituting for your mother at the temple of Rudra by the river for today’s exposition and could pick your brain a bit for that.”
S: “Sure. You’re no different from your daughter in keeping everyone out of your ‘pāka-yajña’.”
LM: “Ask your mother; she would agree too! We certainly believe that too many cooks, especially the guys, spoil the broth. Moreover, we are a bit paranoid over the purity of our yajña-kṣetra.”

S: “So, what have you all been expounding at the temple?”
S: “I’m always inspired by its opening: nama ulūkāya rudrāya aṇuvide |
LM: “I can see that, my dear. It is indeed that opening story that I narrated to them — where Rudra appeared in the form of an owl to teach the doctrine to the great Kāśyapa. In any case, other than that, at some point when all of you kids were out of home, we also combined our efforts to start a second course, a purāṇa exposition. While there are many expositors of Vaiṣṇava-bhakti themes, we decided to go off the beaten track and have readings from the old Skandapurāṇa — to our surprise, it has gathered much greater interest than the philosophical one. It was in that regard that I needed to consult with you regarding the tale of how Skanda helped his mother in creating Vināyaka.”

L: “What is the deal with our old history teacher? As Somakhya would say, most of the teachers and the classmates from school with whom we have lost contact have passed out of our ken like vāsāṃsi jīrṇāni.”
LM: “Somakhya, do you recall anything of your interactions with her? I remember that she was generally, good to my other three daughters, but was among the multiple teachers who would repeatedly complain to me that your wife is rather arrogant due to her varied knowledge that was so atypical of the girls.”
Somakhya looked at Lootika and chuckled: “Actually, she was one of those teachers who was not bad to me at all. She had a Nehruvian bent of mind and liked to downplay the sultanate or the Mogol tyranny and boost the English superiority over our culture. That said, she was rather generous to me and I’d say to Spidery too when we pressed the counterpoints to these issues in class. But why would she want to see us? So much has gone by and we should have passed out of her memory too.”
Lootika waded in: “I sensed that Kallolini seemed to know something more; I was about to ask but they ran away. I’ll call her and ask.”
LM: “Lootika, you can be so callous! They would be busy with patients.”

Lootika’s parents were much like Somakhya’s parents — they felt one should not tarry long at useless conversations. Lootika’s mother got up abruptly and remarked: “Dear kids, I need to get ready to go and teach a couple of classes at the college before I return to leave for the temple. You can keep lazing around here for some time but if you want to meet your old teacher you may either go and see her at school where she would be correcting answer sheets or you can come with me to the temple and catch her there. I suggest the former for that might give you some more time and also still be in a mostly public place. I somehow get the vibe that you should not meet her at her home.”
L: “Why so mom?”
LM: “Dear, do as I say. If you need a ride back home call me or dad and we can pick the two of you up while returning. You can tell me of your encounter then or tonight.” Saying so Lootika’s mother left.

After her mother left, Somakhya and Lootika napped a bit to clear their jet lag. As they were waking up Lootika remarked: “Somakhya, I have a gut feeling that there might be something interesting with the case of our former history teacher. I sensed something in Kallolini’s words and also my mother’s strange remark. Moreover, why would she want to meet us? As you remarked, whatever she might have complained behind my back, she was not bad to us in class, but neither were we particularly memorable to her or those of our classmates whom we mutually found uninteresting.” Lootika then tugged Somkhya’s hand: “I must say that I also feel some vague uneasiness about this.” Somakhya hugged her: “varārohe, at least she was not one of those teachers who complained that you spent too much time with the boys but I agree there might be something more than the mundane here. Let us go and see her at school.”

It might be about 5 years ago when one evening after an early dinner my husband remarked that he needed to go out to meet with a client. Normally, he met his clients on the ground floor of our two-story house that he used as his office. However, there were some occasions when he would go out to meet his clients — I guess the discreetness of the affair required him to do so. In any case, as I mentioned, I never took a deep interest in the specifics of his cases. Hence, I thought it was just one of those days and simply asked when he expected to be back. He said it might take him at least a couple of hours. Four hours passed and he did not return. I tried calling him, but his phone did not ring. I became tense and was wondering what I should do. Just then, I heard a knocking on the door. It was very unusual for anyone to visit us that late, so I looked through the peephole full of apprehension and saw a girl who seemed vaguely familiar. Surprised to see a little girl all by herself knocking at my door at night I wondered if she was being used as a decoy. Hence, I did not open the door but went up to the balcony overlooking the door and looked all around. The street was empty, and her knocking continued at the door. This combined with the vague foreboding from my husband’s failure to return made me run to the door and open it. The girl asked me to follow her. I was pretty sure it was some kind go trap and asked her who she was and what she wanted. She simply said it was very important and asked me to follow her. I said would call the police if she did not tell me. She simply said the police will not be able to help me without her and I could be in trouble if I took the police along without following her. I was surprised that a little girl would talk that way and shut the door and went inside to ring the cops. The next thing I knew I was waking up in the morning. I looked around and was still alone in the house — evidently, I had fallen into an inexplicable sleep when I tried to contact the police the previous night. I recalled the events of the night and in sweat called the police to report that my husband was missing.

The search went on for days and I was repeatedly interviewed by them, subject to lie-detectors tests and what not but nothing substantial ever came out of it and the case went cold like one of the cases he might have liked to look at. In the meantime, the girl kept making her appearance every now and then at intervals of a month and kept asking me to follow her. One night something seized me and I followed her and as I was doing so she tugged me to show the way and I realized it was no real girl — you may think I have gone cuckoo — but a phantom. Her grip had no substance at all to it!”

L and S: “Ma’am we understand this is exactly why you wanted to talk to us. So, no worries. We should not rule out any explanation but tell us your story as you experienced it. Before you continue could you please tell us the dates on which the phantom girl arrives?” The history teacher looked up her phone and gave them the dates. S: “Lootika, they all seem to be trayodaśī nights. Ma’am was that the day your husband regrettably vanished?”

H.T: “I do not keep track of the lunar calendar, but I have a strong feeling that it was not the case. The periodicity set in only after the second visitation.”
L: “Pray, continue with your story.”
S: “Is there anything more you would like to add to your story?”
H.T: “Yes. I did follow the phantom girl on two other occasions. The second time she led me again towards the Hanūmat shrine but instead of going inside vanished at the cart of the śṛṇgāṭaka vendor who stands at the street just where the bridge over the river ends. The third occasion was the most frightening. She led me by the railway station and my legs were wearying from the long walk. She then turned into a narrow alley with Pakistani flags fluttering on either side. A cricket match seemed to be going on and the dwellers were all chanting in unison for the victory of Pakistan. She then stopped at the break in the wall that would take you right to the train tracks and beckoned me to enter. I just could not get myself to proceed any further and turned around to scoot out of that alley known to be a dangerous place at night and call a cab. The phantom girl cried out there!’ pointing beyond the broken wall and vanished.”

L: “I commiserate with you. This is rather bizarre ma’am but what exactly do you want of us?”
H.T: “What does all this mean? Is my husband dead or alive? Can I have the ghost girl stop visiting me?”
L: “We cannot easily answer or solve all of those issues, but we can try to obtain some information about your husband’s fate. Would you be prepared to receive it?”
H.T.: “I’ve resigned myself for the worst. But when the law cannot give an answer; I still would like an answer. Even if the worst has happened why would he visit me from beyond in the form of a little girl? Would I not receive a more understandable signal!”
L: “OK. Could you please look straight forward and look at your eyebrows without moving your head and act as though you are seeing through them to the top of your head. Now close your eyes and open them.” Lootika noticed that the teacher’s eyeballs seemed to roll inwards.
S: “Spidery, I guess you can deploy the siddhakāṣṭha effectively.” Lootika whipped out her siddhakāṣṭha sanctified as per the traditions of bhairavācāra from her bag and deployed it on her former history teacher: “Ma’am be calm; you will see some visions and they might give you an answer. Please note everything you see carefully.” After two minutes in a trance from the kāṣṭha-prayoga, the teacher returned to her senses utterly dazed and shocked.
H.T: “I fear the worst has been confirmed.”
S: “Please tell us whatever you witnessed, however, painful that might be. It would bring you some relief at the end.”
H.T: “I saw that phantom girl being lifted by a man with a beard and thrown down a manhole-like dark abyss. Then, I saw my husband being thrown into the same. Then I saw a holy fuckeer mumble some mellifluous words, I guess in the Urdu and everything goes black. Then I heard a train passing by.”
S: “Indeed! I fear the worst is confirmed. You will have to live with this but the incubus will be lifted in part. Lootika?” Lootika deployed her siddhakāṣṭha again and their former history teacher awoke from another short trance in a state of peace.
L: “We can also perform a bhūtabandha and block that ghost girl from coming to your house.”
H.T: “That would be brilliant. I already feel a strange calm within me for the first time.”
S: “No Lootika! That might temporarily relieve you ma’am, but it could be utterly dangerous for Lootika if she tried the bandha and will not solve your problem for good. That ghost girl is a positive element. When you go to the Śivālaya simply offer some ghee or black sesame oil by drawing this diagram on the riverbank and uttering the following incantation: yathāsthānaṃ sukhaṃ tiṣṭhatu |”. Saying so Somakhya wrote down the incantation and drew the diagram and explained to her how to do the same. He said the ghost girl will return and you might see her sitting outside your house periodically, but she might not knock frequently.”
L and S: “Now may we kindly take leave. At some point sooner or later we hope you get a more complete relief and closure.”

Evening had set in and the streets were filling with a great mass of humanity. The constant blare of horns and their reverberation rent the polluted air as a throng of officer-goers made their way back home after a soul-crushing day in the service of some mahāmleccha overlord. Buses, trucks, cars, rickshaws and motorcycles competed for a sliver of the road in many a near-collision event. The sides of the streets filled with diners looking for some tongue-tickling deep-fried delicacy or sugar-laden treat. The odors wafting from these productions mixed with the fumes of the vehicles and the products of anaerobic fermentation from a distant gutter. Those awaiting their culinary orders to arrive were lost in the virtual worlds beamed on their faces from their phones. Only subliminally recording these, leaving their former teacher to her pile of answer sheets, Lootika and Somakhya started walking back home, each in their own meditation. Lootika broke the silence: “Priyatama, it strikes me that you are quite convinced that the ghost girl and the teacher’s husband had some direct connection beyond merely coming to meet their end in the same pit.” S: “Did you think otherwise?” L: “Truth to be told, I saw it differently. But now I see your line might explain somethings. Perhaps, a marūnmatta is the edge connecting the two in death.” S: “Indeed, that is how I see it. There is a lot more real detective work needed to fill in the rest of the story and neither of us is a lokasaṃcārin to get to the bottom of that. There are too many links for which we lack the right subjects to get a handle. Moreover, a bhūtabandha by us, in this case, would mean wading into territory that we don’t fully understand and the unfulfilled bhūta filled righteous indignation could turn on you. We should pass this by some of our lokajña friends — they might be able to throw more light on the aspects of it belonging to this world. L: “While not performing a bandha, I think still we should attempt a bhūtanivaha of that ghost-girl tonight to make her tell us her story.” S: “Not sure we have all the leads to pull that off successfully…”

Somakhya and Lootika with their kids as also Varoli and her family were visiting Vidrum at the rural paradise he had set up. They had spent the first part of the night stargazing. They would have gone on but realized that Vidrum and Kalakausha were not really as excited as they were by the brilliant skies. So, they decided to be a bit more involved with their hosts and settled down to yarn about the old days in the pleasant breeze of the southern country. Vidrum: “I don’t think I ever told you that the mysterious matter of our former history teacher cleared itself up rather dramatically — you had left the country by then. However, I wondered if you might have played a role in that regard.” S and L: “Truth to be told, this is news to us. Please tell us the story.”

Somakhya: “Ah. Pretty Lootika might have indeed had a hand in that last part about the fuckeer spilling the beans. She tried to draw that ghost-girl to make her speak but someone was blocking that phantom from speaking. So, she broke the block with her prayoga and instigated the phantom against the blocker.”

Bhāskara’s dual square indeterminate equations

Figure 1. Sum and difference of squares amounting to near squares.

In course of our exploration of the bhūjā-koṭi-karṇa-nyāya in our early youth we had observed that there are examples of “near misses”: $8^2+9^2=12^2+1$. Hence, we were excited to encounter them a little later in an interesting couple of indeterminate simultaneous equations in the Līlāvatī. Exhibiting his prowess as both a kavi and a mathematician, the great Bhāskara-II furnishes the following Vasantatilakā verse in his Līlāvatī:

rāśyor yayoḥ kṛti-viyogayutī nireke
kliśyanti bījagaṇite paṭavo’pi mūḍhāḥ
ṣoḍhokta-gūḍhagaṇitaṃ paribhāvayantaḥ || L 62||

Tell me, O friend! those 2 [numbers], the sum and difference of whose squares
reduced by one result in square numbers, wherein even experts in algebra who
keep dwelling upon the mysterious mathematical techniques
stated in six ways, come up as dim-witted [in solving this problem].

-Translation adapted from that conveyed by paṇḍita Ṛāmasubrahmaṇyan, a learned historian of Hindu mathematics

In terms of his kavitvam, Bhāskara-II abundantly illustrates the use of figures of speech, such as the yamaka-s or alliterative duplications. Ṛāmasubrahmaṇyan also mentions that he uses the figure of speech termed the “ullāsa” via the opposition of “paṭavaḥ” and “mūḍhāḥ” in the same verse to bring out the wonder associated with this problem, i.e. it is difficult even for those who are adept at the 6 operations of traditional Hindu mathematics: addition, subtraction, multiplication, division, squaring and square-root-extraction. The problem, put in modern notation goes thus:

Let $x, y, a, b$ be rational numbers with $y>x$. Then,

$x^2+y^2-1=a^2$
$y^2-x^2-1 =b^2$

When we first encountered it, we wondered if it was really that difficult but soon our investigation showed that it was hardly simple for us. To date, we do not have a general solution in integers, placing us squarely among the dull-witted. However, in the course of our study of the integer solutions, we discovered parametrizations with interesting connections beyond those provided by Bhāskara. We suspect he was aware of one or more of these, which is why he termed it a difficult problem for even those well-versed in arithmetic operations. In essence, the problem the generation of 2 new squares plus a unit square for each of them from the sums and differences of the areas of 2 starting squares (Figure 1). Before we consider the integer solutions, let us see the parametrizations offered by Bhāskara to obtain rational fractional solutions. By way of providing several numerical examples (a white indologist of the German school but with a style more typical of the American school had once stated with much verbiage what essentially amounts to “Hindoos must be idiots” for presenting such repetitive examples. He evidently forgot the fact that it was also the style of the great Leonhard Euler), he says:

atra prathamānayane $\tfrac{1}{2}$ kalpitam iṣṭam | asya kṛtiḥ $\tfrac{1}{4}$ | aṣṭa-guṇojātaḥ 2 | ayaṃ vyekaḥ 1 | dalitaḥ $\tfrac{1}{2}$| iṣṭena $\tfrac{1}{2}$ hṛto jātaḥ | asya kṛtiḥ 1 | dalitā $\tfrac{1}{2}$ saikā $\tfrac{3}{2}$ | ayam aparorāśiḥ | evam etau rāśī $1, \tfrac{3}{2}$ || evam ekena+ iṣṭena jātau rāśī $\tfrac{7}{2}, \tfrac{57}{8}$ dvikena $\tfrac{31}{4}, \tfrac{993}{32}$ || (Parametrization 1)

atha dvitīya-prakāreṇa+ iṣṭaṃ 1 anena dvi-guṇena 2 rūpaṃ bhaktam $\tfrac{1}{2}$ | iṣṭena sahitam jātaḥ prathamo rāśiḥ $\tfrac{3}{2}$ dvitīyo rūpam 1 evaṃ rāśī $\tfrac{3}{2}, 1$ || evaṃ dvikena+ iṣṭena $\tfrac{9}{4}, 1$ | trikeṇa $\tfrac{19}{6}, 1$ try-aṃśena jātau rāśī $\tfrac{11}{6},1$ || (Parametrization 2)

The second parametrization he offers is rather simple:

$x=1; y=\dfrac{2t^2+1}{2t}$

He illustrates it with the integers $t=1, 2, 3...$ and $t= \tfrac{1}{3}$. With integers, we see that $y$ is defined by a fractional sequence whose denominators are the successive even numbers and whose numerators are defined by the sequence $2n^2+1: 3, 9, 19, 33, 51...$. This sequence has interesting geometric connections. One can see that it defines the maximum number of bounded or unbounded regions that a plane can be divided into by $t$ pairs of parallel lines. Thus, one can see that the $y$ of Bhāskara’s second parametrization provides the ratio the maximum partitions of a plane to the total number of parallel lines drawn in dyads used for the purpose.

Figure 2. Division of plane into regions by parallel lines: 3, 9, 19… regions by 1, 2, 3 pairs of parallel lines.

Bhāskara’s first fractional parametrization takes the form:

$x=\dfrac{8t^2-1}{2t}$

$y= \dfrac{x^2}{2}+1 = \dfrac{64t^4-8t^2+1}{8t^2}$

He illustrates this with $t= \tfrac{1}{2}, 1, 2$, which respectively yield the pairs of solutions $(1, \tfrac{3}{2})$, $(\tfrac{7}{2}, \tfrac{57}{8})$, $(\tfrac{31}{4}, \tfrac{993}{32})$.

The problem can also be seen as that of finding the intersection between coaxial circles and hyperbolas. Restricting ourselves to intersections in the first quadrant, we can see that the general form of the solutions would be:

$x=\sqrt{\dfrac{a^2-b^2}{2}}$; $y=\sqrt{1 + \dfrac{a^2+b^2}{2}}$

From the above, for integer solutions we can say the following: 1) Given that $a^2+b^2$ must be an even number, $a, b$ should be even. 2) Hence, from the original pair of equations, we can say that $x, y$ must be of opposite parity with $x$ being even and $y$ odd. 3) Further, for $x$ to be an even number it has to be divisible by 8. 4) Hence, $y \mod 8 \equiv 1$. Therefore, all solutions should be of the form $x=8m, y=8n+1$, where $m, n$ are integers. Beyond this, not being particularly adept at mathematics, to actually solve the equations for integers, we took the numerical approach and computed the first few pairs of solutions. It was quite easy to locate the first solution $(8, 9)$ (Figure 1) which in a sense is like the most primitive bhujā-koṭi-karṇa triplet (3, 4, 5). The first few solutions are provided below as a table and illustrated as a $x-y$ plot in Figure 3.

Figure 3. First few integer solutions.

Table 1

We quickly noticed that there is one family of solutions that lie on a clearly defined curve (dark red in Figure 3).

Family 1. This family has the convergents: $\tfrac{y}{x} \rightarrow$ 1, 2, 3, 4 $\dots$. We can easily obtain parametrization defining this family to be:
$x=8t^3$
$y=8t^4+1$

This yields (8,9); (64,129); (216,649); (512,2049); (1000,5001); (1728,10369); (2744,19209); (4096,32769); (5832,52489); (8000,80001). This corresponds to the the third parametrization offered by Bhāskara that may be used to obtain rational fractional or integer solutions:

athavā sūtram –
iṣṭasya varga-vargo ghanaś ca tav aṣṭa-saṅguṇau prathamaḥ |
saiko rāśī syātām evam vyakte+ atha vā avyakte || L 63
Or the sūtra: Square the square of the given number and the cube of that number respectively multiplied by 8, adding 1 to the first product, the solutions are obtained both for arithmetic examples or as algebraic parametrization.

iṣṭam $\tfrac{1}{2}$ asya varga-vargaḥ $\tfrac{1}{16}$ aṣṭaghnaḥ $\tfrac{1}{2}$ saiko jātaḥ prathamo rāśiḥ $\tfrac{3}{2}$ punar iṣṭam $\tfrac{1}{2}$ asya ghanaḥ $\tfrac{1}{8}$ aṣṭa-guṇo jāto dvitīyo rāśiḥ 1 evaṃ jātau rāśī $\tfrac{3}{2}$, 1 | atha+ ekena iṣṭena 9, 8 | dvikena 129, 64 | trikeṇa 649, 216 | evaṃ sarveṣv api prakāreṣv iṣṭa-vaśād ānantyam ||

By taking $\tfrac{1}{2}$ as the given, the square of the square of the given number is $\tfrac{1}{16}$, which multiplied by 8 is $\tfrac{1}{2}$. This plus 1 yields the first number of the solution $\tfrac{3}{2}$. Again given $\tfrac{1}{2}$, its cube is $\tfrac{1}{8}$ which multiplied by 8 yields the second number of the solution, 1. Thus, we have the pair (1, 3/2). Now with 1 as the given we get (8, 9); with 2 we get (64, 129); with 3 we get (216, 649). Thus, with each of these parametrizations (i.e. all the 3 he offers) by substituting any number one gets infinite solutions.

However, this parametrization hardly accounts for all the solutions. Through analysis of the remaining solutions, we could discover several further families with distinct more complex parametrizations. They are:
Family 2. This family has the convergent $\tfrac{y}{x} \rightarrow 1$

$x= (T_t(3))^2-1$
$y=(T_t(3))^2$

Here, $T_t(x)$ is the $t$-th Chebyshev polynomial of the first kind that is defined based on the multiple angle formula of the cosine function:
$\cos(x) =\cos(x)$; $\cos(2x)=2\cos^2(x)-1$; $\cos(3x)= 4\cos^3(x)-3 \cos(x)$
Thus, we get the Chebshev polynomials $T_n(x)$ as:
$T_1(x)=x$; $T_2(x)= 2x^2-1$; $T_3(x) = 4x^3-3x \dots$

Thus, we get the pairs (8,9); (288,289); (9800, 9801); (332928, 332929). All these points lie on the line $y=x+1$

One observes that $\tfrac{x}{2}=k^2$ where $k \rightarrow$ 2, 12, 70, 408 $\dots$

From the above, it is easy to prove that the sequence of fractions $\tfrac{\sqrt{y}}{k}$ are successive convergents for $\sqrt{2}$. For $t=4$ we get $\tfrac{577}{408 } = 1.41421 \dots$, which is Baudhāyana’s convergent approximating $\sqrt{2}$ to 5 places after the decimal point.

Family 3. This family has the convergent $\tfrac{y}{x} \rightarrow \tfrac{4}{3}$ (green line in Figure 3). It can be parametrized thus:

$x=8 \left \lceil \dfrac{(9 - 3 \sqrt{7}) (8 + 3 \sqrt{7})^t}{28} \right \rceil$

Thus, the ratio of successive $x$ converges to $8 + 3 \sqrt{7}$, a number which is again related to Chebyshev polynomials of the first kind evaluated at 8:

$\displaystyle \lim_{n \to \infty}\dfrac{T_n(8)}{T_{n-1}(8)} = 8 + 3 \sqrt{7}$

$y=\left \lfloor \dfrac{32}{3} \left \lceil \dfrac{(9 - 3 \sqrt{7}) (8 + 3 \sqrt{7})^t}{28} \right \rceil \right \rfloor -1$

Thus, we arrive at the pairs constituting this family as: (8,9); (80,105); (1232,1641); (19592,26121) $\dots$

Family 4. This family has the convergent $\tfrac{y}{x} \rightarrow \tfrac{\sqrt{5}}{2} \approx 1.1180339$. It can be parametrized thus:
$x= M(6t)$, i.e. $x$ is every 6th term of the mātra-meru sequence $M(n): 1, 1, 2, 3, 5, 8, 13...$
Thus, we can also express $x$ using the Golden ratio $\phi \approx 1.61803 \dots$:

$x= \dfrac{\phi^{6t}-\phi^{-6t}}{2\phi-1}$

Thus, the ratio of successive $x$ converges to $\phi^6=9+\sqrt{80}$; we can also write $x$ using the hyperbolic sine function:

$x= \dfrac{2 \sinh(6t\log(\phi))}{2\phi-1}$

Similarly, we get:

$y=\dfrac{\phi^{6t}+\phi^{-6t}}{2}$

As with $x$ we can also get a hyperbolic trignometric expression for $y$:
$y= \cosh(6t \log(\phi))$

Finally, we can also write $y$ compactly in terms of Chebyshev polynomials of the first kind:
$y= T_t(9)$

Thus, the first few members of this family are: (8,9); (144,161); (2584,2889); (46368,51841) $\dots$

Family 5. This family has the convergent $\tfrac{y}{x} \rightarrow \tfrac{\sqrt{65}}{4} \approx 2.0155644$. It can be parametrized using the continued fraction expressions of the convergent:
$x \rightarrow$ denominators of odd terms of the continued fraction convergents of $\tfrac{\sqrt{65}}{4}$
$y \rightarrow$ numerators of odd terms of the continued fraction convergents of $\tfrac{\sqrt{65}}{4}$

The relevant partial convergents are $\tfrac{129}{64}$; $\tfrac{33281}{16512}$; $\tfrac{8586369}{4260032}$; $\tfrac{2215249921}{1099071744} \dots$

Thus, we see that the convergents with odd numerators and even denominators yield the $(x,y)$ corresponding to this family, with the first term matching the second term of family 1; hence, it may be seen as branching from family 1. In practical terms, one can obtain these values using the below 2-seeded recursions:
$y \rightarrow f[n] = 258f[n-1] - f[n-2]; \; f[1]=0, f[2]= 64$; second term onward
$y \rightarrow f[n] = 258f[n-1] - f[n-2]; \; f[1]=1, f[2]= 129$; second term onward

Family 6. This family has the convergent $\tfrac{y}{x} \rightarrow \tfrac{\sqrt{689}}{20} \approx 1.31244047$. It can be parametrized using the partial convergent fractions approximating the convergent.

$x \rightarrow$ denominators of partial convergents of $\tfrac{\sqrt{689}}{20}$ divisible by 8
$y \rightarrow$ numerators of partial convergents of $\tfrac{\sqrt{689}}{20}$ $\mod 8 \equiv 1$

The relevant partial convergents are $\tfrac{105}{80}$; $\tfrac{22049}{16800}$; $\tfrac{4630185}{3527920}$; $\tfrac{740846400}{972316801} \dots$

The first term is the same as the second term of family 4. In practical terms, one can obtain these values using the below 2-seeded recursions:

$y \rightarrow f[n] = 210 f[n-1] - f[n-2]; \; f[1]=0 f[2]= 80$; second term onward
$y \rightarrow f[n] = 210 f[n-1] - f[n-2]; \; f[1]=1 f[2]= 105$; second term onward

In the case of families 4, 5 and 6, we observe that the sum and the difference of the squares of the numerators and denominators yield perfect squares. Further, the denominator is always an even number and the numerator a surd of the form $\sqrt{k^2+1}$ or $\sqrt{k^2+k/2}$. This suggests an approach for discovering new families. Our search till 100000 uncovered 2 more families of this form

Family 7. This family has the convergent $\tfrac{y}{x} \rightarrow \dfrac{\sqrt{29585}}{104} \approx 1.6538741$; $29585 = 172^2+1$

It can be parametrized using the partial convergent fractions approximating the convergent thus:

$x \rightarrow$ denominators of partial convergents of $\tfrac{\sqrt{29585}}{104}$ divisible by 8
$y \rightarrow$ numerators of partial convergents of $\tfrac{\sqrt{29585}}{104}$ $\mod 8 \equiv 1$

The relevant partial convergents are:
$\tfrac{59169}{35776}$; $\tfrac{7001941121}{4233660288}$; $\tfrac{828595708317729}{501002891125568} \dots$

In practical terms, one can obtain these values using the below 2-seeded recursions:

$y \rightarrow f[n] = 118338 f[n-1] - f[n-2]; \; f[1]=0 f[2]= 35776$; second term onward
$y \rightarrow f[n] = 118338 f[n-1] - f[n-2]; \; f[1]=1 f[2]= 59169$; second term onward

Family 8. This family has the convergent $\tfrac{y}{x} \rightarrow \dfrac{\sqrt{44945}}{208} \approx 1.0192421$; $29585 = 212^2+1$

It can be parametrized using the partial convergent fractions approximating the convergent thus:

$x \rightarrow$ denominators of partial convergents of $\tfrac{\sqrt{44945}}{208}$ divisible by 8
$y \rightarrow$ numerators of partial convergents of $\tfrac{\sqrt{44945}}{208}$ $\mod 8 \equiv 1$

The relevant partial convergents are:
$\tfrac{89889}{88192}$; $\tfrac{16160064641}{15854981376}$; $\tfrac{2905224100939809}{2850376841726336} \dots$

In practical terms, one can obtain these values using the below 2-seeded recursions:

$y \rightarrow f[n] = 179778 f[n-1] - f[n-2]; \; f[1]=0 f[2]= 88192$; second term onward
$y \rightarrow f[n] = 179778 f[n-1] - f[n-2]; \; f[1]=1 f[2]= 89889$; second term onward

In families 7, 8 there are no small terms that connect them to any of the other families; keeping with Hindu love for big numbers, they start relatively large and grow rapidly. These 8 families cover all the solutions in Table 1 and Figure 3 $(\le 10^5)$. The relationships between them are shown in Figure 4.

Figure 4. The relationship between families.

Is there a general way to obtain all parametrizations for the integer solutions of this pair of indeterminate equations? Perhaps this has already been answered by mathematicians or perhaps not. In any case, as Bhāskara had stated, the solutions to this couple of equations is not an entirely trivial problem and sufficiently absorbing for an enthusiast of arithmetic.

Pandemic days-6: Genetic risk factors

The coronavirus that made its way to humans aided by the Cīna-s at Wuhan has now been with us for nearly an year. Right from the early days of this outbreak, one thing has been notable about this virus: some people got very ill from it, while others breezed through a relatively mild or supposedly “asymptomatic” infection (though we still do not know the long term consequences of the mild infection). This made the disease way more deadly than its cousin SARS as potentially infectious individuals with the mild form of the disease could wander about spreading it. As a result, at the time of writing, at least 1,085,000 people have died from it the world over, and anywhere between 40-300 million could have been infected by it. Some factors affecting the differential outcome were clear even when the virus was still only with Cīna-s: it affected older people and men more severely. In the early days of the pandemic, several other factors were also proposed to affect the outcome of the disease, like temperature extremes, humidity, prior vaccination with BCG. However, these, especially the environmental ones, have not been supported by the data coming from the explosive pandemic that followed. It was also clear that there were going to be genetic factors that influence the outcome. These are becoming clearer only now and are the topic of this note. This note is based on data from several recent studies that have tried to identify genetic risk factors in various populations. What we do here is to briefly look at the genes that have been identified and give some commentary on them and what can be inferred from them.

The first set of studies by Bastard  (yes, that is the author’s name; not an easy one to bear in the English-speaking world) and Zhang et al took a directed approach to look at 13 genes in the Toll-like receptor-3 (TLR3)- type-I interferon system. Mutations in these genes have previously been implicated in severe influenza with involvement of the lower respiratory tract and other viral diseases. They found that potential loss-of-function variants in these genes were enriched in patients with a severe outcome of the Wuhan disease. In a related study they found, that an autoimmune condition with antibodies against the type-I interferons also correlated with a similar outcome as the potential loss-of-function mutations. This supported the idea that defects in the interferon-I (IFN) system are a predictor of disease outcome even in the case of the current coronavirus. This is rather interesting as the bats show distinct alternations to their IFN-I system relative to other mammals. First,  black flying foxes have been shown to have a higher and potentially constitutive expression of IFN-I genes. Second, the Egyptian fruit bats show and expansion of the IFN-I genes, especially the subtype IFNW (interferon $\omega$). These observations, together with the fact that bats have a high level of tolerance to SARS-like CoVs (and other viruses) support the idea that the type-I IFN system is important in surviving not just SARS-CoV-2 but also other viruses.

As a simple caricature, the following pathway describes the role of products of the 13 genes in the IFN-I system in cells infected by a virus (say the respiratory epithelial cells) or specialized blood cells, which are part of the immune system, that sense the virus (plasmacytoid dendritic cells):

1. Recognition of the invading virus by the leucine-rich repeats of the TLR3 protein triggers a signaling response that additionally involves TRIF, UNC93B1, TRAF3, TBK1 and NEMO proteins which ultimately results in activating of a transcription factor IRF3 in the nucleus.
2. Consequently, IRF3 induces the transcription of IFN-Is, which is further amplified by a related transcription factor IRF7 which is induced by IRF3.
3. The secretion of INF-Is is followed by their binding of receptors on other cells like epithelial cells in the respiratory tract. The receptors are dimers of the two paralogous proteins IFNRA1 and IFNRA1.
4. The receptors activate the associated transcription factors STAT1 and STAT2, which then associates with another transcription factor IRF9 (a paralog of IRF3 and IRF7) to activate the interferon-stimulated genes that mediate the immune response to the virus.

This is the well-known INF-I immune response. Of these proteins, the TLR3 and TRIF/TICAM1 are proteins with TIR domains, which we had earlier shown to have very ancient roots in the immune response of bacteria against the viruses that infect them. UNC93B1 is a membrane protein involved in the trafficking of the TLR3 protein from the endoplasmic reticulum to endolysosome where it can intercept the endocytosed virus. TLR3 additionally has the receptor portion in the form of leucine-rich repeats that recognize the invasive virus. TRIF has an $\alpha$-helical tetratricopeptide repeats that keep its TIR domain inactive till TLR3 is activated. At that point, it associates with TLR3’s TIR domain. TRIF also has an RHIM motif, a short sequence that allows the protein to form homotypic oligomers which are important for the downstream signaling. Thus, it serves as a platform for initiating a signal with the cell in response to the sensing of the virus by TLR3. The signal is set off first by TRAF3 which is an E3 ubiquitin-ligase that is recruited to the platform formed by TRIF. It consequently conjugates Lysine-63 ubiquitins to its targets. This signal is transmitted further via the kinase TBK1, which associates with NEMO to form a signaling-kinase complex similar to the kinase complex that activates the inflammatory transcription factor NF$\kappa$B by phosphorylating its inhibitor IKK. TBK1 in addition to its kinase domain has a Ubiquitin-like domain that we had discovered a while back.  The presence of a ubiquitin-like domain in TBK1 allows it to associate with the ubiquitins conjugated by TRAF3. As a consequence of this interaction via its ubiquitin-like domain, it becomes functionally active to phosphorylate the DNA-binding transcription factor IRF3. This then dimerizes to activate the transcription of the interferon genes. This response to the virus can be triggered in different ways but this is the typical mechanism for the RNA viruses like influenza or DNA viruses like Herpes simplex virus. Thus, mutations in this system have previously shown to impair the response to influenza resulting in severe pneumonia or HSV resulting in encephalitis.

The second part of this response is signal transduced by the IFN-I via its receptor. This is via the famous JAK-STAT pathway that involves the kinases JAK which phosphorylate the STATs. These and their partner IRF9, all DNA-binding transcription factors, induce the IFN-I stimulated genes, many of which are the “sword-arm” of the antiviral defense. Thus, mutations in the two IFNAR genes, IRF9 and STATs also result in negative outcomes from viral infections and adverse reactions to live measles and Yellow fever vaccines. However, interestingly,  a mutation in the IFNAR1 gene resulting in an impaired receptor that binds the type-I IFN, IFNB, weakly results in greater resistance to tuberculosis. This is rather striking as, unlike with the viral diseases, it selects in the opposite direction for the strength of IFN-I signaling. The complexity of this situation even with SARS-CoV-2 is suggested by reports that the localized hyper-expression of type-I and III IFNs in the lung results in a more severe disease poor lung-repair. However, in contrast, reduced IFN-I production by peripheral blood immunocytes is associated with a severe form of the disease. Thus, over the IFN-I is important for the defense against SARS-CoV-2 but the location of over-expression seems to matter.

A notable point is that while both the life-threatening and benign forms of the disease are fairly uniformly distributed across populations with diverse ancestries, these IFN-I related loss-of-function variants reported by the authors are primarily found in Europeans, with some presence in diverse Asian populations (Figure 1). While the numbers are small, it is still significant that they did not get any of these variants in Africans. This is striking given that, another study found that in the USA infection and death rates are 2 to 3 times higher in people of African ancestry than their proportion of the population. This, suggests that in Africa there has possibly been selection against these variants due to pressure from other viruses which are prevalent there. Indeed, the related coronavirus MERS might have had its ultimate origins in Africa even suggesting direct events of selection by coronaviruses in the past. However, notably, the researchers found that African ancestry people in the US have significantly higher expression in the nasal epithelium of the transmembrane serine protease 2 (TMPRSS2) which along with the other protease ACE2 is a receptor used by SARS-CoV-2 to invade target cells.

Also related to the above complex of 13 genes, was a small study by van der Made et al based on exome sequencing that identified rare loss-of-function mutations in TLR7 in 4 young men with severe disease. This resulted in defective type-I and type-II interferon production. While a small study, it is notable that it recovered these mutations in TLR7. This gene is in a cluster with its paralog TLR8 on the X-chromosome; hence, males have only one copy. Importantly, both of them, like TLR3 are sensors the detect viruses which enter cells via endocytosis. It specifically senses single-stranded RNA fragments that are enriched in guanine and uracil in the endosome of plasmacytoid dendritic cells and B cells, raising the possibility that impairment of these virus-specific TLRs might be part of the increased susceptibility to SARS-CoV-2 of males.

Figure 1. The mapping of different forms of the disease on to the 1000 genomes populations modified from Zhang and Bastard et al. LOF are the loss-of-function variants they identified.

The next study by Zeberg and Pääbo discovered a genomic segment of $\sim 50$  kb that confers an elevated risk of severe disease which is inherited from Neanderthals. This region on chromosome 3 kept coming up repeatedly in multiple investigations for genetic determinants of disease severity. This core region of 49.4 Kb and the larger surrounding region of ~333.8 Kb shows strong linkage disequilibrium and appears to have introgressed from a Neanderthal ~60-40 Kya. This region is rather interesting because it encodes 5 chemokine receptor genes, namely XCR1, CXCR6, CCR9, CCR1 and CCR3. These are all receptors for the signaling proteins known as chemokines, which transmit various immune signals such as in the recruitment of effector immunocytes to the site of inflammation (e.g. various lineages of cytotoxic cells and antibody-producing B-cells) or in directing T-cells to guard different parts of the lungs. Gene-knockouts pf CCR1 suggest that it plays a role in protecting against inflammation and increases susceptibility to fatal infection of the central nervous system by the coronavirus MHV1  in mice. Reducing signaling via this receptor has also been shown to increase susceptibility to the herpes simplex virus type 2. Some chemokine receptors are used by viruses and other pathogens to enter the vertebrate cells. For example, CCR3 and CXCR6 from this locus code for the co-receptor for the AIDS virus HIV-1 and/or SIV. The human herpesvirus 8 encodes its own chemokine vMIP-II, which targets the protein XCR1 encoded by this locus and blocks signaling via it. Thus, the chemokine receptors are a central part of the immune response of jawed vertebrates and under strong selection from the host-pathogen arms race.

What is most striking about this region is that it is elevated in frequency in the Indian subcontinent (~50%; It is found in ~16% of Europeans) while absent or rare in East Asia.  Indeed, after the mating with Neanderthals, the introgressed regions from them have been routinely purged off the genome of Homo sapiens suggesting a degree of incompatibility with the sapiens alleles. This is consistent with the association of Neanderthal alleles with certain immune dysfunctions. However, this region has followed the converse pattern. If it has been retained after coming from a Neanderthal ancestor and elevated in frequencies it must be due to selection for it in the subcontinent likely due to some relatively recent or extant pathogen. The region has been previously noted as being under selection in East Bengal. This raises the possibility that it could have conferred an advantage to diseases such as cholera. However, it is rather notable that despite gene flow between and geographic proximity it is so rare in East Asia. We and others have long held that several extant CoV diseases (today relatively mild) have originated in East Asia, likely China, potentially as a side effect of their culinary habits. This would imply that there was strong selection from these CoVs against this Neanderthal-derived variant in East Asia when those CoVs were still severe, even as it was selected for in India by other pathogens. Thus, it is a classic evolutionary phenomenon of bidirectional selection in action. Such selection events often leave their mark in immune molecules driving them in different directions. The Duffy Chemokine receptor by which the Plasmodium vivax and P. knowlesi malarial parasites enter cells is likely to be another such. Loss or reduced expression of the Duffy receptor favors resistance to vivax malaria. But the protein is retained widely in humans suggesting some immune function.

Figure 2. Distribution of Neanderthal variant across populations from Zeberg and Pääbo .

Finally, another set of genome-wide association studies by Ellinghaus et al and Roberts et al identified multiple single nucleotide polymorphisms (SNPs) associated with a severe form of the disease. One of these in chromosome 3 corresponds to the same region as identified by the above study as coming from the Neanderthals. Another SNP was identified on chromosome 9 which is in the vicinity of the ABO gene that determines the ABO blood type. The ABO blood group is determined by the oligosaccharide synthesized by 4 glycosyltransferases: the two closely linked paralogs FUT1 and FUT2 make the base oligosaccharide by adding a fucose. This is modified further by the products of the ABO gene, the A-variant glycosyltransferase which adds an $\alpha$ 1-3-N-acetylgalactosamine and the B-variant glycosyltransferase which adds a 1-3-galactose. This oligosaccharide is the conjugated to lipid head-groups and proteins (as on the RBC surface) to give rise to the A/B/AB antigen. If this gene is dysfunctional, it results in O where neither sugar is added. These sugars are believed to play a role in cell-cell adhesion. The polymorphism in ABO across humans suggests that it has been under some kind of immune selection. Indeed, there have been studies claiming an association of this gene with susceptibility to various bacterial and viral infections (noroviruses and rotaviruses). Interestingly, a knockdown of the ABO gene has been reported to inhibit HIV-1 replication in HeLa P4/R5 cells. This could be because of multiple reasons: 1. pathogens specifically binding cells with glycoproteins decorated by particular versions of the sugar. 2. Viruses themselves possess various glycoproteins against which antibodies develop. These could cross-react with the host glycoproteins exerting selection via autoimmunity. Alternatively, the absence of a certain modification on the host protein could help the host to develop better neutralizing antibodies against certain viral glycoproteins. It has been suggested that the influenza viral glycoproteins and ABO locus might be in some such evolutionary interaction. 3. Immunocytes localize to specific parts of the body by recognizing the sugars on surface proteins and lipids. These might play a role in response to pathogens. Indeed, other than the ABO (H included) blood group, other blood group systems are also based on polymorphisms of glycosyltransferases (PIPK, Lewis, I, Globoside, FORS, Sid) or extracellular ADP-ribosyltransferases (Dombrock) suggesting that such evolutionary entanglements between pathogens and cell-surface modifications might be more widespread. However, the role of ABO in susceptibility to SARS-CoV-2, even if plausible, remains unclear.

The Roberts et al study also identified a SNP on chromosome 22 possibly associated with the $\lambda$-immunoglobulin locus that codes for the antibody light chains. This is again consistent with a defect in antibody production by B-cells. Another SNP identified by them lies on chromosome 1 in the vicinity of the  IVNS1ABP gene. The SWT1 gene also lies some distance away from the former gene. Interestingly, IVNS1ABP has been shown to interact with the influenza virus NS1 protein. This NS1-IVNS1ABP  complex targets the mRNA of another influenza gene M1 to nuclear speckles enriched in splicing factors for alternative splicing. The result is an alternatively spliced mRNA M2 that codes for a proton channel needed for acidification and release of viral ribonucleoproteins in the endosome during invasion.  Interestingly, IVNS1ABP belongs to large class POZ domain proteins with central HEAT and C-terminal Kelch repeats that also function as cullin-E3 ubiquitin ligases, several of which have antiviral roles. Hence, it showing up in the context of SARS-CoV-2 is rather interesting as it raises multiple possibilities: 1. Is it involved in the trafficking of viral mRNA as in influenza? 2. Is it an intracellular antiviral factor that recruits an E3 ligase complex for tagging viral proteins for destructions?

It is also possible that this SNP affects the nearby SWT1 gene. Sometime back we had shown that this protein contains 2 endoRNase domains. It prevents the cytoplasmic leakage of defective unspliced mRNAs by cleaving such RNAs at the nuclear pore. It is hence possible that this protein also interacts with viral RNA in someway. In either case, it is notable that this SNP is associated with disease severity only in males and not females. The cause for this again remains a mystery. Finally, this screen recovered a SNP on chromosome 1 close to the SRRM1 whose product is also involved in pre-mRNA splicing. This again raises the possibility of interaction with viral RNA.

In conclusion, the risk factors have pointed in many different directions, some relatively well understood from susceptibility to other viruses and yet others which remain murky. Evidently, there will be more rare ones which remain to be uncovered. However, even with the current examples, there are hints of bidirectional selection at multiple loci suggesting that sweeps of dominant pathogens have optimized our immune systems in different directions. The victory against one could leave one susceptible to another.

https://science.sciencemag.org/content/early/2020/09/29/science.abd4570

https://science.sciencemag.org/content/early/2020/09/23/science.abd4585

https://jamanetwork.com/journals/jama/fullarticle/2768926

Counting pyramids, squares and magic squares

Figure 1. Pyramidal numbers

The following note provides some exceedingly elementary mathematics, primarily for bālabodhana. Sometime back we heard a talk by a famous contemporary mathematician (M. Bhargava) in which he described how as a kid he discovered for himself the formula for pyramidal numbers (i.e. defined by the number of spheres packed in pyramids with a square base; Figure 1). It reminded us of a parallel experience in our childhood, and also of the difference between an ordinary person and a mathematician. In those long past days, we found ourselves in the company of a clansman who had a much lower sense of purpose than us in our youth (it seems to have inverted in adulthood). Hence, he kept himself busy by leafing through books of “puzzles” or playing video games. He showed us one such “puzzle” which was puzzling him. It showed something like Figure 1 and asked the reader to find the total number of balls in the pile if a base-edge had 15 balls. We asked him why that was a big deal — after all, it was just a lot of squaring and addition and suggested that we get started with a paper and pencil. He responded that he too had realized the same but had divined that what the questioner wanted was a formula into which we could plug in a base-edge with any number of balls and get the answer. We tried to figure out that formula but failed; thus, we sorted with the mere mortals rather than the great intellectuals.

Nevertheless, our effort was not entirely a waste. In the process of attempting to crack the formula, we discovered for ourselves an isomorphism: The count of the balls in the pyramid is the same as the total number of squares that can be counted in a $n\times n$ square grid (Figure 2). In this mapping, the single ball on the top is equivalent to the biggest or the bounding square. The base layer of the pyramid corresponds to the individual squares of the grid. All other layers map onto interstitial squares — in Figure 2 we show how those are defined by pink shading and cross-hatching of one example of them. In this mapping, the entire pyramid is mapped into the interior of the apical ball, which is now represented as a sphere. Thus, the number of balls packed into a pyramid and the number of squares in a $n\times n$ square grid are merely 3D and 2D representations of the same number, i.e. the sum of squares $1^2+2^2+3^2...n^2$

Figure 2. The total number of squares formed by contact in a square grid.

We got our answer to this a couple of years later when we started reading the Āryabhaṭīyam of Āryabhaṭa, one of the greatest Hindu scientists of all times. He says:

varga-citi-ghanaḥ sa bhavet citi-vargo ghana-citi-ghanaś ca || AB 2.22

The sixth part of the product of the three quantities, viz. the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the (original) series is the sum of the cubes. [vide KS Shukla].

In modern language, we would render the first formula, which concerns us, as:

$\displaystyle \sum_{j=1}^{n} j^2 = \dfrac{n(n+1)(2n+1)}{6}$

This is the formula for the figurate numbers known as the pyramidal numbers as they define square pyramids (Figure 1). A pratyakṣa geometric proof for this offered by the great Gārgya Nīlakaṇṭha somayājin (Figure 3). While this proof appears in Nīlakaṇṭha’s bhāṣya on the Āryabhaṭīyam, it is likely that some such proof was already known to Āryabhaṭa.

Figure 3. The formula for pyramidal numbers or the sum of squares of integers.

For bālabodhana:

1) He first asks you to lay a rectangular floor of $(2n+1)(n+1)$ cubic units.

2) Then you erect the walls on 3 of sides of the floor of height $n$ cubic units, namely the 2 sides of length $n+1$ and 1 side of length $2n+1$.

3) The shell thus constructed has:

$(2n+1)(n+1)=2n^2+3n+1 \rightarrow$ floor

$(2n+1)n-(2n+1)=2n^2-n-1 \rightarrow$ backwall

$2(n^2-n)=2n^2-2n \rightarrow$ sidewalls

i.e. a total of $6n^2$ cubic units or bricks.

From the figure, it is apparent that the shell can accommodate another shell based on $(n-1)$, which in turn can accommodate one based on $(n-2)$ units and so on till 1. Thus, we can fill a cuboid of volume,

$\displaystyle n(n+1)(2n+1)= 6\sum_{j=1}^{n} j^2$,

This yields Āryabhaṭa’s formula from which we can write the sequence of pyramidal numbers $Py_n$ as:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240…

We had earlier seen Āryabhaṭa and Nīlakaṇṭha’s work on triangular numbers (sum of integers) and tetrahedral numbers (the sum of the sum of integers) [footnote 1]. From that, we know the formula for tetrahedral numbers to be:

$Te_n=\dfrac{n(n+1)(n+2)}{6}$

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680…

We see that $Py_n=Te_n+Te_{n-1}$. This can be easily proven by seeing that merging two successive tetrahedral piles of spheres we get a square pyramid pile of balls (see figure in [footnote 1]). Shortly thereafter, this led us to finding for ourselves the space occupancy or density constant for atomic packing. Consider uniformly sized spherical atoms to be packed in a pyramid, like in Figure 1. Then the question is what fraction of the volume of the pyramid will be occupied by matter. We know that the volume of the pyramid whose side length is $l$ is $V = \tfrac{l^2a}{3}$, where $a$ is its height. From the bhujā-koṭi-karṇa-nyāya we have its height as $\tfrac{l}{\sqrt{2}}$. Hence,

$V = \tfrac{l^3}{3\sqrt{2}}$

Now the volume occupied by the atoms from Āryabhaṭa’s series sum is:

$V_a=\dfrac{2n^3+3n^2+n}{6}\cdot \dfrac{4\pi r^3}{3}$

The radius of each atom is $r=\tfrac{l}{2n}$. Plugging this in the above we get:

$V_a=\dfrac{2n^3+3n^2+n}{6}\cdot \dfrac{4\pi l^3}{24n^3}$

Simplifying we get:

$V_a = \left( \dfrac{1}{18}+ \dfrac{1}{12n}+\dfrac{1}{36n^2}\right)\pi l^3$

Since the atoms have infinitesimal radius we can take the limit $n \to \infty$ and we are left with:

$\displaystyle \lim_{n \to \infty} V_a =\dfrac{\pi l^3}{18}$

Thus, we get,

$\dfrac{V_a}{V}=\dfrac{\pi}{3\sqrt{2}} \approx 0.7404805$

Hence, little under $\tfrac{3}{4}$ of the space occupied by solid matter is filled by uniform spherical atoms. This meditation on atomic packing led us to another way of counting squares. Imagine circles packed as in Figure 4. The circles can then be used to define squares. The most obvious set of squares is equivalent to the $n \times n$ grid that we considered above. Here the smallest squares of the grid are equivalent to those circumscribing each circle, or alternative inscribed within it as shown in the example with just 1 circle. We can also join the centers of the circles and get bigger squares. If we instead circumscribe the circles we get an equivalent number corresponding to the bounding and interstitial squares of the $n \times n$ grid. However, we notice (as shown in the $3 \times 3$ example) that we can also get additional squares by joining the centers cross-ways. So the question was what is the total number of squares if we count in this manner?

Figure 4. Squares defined by packed circles.

We noticed that the first two cases will have the same number of squares as the pyramidal number case. However, from the $3 \times 3$ case onward we will get additional squares. We noticed that for $3 \times 3$ we get one additional square beyond the pyramidal numbers; for the $4 \times 4$ case we get 4 additional squares. It can be seen that the number of additional squares essentially define tetrahedral numbers; thus, we can write the sequence $S_n$ of this mode of counting as below:

$S_1=Py_1, S_2=Py_2$, when $n\ge 3, S_n=Py_n+Te_{n-2}$.

$\therefore S_n=\dfrac{n(n+1)(2n+1)}{6} + \dfrac{n(n-1)(n-2)}{6}= \dfrac{n^3+n}{2}$

$S_n:$ 1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695…

We can also derive this sequence in another way. Write the natural numbers thus:

(1); (2,3); (4,5,6); (7,8,9,10); (11,12,13,14,15);…

If we then take the sum of each group in brackets, which has $1, 2, 3 ... n$ elements, we get $S_n$.

We observe that from the 3rd term onward this sequence remarkably yields the magic constants $M$ (row, column and diagonal sums or the largest eigenvalue of the matrix defined by the magic square) for the minimal magic squares (bhadra-s) i.e. magic squares made of numbers from $1:n^2$ where $n$ is the order or the side length of it [footnote 2]. We also realized then that this was the basis of the “magic choice” property which we used in a schoolyard trick. That is illustrated in Figure 5.

Figure 5. Magic choice.

Write a $n \times n$ square of all integers from $1...n^2$. Ask a person to silently randomly choose $n$ numbers such that each row and column of the square is represented once (orange circles in Figure 5) and sum them up. Then, without him revealing anything you tell him the sum. The sum will be $S_n$. This is the weaker condition which can be converted to a magic square for all $n\ge 3$.

Finally, we will consider another sequence that can be derived like the above. It is the simple sum of pyramidal and tetrahedral numbers without shifting the latter by 2 terms as we did above to get $S_n$. Thus, this new sequence is:

$I_n = \dfrac{n(n+1)(2n+1)}{6} +\dfrac{n(n+1)(n+2)}{6} =\dfrac{n(n+1)^2}{2}$

$I_n:$ 2, 9, 24, 50, 90, 147, 224, 324, 450, 605, 792, 1014, 1274, 1575, 1920, 2312, 2754, 3249, 3800, 4410…

We observe that this sequence defines the sum of the integers in the interstices between triangular numbers (Figure 6). Further, it also has a geometric interpretation in the form of the area of the triangular number trapezium (Figure 6). Successive, triangular number trapezia are defined by the following 4 points: $(0,T_n); (T_n, T_{n+1}); (T_{n+1},T_{n+2}); (T_n+1, 0)$. These trapezia always have an integral area equal to $I_n$ starting from 9.

Figure 6. Triangular number interstitial sums and integer area trapezia.

The tale of the dolmen-dweller

The exams were over and the last semester of toil was all that stood between Vidrum and the dim glimmer he saw at the end of his metaphorical tunnel. He finally had some free time that he wished to savor in full. He had been so busy with his studies that he was still unclear as to how to spend that pleasant autumn morning. Just then he got a message from his friends Sharad and Murund that they would like to drop by. Seeing the opening for some activity he asked them to come over quickly and get some breakfast along for him. As they filled themselves with the viands at Vidrum’s place Sharad and Murund took off on politics. Ere long Sharad had launched into an interminable exposition of the electoral politics of the day: “The Tiger Party has won the elections for the municipal corporation in the South Visphotaka constituency. Vidbandhan Singh of the Kangress-Secular Party has won the elections for the Zilla Parishad at Shengaon. Lakkiraju of the Cycle Party has beaten the 5 times incumbent Potturaju for Sarpanch at Sarvepalli.” Thus, he went on and on. Murund: “The education policy of the Hindu-dal sucks. I don’t know why you still support it Vidrum. I am sure it is under the influence of Somakhya and Sharvamanyu.” Sharad: “Indeed, I believe it will be a major factor in the Mahanagarapalika elections in Turushkarajanagara where the Moslems might form an alliance with 4 other backward caste jāti-s because of the introduction of the 50-50 marks policy for the local language and national language.”

Vidrum was trying hard to be polite but the blow-by-blow analysis being presented by Sharad with interjections from Murund was getting too much for him to bear. Just then Sharad was about to launch into his speculations regarding the upcoming speech of Ram Mandir Mishra on “Hinduism as Secularism”. As though to show the sign of participation, Vidrum knowledgeably asked: “Is that election for the Mahanagara Nigama at Surat going to be indicative of the success of the Hindu-dal? in the national elections?” Sharad’s jaw dropped at his host’s ignorance and he went quiet for a few seconds trying to recover. Taking advantage of the silence Vidrum remarked: “Friends, I need to go to college to retrieve something I left behind.” Perhaps, it hit Sharad that his grand lecture was a waste or perhaps he felt a bit let down by Vidrum’s disinterest; whichever was the case, he and Murund decided to leave along with Vidrum and go their own way. In reality, Vidrum had nothing to retrieve — it was just a ruse to end the incessant patter of Sharad’s monologues. In any case, Vidrum was a bit of a changed man these days and he thought he could just do some calming reading in the library, catch some lunch and plan the rest of the day and beyond. There, he came across an article on dolmens and cists to the South West that was located among some reasonable rock-climbing spots. He thought that it might be a great idea to rope in some of his friends to explore that site.

At the library he sighted Vrishchika and went up to her: “Vrishchika, thanks for all those cheat-sheets you generated for pharmacology and biochemistry — it really clinched the day for me”. Vri: “Well, is good to hear that. Unlike you, I still have a couple of stupid exams to finish before I’m a free bird.” Vid: “Why would you need to study?”. Vri: “Well I could certainly pass them without studying but if you are in this business of beating others it takes more effort, just as the guys say you need to keep up the practice if you have to be in fighting shape.” Vid: How is your sister Lootika? is she done with her exams?” Vri: “Yes, if you wait for half an hour, you will see her swing by — today we are riding back with our father.” Vid: “What happened to her aśva?”. Vri: “You wouldn’t believe it. We were attacked two days ago by those loutish gangster boys Samikaran et al. They broke the spokes of her aśva; so, it is under repair now.” Vid: “That is awful of them. What happened?” Vri: “When we were returning home they whistled at us and displayed some lewd gestures. Lootika asked me to ignore them but I lost my cool and lunged at them with my bike and hit one. Then one of them drew a knife then, and Lootika tried to hit him with her bike when Samikaran drew a stick and skewered her spokes. She fell off her aśva but both of us recovered our bikes, sprayed the camphor-mangrove juice mixture we have developed on their faces, and were able to make away swiftly.” Vid: “Wow — well done — that is quite an improvement on your part!” Vri: “It was a bit of a risk but I think quite a few of the public saw what happened and they might be a bit wary now on. We will be banking on your help if we need it.” Vid: “Not exactly in a mood for gangster fighting but don’t worry — if required my billhook will be there to help you.”

A little while later Lootika came by to take her sister along. Vidrum inquired regarding her bike and mentioned to her his plan to explore the place with dolmens and cists to the southwest: “Do try to come along tomorrow. I’m now heading to tell Somakhya and some others who are likely to be interested.” L: “Sounds exciting, see you tomorrow at the railway station.”

The next day Vidrum with a band of 7 of his friends were exploring the strange landscape near Siddhakoṭa, which they had reached after a 1.5 hr train ride and an hour’s walk thereafter. Many interesting structures caught the eye of Somakhya and Lootika though it did not interest the rest too much. However, as was usual, Vidrum and Sharvamanyu hung around with them listening to the comments they might make. They soon found a series of menhirs that seemed to trace out a winding path along a slope leading to a shelf in the basaltic rock. Somakhya: “This is a sign of megalithic settlement in the area. From the irregular shapes of the menhirs, the site seems to be from an earlier megalithic phase predating the Aryan contact and the Dravidian expansion.” Lootika sighted a strange painting on the wall with circles and lemniscate-like figures: “Hey, this might be a sign of an even earlier settlement”. Somakhya then shouted out to the rest as he found a chalcedony microlithic core: “Indeed, the art on the rock might have been as early as the Mesolithic”. Then Sharvamanyu went to join the rest as someone called him and Vidrum took a little detour with Somakhya and Lootika to climb some cliff faces. Vidrum made an improbable climb and reached a ledge with an overhang. He shouted out to his companions that he had found an inscription with more rock art. Neither Somakhya nor Lootika could go up the way Vidrum had done but after a while, they found another easier path up and joined Vidrum on the ledge. They remarked that they could not read the script but it seemed to be of early Cālukyan provenance by its form. They then saw the rock art Vidrum had found — it depicted an elaborate battle scene with elephants, horses and headhunters — clearly of an age far removed from the Mesolithic art they had seen earlier. Somakhya wondered if given the inscription and the location it marked a record of a historic battle fought between the Cālukya-s and the Pallava-s.

The other clump of the remaining five friends headed up a tumulus adjacent to the rock faces the three were exploring. There, Sharvamanyu found a strange rock with cupules. He remembered Somakhya and Lootika showing him such a rock in the past that made musical notes when struck. He tried the same with this rock and it gave out a sonorous jangle. Soon the rest of them were striking a rock trying to make music with the rock. Bhagyada, one of the five, suddenly said that she heard Lootika call her and ran towards a dolmen that lay just beyond the singing rock. The music they were making reached the ears of Vidrum and his two companions; Vidrum: “I presume they have found a singing rock like the one you had shown us near Vināyakakoṭa.” Suddenly, the air was pierced by a shrill cry of horror and pain and everything went silent. Lootika: “Friends let us get back to the rest. That is the yell of my friend Bhagya. I fear something terrible has happened to her.” It took them some time to join the rest because Lootika in her disquiet for her friend almost slipped and fell while getting down from the ledge. When they reached the rest of their companions, they saw them clustered around Bhagyada and fanning her. Vidrum stepped forward and checked her pulse and sprinkled water on her from a water bottle. As he was attending to her, Sharvamanyu remarked: “Lootika, this is strange. She said she had heard you call her and went under that dolmen. We then heard her utter a cry and found her collapsed. When we tried to get her out of the dolmen, I swear to you, even though the roof is high enough for us to get beneath it we felt as though someone had given us a hard knock on the head. It is still aching a lot.” Lootika: “That is strange indeed! We were a bit of a distance away and I did not call anyone — we heard your music and her cry but, as you saw, it took us some time to reach you all.”

In the meantime, Bhagyada had woken up but was uttering something that sounded like gibberish to most. Somakhya: “That is very strange. It is pretty linguistic though not understandable — it sounds like some type of an unknown Dravidian language.” Seeing Lootika clutching her friend and trying to calm her down Somakhya took out a powder made of theanine, a cactus and Brāhmī handed it to Lootika: “Gautamī, make her a tea from this.” Some time after taking that tea she gradually stopped uttering the copious gibberish and seemed to slip into a dazed trance. Vidrum came up to Lootika who was still holding Bhagyada: “Lootika this seems to be something in your realm — I don’t know of her ever having any such problem in all these years.” Suddenly, something clicked in Lootika’s mind and she sprang up and ran under dolmen. She instantly reco