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\kappaB 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.

Wuhan_susceptibilityFigure 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.

Neanderthal_locusFigure 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.

Further reading:
https://jamanetwork.com/journals/jama/fullarticle/2770682
https://www.medrxiv.org/content/10.1101/2020.10.06.20205864v1.full.pdf
https://www.nature.com/articles/s41586-020-2818-3

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

Posted in Scientific ramblings | Tagged , , , , , ,

Counting pyramids, squares and magic squares

pyramidal_numbers

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

pyrFig2Figure 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:

saika-sa-gaccha-padānām kramāt-tri-saṃvargitasya ṣaṣṭho .aṃśaḥ |
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.

pyramidal_numbers_nIlakaNTha

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?

pyrFig4Figure 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.

pyrFig5

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.

pyrFig6A

pyrFig6BFigure 6. Triangular number interstitial sums and integer area trapezia.


Footnote 1: https://manasataramgini.wordpress.com/2017/10/23/triangles-hexes-and-cubes/
Footnote 2: https://manasataramgini.wordpress.com/2017/05/14/the-magic-of-the-deva-ogdoad/

Posted in Heathen thought, Life, Scientific ramblings | Tagged , , , , , , , , ,

The tale of the dolmen-dweller

bhagya3

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. Another 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 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 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, it 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.

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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 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 recoiled holding her chest, stumbled out of it and ran back to Somakhya’s side: “Somakhya, its occupant seems pretty aggressive and I’d say malevolent — see how he tricked Bhagya into his lair by mimicking me. We need to subdue this guy forcefully!” Vidrum: “Will Bhagya continue to be in this trance-like state? What do we do now, we need to get back too!” Somakhya: “Give us half an hour; hopefully we can restore her to normalcy.” Drawing Lootika aside, he pointed to a spot under a tamarind tree and whispered: “It is very opportune that we have a tamarind tree there. Go under it and call upon the West-facing Sadyojāta-rudra surrounded by the 24 yoginī-s: Śarabhānanā, Suvīra, Vajribhā, Rāśibhā, Cakravartī, Śauṇdī, Khadgakarṇā, Mahātapā, Cakravegā, Mahāyāmyā, Subhadrā, Gajakarṇikā, Carā, Somādevī, Gavākṣī, Vāyuvegagā, Airāvatī, Mahānāsā, Daṃṣṭrālī, Sukarkaśa, Vedhanī, Bhaṭṭā, Droṇā and Kākenakā. I will call upon the Skanda in the midst of the four Vināyaka-s under that kadamba tree. Then we shall go in.”

For half an hour, which seemed like a whole day, the rest of the group was impatiently and tensely milling around their incapacitated friend constantly remarking that they were taking a risk with Somakhya and Lootika’s hocus-pocus. They suggested that they should try to support her begin the gradual trudge back to the station for it would take a while with Bhagyada in such a condition. However, Vidrum and Sharvamanyu said they should give the two their chance and in any case giving her some time to recover might not be a bad idea as her vitals seemed alright beyond some elevation of her pulse rate. Having completed their dhyāna, Somakhya and Lootika entered the spacious dolmen. They felt some invisible barrier trying to keep them out but Lootika remarked: “I think the great yoginī-s have drawn him into our control” as they pushed through. Somakhya: “For a good mantravādin well-versed in the Yoginīsaṃcara, there is no bhūta that cannot be subdued with Rudra and the 24 devī-s emanating from the arṇa-s of the Tatpuruṣa-ṛk. But a lesser mortal might need other mechanisms like the śakti of Kumāra with the Vināyaka-s, or Khaḍgarāvaṇa or Caṇḍāsi.” Once inside, they saw a peculiar figure drawn on the ceiling — it almost seemed like a hybrid of old rock art on which was superimposed a more recent marking of letters. The two made an etching of it on paper and placed it in the circle of the 24 and performed bhūta-bandhana. Just as they were finishing, they heard the relieved shouts of their companions. As they emerged from the dolmen, they saw Bhagyada quite completely recovered. Bh: “I think I might have hit my head on the rock and lost my wits from trauma.” Heaving a collective sigh of relief, they decided to return from that place which had clearly frightened most of them. Lootika placed the etchings in her backpack and told Somakhya that she would make a fair copy and send it over to him.

bhagya1

As they hopped off the train at their home station, they shouted out goodbyes and reached for their local conveyances to get back home. Bhagyada gave Lootika a ride home on her motorized two-wheeler. She hung around with Lootika for a while yarning about the strange incident whose focus she was. As Lootika chatted with her friend, she made a clear color copy of the etching and handed it over to Bhagyada as she was parting: “Bhagya, you stay close to Somakhya’s house — just drop this with him as you head home. I’m going to be busy with some experiments next two days but we’ll go to the clothing-stores on Friday afternoon.” Bhagyada was flush with some cash from her internship. Hence she decided to call her friend Charusmita to accompany her to an eatery for dinner. Later that night, as Somakhya was writing an entry of the day’s adventures in his scrapbook, his mother came into his room. She seemed a bit agitated: “Do you know of the whereabouts of your classmate Bhagyada. Her mother is on the phone — she has apparently not returned home.” S: “Call Lootika’s mom — she was to drop off Lootika at her house.” Somakhya overheard his mother speaking to Lootika’s mother and learning that Bhagyada had left her home a long time ago and suggesting that they file a missing person report.

The next day Lootika was to help her mother in teaching the secret kula-prayoga of the Śrī-sūkta to her sisters Varoli and Jhilleeka. She was excited about it as she had obtained a picture from Somakhya’s mother of an archaeological site in the North that informed her of its unexpectedly ancient origins. However, Lootika found her mother was rather restless with the case of her friend’s disappearance. Just then Vrishchika came out of her study and gave them the news of what had happened. As Bhagyada had said goodbye to her friend after dinner and was headed to the parking lot she was accosted by Samikaran and his friend Mohammad Omar. They asked her to join them for some recreational substance inhalation. She refused but they kept engaging her and preventing her from leaving the spot. After some time it got more threatening and an aggressive encounter ensued. She tried to reach her vehicle and flee but her accosters pulled out its carburetor and threw it away into a gutter. She then tried to make a call when they seized her phone and backpack. Just then a jeep of the cops passed by and they fled with her belongings after pushing her into the drain. Eventually, the cops brought her home but told her that any investigation of Mohammad Omar without stronger evidence would not be easy as he was connected with the much-feared boss of the Majlis Party and that Samikaran was a respectable medical intern. L: “Oh dear! That’s awful. But Vrishchika you are such a lokasaṃcāriṇi. How did you get to know this — she is my classmate after all.” Vrishchika smirked at her sister and said: “They had taken her to the college hospital for a checkup; thus, I learnt from my sources.” L: “Let me go and check on her.” Her mother was a bit alarmed: “Lootika, why don’t you do that later. Let us finish the lesson now. I don’t want you or Vrishchika picking up any more scraps with those rowdies. This or worse could have happened to both of you’ll a few days back. These are dangerous guys and nobody can save you if they kidnap you all.”

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Several years had passed since those incidents. Dr. Samikaran was preparing for a beef party at his house along with Rabri Chatterjee of the Maoist Party. The big man in attendance was going to be the rising star, Mohammad Omar. He had just been appointed as the country head of the multinational, Social Platforms, by its mleccha bosses Joe Dremel and Joanna Ting. In the honor of his guest, Samikaran had ensured that there was going to no alcohol, and the beef was halal; nevertheless, to sweeten the deal for all the party-goers, different recreational substances were being provided. Soon Mohammad Omar was in attendance and he made a speech about how his company was going to make a big donation for Samikaran’s Bamman Haṭāo organization for its drive for medical assistance to the hill-tribes of the Nahali territory. He received a long ovation and many a young woman crept up beside him seeking his attention. He occasionally complimented one or the other of them for their progressive ideals and asked them to drop him a call and send along their CVs. Some minor players from the movie industry would also mill around him begging for a special account on Social Platforms. Thus, they partied hard till little after midnight, when Samikaran and Mohammad Omar rounded up some of the women whom they had picked and asked them to continue partying in a smaller room to the side. They said they would join them a little later as they had to briefly attend to some official issues.

Locking up the women in that room Chatterjee, Samikaran and Mohammad Omar retired to a secret chamber. There MO guffawed loudly and stroking his ample beard remarked that the party was great and that Samikaran had done a good job with the guest list. He then said that now that he was in control he had a mechanism to ban all Hindutva accounts on Social Platforms. He also said he had legion of “players” who would set up fake Hindutva accounts and make them a “heck of a laughing stock very soon, all the way to the prime minister.” He said he had the full secret backing of justice Shashi Yabhak and his network of contacts if it ever got legal. Samikaran smiled in acknowledgment even as Chatterjee asked him about the planned riots to disrupt the speech of the Finance Minister Danesh Gupta. MO declared that it would be a riot like none before and the government will be embarrassed beyond words before an international audience. He then excused himself with a wink saying he “wanted to spend some time with the ladies”. Samikaran said that was what the party was for and asked him to have fun. However, Mohammad Omar was soon rudely interrupted in his fun as Chatterjee rushed in and told him that Samikaran wanted him back as there was something very serious happening.

MO: “Why the #*%! did you want me, Sami?” Smk: “Listen, this is super-serious stuff. My spies have just informed me that the Special Task Force has been activated to arrest you for the killing of the intelligence officer during the March riots and are headed this way.” MO: “How the hell could they know?” Smk: “See, I’ve told you to be careful with the girls. I believe it is one of them who was a mole.” MO: “I’ll burn her alive.” Smk: “We will do that later; now you need to run. I suggest you quickly get on to the mofussil road leading to Amirpur and make your way to the bunker I’ve installed under Sultanganj Mohalla with the help of our Chinese comrades. Our agents can then sneak you out of the country.” MO: “Bro, why panic so much. My lawyers and his honor’s network will get me out in hours. I can then use that as good propaganda for our cause.” Chatterjee: “This is serious. They know that you have done that before. This time around they might either “encounter” you or have some thug bump you off as soon as you’re in jail. So, listen and run.” MO: “What about you guys?” Smk: “Nobody can do much to a respectable MD whom even politicians consult.”

Suddenly, it dawned on Mohammad Omar that the noose might indeed be tightening around him. He quickly jumped into his truck and started driving away as Samikaran had suggested.

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Lootika suddenly sat up on the bed, as though startled by a dream. Somakhya was half-awakened by it but instinctively pulled wife back on to the pillow and lapsed in slumber again. That morning at breakfast, S: “Gautami, did something bother you in your sleep.” L: “Why? I don’t know. Now that you ask, I think I woke up from a tense dream and then had a really good sleep for some reason. Ah! Now I recall. For some reason, that strange rock art we had seen in the dolmen at Siddhakoṭa flashed vividly in my sleep.” S: “I’d almost forgotten that. Did you ever save the etching we made? That was supposed to be a khārkhoḍa. I wonder what is going to happen with a khārkhoḍa binding an aggressive fellow like that one floating around.” L: “Let me think… I remember now. I did send the khārkhoḍa over to you with Bhagya but that was the day she was assailed by the louts and I believe it was among the items stolen from her and never recovered.”

A little later Vidrum was hosting Bhagyada and her husband Sandeep at his residence. As they were rambling about the old days, the topic veered to the Siddhakoṭa adventure and its aftermath. Bhagyada was still rather shaken by its mention. Vidrum: “I think in the least you might feel some comfort from this news item I saw a little while back.” Let me read it: “ June 27th. The Special Task Force had identified the head of Social Platforms, India, as the mastermind of the recent riots in various cities. It was alleged that he had created a network of university students, doctoral scholars and street ruffians to orchestrate these riots in various cities and that he was directly involved in arranging the assassination of the intelligence officer Mr. Vir Singh. It was also alleged that he had used his power to ban various pro-Hindutva accounts on Social Platforms and spreading false rumors about them and the government. As the Special Task Force headed to arrest him, he is said to have driven off on his truck via the mofussil roads. The observers say that for some unknown reason he suddenly veered his truck on to the Siddhakoṭa road and speed straight to the archaeological site in the hills. He abruptly abandoned his truck and ran up the hills to a dolmen where he was found dead with his skull smashed in. In his truck, several goods were recovered, which the STF claimed to have been stolen over the years from female victims whom he had lured. His case was heard yesterday and Justice Shashi Yabhak dismissed the charges against the deceased as frivolous and his death was ruled a suicide. Social Platforms CEO Joe Dremel condoled the death of his employee and declared him as a greater fighter against the dark forces of fascism, Hindutva and brahminist supremacy that were shredding the democratic fabric of India.” Vidrum: “Bhagya, I wonder if your stolen items were among those recovered in the truck.”

Posted in art, Heathen thought, Life | Tagged , , , , , , , ,

Ruminations on meteorites, organics and water

In our times the Christian Anglo-Saxons were famous for their “war on drugs”. However, in the 1800s, when they lorded over India, they were famous as global drug dealers. On the morning of August 25, 1865 CE around 9:00 AM, one such dealer, Mr. Peppe (titled the sub-deputy opium agent), was making his rounds overseeing his Indian serfs laboring in the poppy fields at Sherghati in Bihar. The still air on that cloudy day was pierced by an earth-shattering detonation — a stone had fallen from the heavens. The laborers reaching the place in the fields where it had fallen recovered a 5 kilogram heavenly stone buried knee-deep in the mud. The stone was conveyed to a senior drug dealer, Costley (titled Deputy Magistrate of Sherghati). After examining it, he did not believe it was a meteorite because it did not resemble that which he had known to have fallen from the skies in Faridpur, in 1850 CE. However, Peppe confirmed that he has seen it fall in the poppy fields; thus, it survived being discarded. Eventually, it landed in the hands of the English tyrants of Bengal who promptly conveyed it to the British Museum in London, where the meteor-collector Mervyn Herbert Nevil Story Maskelyne was greedily gobbling up the meteorites that his agents from India would supply. It so happened that this 5kg stone was a little piece of Martian real estate, which became the founding member of a class of Martian meteorites known as the shergottites.

Having read this tale in our early youth, alongside our first sightings of meteors as though cast from the mouth of the Kṣetrapāla, we became increasingly interested in meteoroids and meteorites. Soon we learnt that these objects offer windows into the solid material of the solar system and the very origin of planets such as earth. Meteoroids have diverse origins. Some, like the progenitors of the shergottites, are pieces of other planetary bodies like Mars, the asteroid Vesta and the Moon. Yet others are the dust of comets or fragments of cometary nuclei. These are behind the meteor showers which occur when the earth crosses a cometary orbit and cometary dust burns up in the atmosphere from friction, resulting in visible meteors. However, the most common are material left from the collisional and accretionary process that formed the rocky planets. Such collisions continue to occur in the asteroid belt generating meteoroids. Some such meteoroids survive the atmospheric burnout and drop to the surface of the earth. Every year a flux of at least 10^7 kg of meteorites reaches the earth and the bigger pieces are much sought after by collectors to this date. In the past 2 centuries, it has become clear that there is much diversity among meteorites. One way of classifying them (by no means exhaustive) is shown Figure 1.

meteoritesFigure 1

The iron meteorites are largely inorganic, \ge 90\% metals like iron and nickel. We read in the memoirs of the Mogol tyrant Jahangir:

“One of the strangest things that happened during this period occurred on the thirtieth of Farvardin of the present year [April 9, 1621] in a village in the pargana of Jalandhar. At dawn a tremendous noise arose in the east. It was so terrifying that it nearly frightened the inhabitants out of their skins. Then, in the midst of the tumultuous noise, something bright fell to the earth from above. The people thought fire was falling from heaven. A moment later the noise ceased, and the people regained their composure. A swift messenger was sent to Muhammad Sa’id the tax collector to inform him of the event. He got on his horse at once and went to the site to see for himself. For a distance of ten or twelve ells [ \approx 11.5-14 m] in length and breadth the earth had been so scorched that no trace of greenery or plants remained and it was still hot. He ordered the earth dug up. The deeper they dug, the hotter it was. Finally they reached a spot where a piece of hot iron appeared. It was so hot it was as though it had been taken out of a furnace. After a while it cooled off, and Muhammad Sa’id took it home with him. He placed it in a purse, sealed it, and sent it to court. I ordered it weighed in my presence. It weighed 160 tolas [ \approx 1.866209 kg]. I ordered Master Daud to make a sword, dagger, and knife of it and show them to me.” (translation by Wheeler M. Thackston)

This is a classic example of an iron meteorite, which was probably the first source of iron used by humans sometime before the regular iron age. In contrast, the stony iron meteorites feature different kinds of mixtures of metals and minerals. The stony meteorites are usually divided into chondrites which have “chondrules” and achondrites which lack them. Chondrules are crystalline material derived from molten silicate droplets. The shergottites and related Martian meteorites are typically classified as achondrites. The stony meteorites often contain organic and inorganic carbon. The presence of organic compounds was first noticed in the carbonaceous meteors by the famous early modern chemists Jöns Jacob Berzelius and Marcellin Berthelot. But the significance of these organic compounds came to the fore only after the dramatic event that took place on 28 September 1969 10:58 AM near a place called Murchison in Australia. A blazing bolide flew into earth breaking up initially into 3 pieces and vanishing with a cloud of smoke followed by an earthquake. One fragment smashed through a barn and fell on the hay without any deaths. A search of the location recovered numerous fragments totaling to over 100 kg. This came to be known as the famous Murchison meteorite. An analysis of its composition revealed that it had >10\% water and \approx 2.2\% carbon by weight, consistent with its smoky disintegration.

Since then the organics of the Murchison meteorite have been intensely studied and the following have been detected: 1) Over 70 distinct amino acids; 2) fatty acids; 3) purines; 4) pyrimidines; 5) A complex mixture of sugars; 6) alcohols; 7) aldehydes; 8) ketones; 9) amines; 10) amides; 11) aliphatic and aromatic hydrocarbons; 12) some heterocyclic aromatics; 13) ethers; 14) organo-sulfur and organo-phosphorus compounds. Some of these are at very low concentrations like amines while fatty acids are quite abundant. The fact that these compounds show a mixture of chiralities and a distinct Carbon 13 isotopic signature showed that they had a celestial rather than an earthly biological origin. Given that the Murchison meteorite represents material as old or considerably older than the Earth, it established beyond doubt that interstellar organic matter was a component of the material that formed the original rocky bodies of the solar system. It also suggests that upon accretion into asteroidal bodies, the originally simple interstellar organic molecules reacted abiotically by the action or water and heat to form the entire range of more complex molecules that could serve as building blocks of life. However, given that life is characterized by homochirality of its building blocks, it also suggested at least a subset of such bodies was not transmitting life itself but only its building blocks.

In our youth, when we first read of the Murchison meteorite, we believed the majority view that meteors such as this had delivered the building blocks of life to the early Earth and it was followed by local pre-biotic evolution resulting in life. However, as our understanding of biology improved we increasingly started moving towards panspermia though cellular life on Earth had a single common ancestor. First, the archaeo-bacterial split implied a certain barrier to lateral gene transfer that had since then broken down on Earth. This could be most easily explained by two seeding events, one which brought the bacteria and the other which brought the archaea. Second, consistent with the above, there are many protein divergences among housekeeping functions that imply divergence time that likely greater than the age of the Earth, even assuming early acceleration in virus-like primitive replicators. Third, genomic analysis strongly favors a heterotrophic ancestral organism. Abundant food sources that would have allowed such organisms to get started before autotrophy evolved are not likely to have existed on the early Earth. Being in the hot inner Solar System it is less likely to have had materials like tholins that would have fed the early heterotrophs. Hence, we again see hints that life emerged elsewhere in a tholin-rich region and secondarily reached Earth. Thus, we came to see the Murchison meteorite and other carbonaceous chondrites as merely part of a spectrum of stony material with interstellar organics and their processed products which went all the way to life. Thus our view was that some such body could have delivered life to Earth from outside.

The key to this, which still remains quite mysterious, is the emergence of homochirality. Laboratory organic syntheses attempting to mimic prebiotic processes do not easily reproduce the homochiral constraint typical of biosynthesis. In our adulthood, even as we were locked in other scientific explorations, new studies on the Murchison meteorite that cleverly avoided the effects of contamination and racemization indicated that there was a \approx 2-9\% excess of L-enantiomers among the amino acids found in it. This supported the hypothesis that there was an initial step wherein a limited anisotropy was established (as seen on the meteorite) followed by an amplification step with selection for one enantiomer. This brought the focus on certain earlier studies in organic chemistry which have shown that asymmetric photolysis by circularly polarized light could produce notable enantiomeric excess (e.g. the photolysis of camphor). Thus, early on, it had been proposed that pulsars emitting circularly polarized synchrotron radiation could have caused the initial enantiomeric anisotropy. But it has been pointed out that such radiation could break up amino acids. Others have hence suggested circularly polarized light scattered from dusty regions in the Milky Way could provide the appropriate light for such reactions. Such light has been directly observed in the reflection nebulae of the OMC-1 star-forming cloud in the Orion region, which is rich in organic compounds. However, the energetics of this proposal remains to be understood because UV radiation could destroy the amino acids. In any case, the photochemical enantiomer selection remains the most likely possibility, and those conditions are not found on Earth. This suggests that enantiomeric excess happened in space. Now one could still argue that after this excess was established such an enantiomerically biased mixture was delivered to Earth by meteorites and that provided the building blocks for life on Earth. However, this does not take into account the racemization and re-equilibration of enantiomers on Earth post-landing. Hence, we see this as additional support to the idea that life formed close to the site where the enantiomeric excess was already established and maintained and then seeded on Earth.

An unintended consequence of the dramatic Murchison fall has been the relative neglect of other types of meteorites. On 26 April 1895 CE multiple detonations were heard over what is today Uttar Pradesh and four stones fell from the sky. Two of the pieces were found at Bishunpur (942 g) and Parjabatpur (97 g), 1.8 km apart. The insatiable appetite of the British mineralogists for meteorites resulted in them being promptly shipped to the British Museum in London to become one of the famous representative specimens of the ordinary class of chondrites. It has been subject to several studies and was reliably shown to contain at least two organics, toluene and dimethyl ethyl naphthalene. Indeed, the ordinary chondrites contain a considerable amount of organic compounds (probably more than Murchison) but their composition and concentrations remain poorly explored, indicating that Murchison-type carbonaceous chondrites cannot be considered the sole candidates for vehicles of organic compounds. Further, the shergottites have been reliably shown to contain aromatic and alkylaromatic hydrocarbons, phenol and benzonitrile. The famous Allan Hills 84001, which was found in Antarctica, also belonging to the Martian class, was initially claimed to contain fossils of bacteria but this claim should be seen as plainly dubious as those structures can also form through inorganic processes. Nevertheless, this meteorite contains polycyclic aromatic hydrocarbons. Given that these compounds are fairly common on rocky asteroid material and comets, it is unlikely that they have special significance for the formation of the building blocks of life. However, a better study of these might still give us clues regarding the possible baselines for organics forming on various rocky planets of the Solar System.

Finally, we come to the question if any of these asteroids resemble the Earth in their composition? The simple answer is no. However, over the years several workers have been invoking the parent-bodies of enstatites, which are magnesium silicate-containing chondrites, as possible candidates for the progenitors of the Earth. While the Earth as it stands is not identical to the composition of the enstatites they do have similar isotopic signatures to terrestrial rocks making them a likely contributor to the origin of the Earth. Astrochemists have invoked their high ^{15}N content to propose that the high carbon content in the enstatite chondrites was derived from an organic precursor. However, having undergone thermal metamorphosis in the inner Solar System it has been mostly converted to a graphite-like material. Recently, enstatites have been in the news because a recent study has shown that they contain sufficient Hydrogen with a specific isotopic signature to have contributed to the emergence of a major fraction of the water on the Earth. This strengthens a big role for the enstatite bodies in the origin of the Earth and its water. Nevertheless, the ocean compositions are not exactly of the isotopic signature one would expect from a purely enstatite origin. The authors of the said recent study admit this fact and propose that carbonaceous chondrites of the Ivuna-type could have supplied additional water after the initial formation of the Earth to result in its current isotopic signature.

What are the implications of this for the origin of life? One possibility is that the secondary delivery of water by carbonaceous chondrites after the initial formation of the earth from enstatite-like material could have also been the vehicle for the seedings of life. The presence of preexisting water from the enstatite building blocks could have provided for already congenial conditions for the seeded life to take root and expand. Of course, an alternative possibility exists. Most proposals do not see the Earth as arising purely from enstatite chondrites. Additionally, the original mix is likely to have had some kind of carbonaceous chondrites. They too could have seeded life. Further, we cannot rule out the role of other rarer bodies involved in the early collisions.

Posted in History, Scientific ramblings | Tagged , , , , , , , , , ,

Winners and Losers

Somakhya and Lootika were visiting the Śūlapuruṣadeśa for work reasons. Unlike their ārya ancestors, they did not like being on the move much. It was a rare occasion that both had been able to travel together and it brought them some welcome relief as they did not have to individually worry about not getting up on time, taking care of their luggage, or be over-vigilant about missing some travel sign in an alien land. Sightseeing made no sense to either without the other as a companion — thus, their largely solitary peregrinations to date had mostly put them past the urge of motivated sight-seeing. If at all they reminisced about such things, their thoughts often went back to the rare memorable occasions when they had been able to travel together to take in the history of the locus and comment on it.

It was a brisk morning in early autumn, still not so cold as to prevent a brief meander before they caught a train to find their way to the flight back home. Lootika suddenly stopped before a clump of lilac trees. She exclaimed and pulled out her phone right away to photograph. S: “What’s the matter ūrṇāyī ?” Pointing to the light violet flowers on the lilac trees she remarked: “Check those flowers out”. S: “Strange ain’t it for these to be blooming on an autumn day like this?” L: “ārya, I’d say that is ominous.” S: “Why?”

L: “Once in our youth my sisters and I had plied the bhūtacakra (planchette) and the pointer was seized by the bhūta of a Hindu soldier from the Marahaṭṭa country who had died from cold in Germany during WW-2. His death narrative had eerie similarities to the story of a Danish author titled ‘The Little Match Girl’. But he made an interesting remark. He said that after this very place had been fire-bombed with white phosphorus by the English marauder Bomber Harris, an agent of the monstrous Churchill, and his merry bombers the lilac trees had mysteriously burst into bloom in autumn. Our Marahaṭṭī was imprisoned nearby and pressed by his German captors to participate in the clearing of the rubble earlier that year.”

As they wandered on a bit along the beautiful street Somakhya remarked: “I’ve not read that tale ‘The Little Match Girl’ but everything seems rather ironic here and I get a vague sense of why you termed it as ominous. Ironically, the Europeans came to know of Phosphorus in this very place due to a śūlapuruṣa alchemist Brandt — who obtained it from nṛmūtra. It is possible he was inspired by the transmission by marūnmatta-s of a similar discovery of the element by rasasiddha-s like Nāgārjuna who in their tantra-s mention a similar substance with a suvarṇa-prabha, i.e. phosphorescence, being extracted from the same source. That very rasa came to destroy the city in which it was isolated!”

L: “Interesting indeed. The tangled connections get ghostly for me because the ‘match’ in the title of the story I mentioned refers to matchstick which again was made from P.” S: “What is the story? Paraphrase it for me if you feel so inclined.” L: “Let us start walking towards the railway station and I’ll give you the gist of it. In short, it is the tale of a little European girl who was sent out to the streets to sell match sticks. While she was doing so on the dark evening of the last day of the year, she lost the big slippers of her mother she was wearing as they slipped off her feet while evading traffic on the street. With snow, she started getting hypothermic and was afraid to return home as she might be abused for not having sold a single match that day. Reaching the end of her endurance from the cold, she sat down near a house and lit a match to warm herself — then another and so on. Each time she lit a match she had a phantasmagoria before finally perishing from the cold at the climax of her visions. The author concluded by stating that when her corpse was found the next day the people had no clue of the pleasant phantasmagoria she had witnessed before dying. Verily, my sister Vrishchika would paradoxically remark ‘for some the coming of Vivasvān’s black son is the most pleasant climax of their existence’. Now, that Hindu soldier who died in somewhere in these regions also had run of phantasmagoria before meeting his end which he narrated to us — it certainly sent a chill through me before I performed a śamanam to set him at peace and it positively shook my sisters to the core for several days.”

S: “Ah, the tale reminds me of one we had to study for an apabhraṃśa examination titled diyāsalāī kī kahānī. I wonder if the writer was inspired in some way by that of the Dane. In any case, you have to tell me of the Marahaṭṭa’s phantasmagoria.”

By then they had reached a souvenir shop near the railway station. There, they were greeted by a Japanese man whom they had made acquaintance of at the meeting for which they were there. He too was taking the same train as them; so, they did decide to go together. He asked Somakhya and Lootika to join him in checking out the souvenirs as he wanted to buy something for his kids. As they were doing so they were joined by a mahāmleccha who too they had made acquaintance of at the past meeting. He too went to the shop to purchase some porcine meat and a bottle of an alcoholic beverage. He was headed towards Trivargadeśa and somewhat nervously remarked: “I hope my train arrives on time.” The Japanese man responded: “While not as punctual as in Japan the trains here usually keep good time.” The mahāmleccha responded with a smirk: “Thanks, I know you guys have completely outdone your erstwhile allies even as we dismantled our own railroads as though we taking the Nazi supply chains to the scrapyard.” The Japanese: “We were not really allies and for our population density a good rail system is very important unlike in your part of the world.” The mahāmleccha: “Sorry, I was just joking.” With that, they all headed into the station and the mahāmleccha soon caught his train to head for his destination.

Soon the two with their ati-prācya fellow traveler found themselves on speeding away on the train from their kṣetra of the past week. Perhaps due to the phlegmatic disposition of their fellow traveler, they remained quiet for some time before he suddenly pointed to a decrepit monument that sped past them. With dark clouds hanging about it, it presented a melancholy specter. The J: “That is a ruin from the fire-bombing during WW-2. Even Tokyo was similarly reduced to ashes. It was worse there because we used to build almost everything from wood and paper. Here at least this solid stone structure is standing.” S and L: “We were just talking about something related before we saw you. Bad as this was, the burning down of Japan was evidently worse.” The J: “The American Demon LeMay who was the mastermind of the bombing of Japan first learnt his tricks here. But for him the Japanese were merely cockroaches, so even if there might have been some restraint here none of it was there when the demon came east.” S: “He was indeed a psychopathic war-criminal but what matters is victory. If one wins then even your psychopaths and mass-murders will be hailed as national heroes. That was so or demon LeMay or the other mass-murderer who likely inspired him, bomber Harris.” The J: “That is true. The biggest humiliation for us when the government was obliged to confer on the demon the highest Japanese honor for a foreigner! People have still not forgotten that.”

Then, they lapsed into a silence punctuated by an occasional conversation on more placid topics like the little rice-growing town their interlocutor was born in with schools that hardly had 10 students or the rare festival they still celebrated to the lord of the universe, Mahendra. Thus, they passed the rest of their journey to the airport to catch their flights. After the usual annoying grind of long ques, probing harassment by security and a cab-ride lasting a few hours, Somakhya and Lootika finally reached the house of the former’s parents where they had left their kids. After they had settled in, Somakhya’s mother remarked to Lootika: “Your sister Varoli was very insistent that you all go to her place for dinner but I asked her to join us with her family; that way it might be more fun.” L: “That’s great, it resolves everything for me.” As was typical among the caturbhaginī, when Varoli arrived with her husband Mitrayu and her kid, Lootika gave her a detailed account of their visit to the land of the śūla-jana-s. Concluding with the lilac flowers, reminiscences of the phantom of the Marahaṭṭi soldier and their encounter with the atiprācya, Lootika remarked: “I forgot to ask the atiprācya about why he said the Japanese and the Nazis were not true allies. He seemed sincere about it though the mahāmleccha took it as a sign of being coy of the casual association he had made between the Japanese and the Reich. Somakhya, I’ve heard you say that it was more an alliance of circumstance.”

S: “Yes, he did seem sincere and is perhaps one of the better informed of his people. From the perspective of the outsider, the Germans of the age adopted an essentially Galtonian framework of dealing with the other, which accorded a certain hierarchy to races. They felt they as Germanic people (the English included) were at the top of it, with the Rus and the Slav below them and the melanistic peoples of the world in the lower rungs. As for the East Asians, the cīna-s and the uṣāputra-s they were ambivalent. But for the most part the Germans, Hitler included, saw them with a degree of respect, unlike what mahāmleccha commentators say. This was especially so given that the Japanese had defeated the Germans in China during WW-1 and before that the Rus whom the Germans had backed. Thus, they had ‘earned some respect’ of the mleccha-s by showing themselves capable of defeating them on their own. While in WW-1 the J had fought on the side of the English it was not due to any particular friendship with them. Their strategic objectives were to keep the mleccha-s out of their long-desired sphere of influence, i.e. mainland China, which was the target of the mighty J lords since Hideyoshi and his audacious attack on the Koreans and the Ming. Thus, their victory against the Germans in WW-1 allowed them to conquer German colonial and vassal territory in the east. Hence, after WW-1 when the J began their conquest of China the Germans naturally supported the cīna-s. It was only later they started supporting the J seeing the practical advantage of having a power like Japan to aid them in the East against the English and American might. Even then the śūlapuruṣa-s kept their new J allies in the dark about their dealings with the Soviet Rus. This of course made the J wary of the German intentions for they greatly feared the indomitable Soviet Rus. Finally, the alliance was sealed only because the Japanese seeing that their energy supply was being constricted by the Americans and English decided that the way out was to attack the Anglo-Saxon Christian powers. The J had been generally wary of mleccha-s operating in the East as they had learnt how destructive they could be from their actions on us and the cīna-s. So, indeed their alliance with the śūlapuruṣa-s may be seen as a convergence of interests due to the emergence of the English-German conflict and the English hope of continuing their world dominance by constricting the rising Japan.”

Varoli: “That might help make sense of something which always puzzled be: if the atiprācya-s and the śulapuruṣa-s were such close allies why did they not attack the Soviet Rus empire in conjunction with the Germans? That could have considerably weakened the chances of the ultimate Rus victory in the war. When I used to ask such things in our infamous college Right-Wing Debate club it would invariably evoke responses like the Germans did not trust the ‘yellow race’ or that the śūlapuruṣa-s were too proud to seek assistance from the Untermenschen of the East.”

Mitrayu: “As ever these types never got it that the situation on the ground was more complicated. As Somakhya noted the śūlapuruṣa-s had hardly been open with the atiprācya-s about their own pacts with the Soviets. If that were the case, then the J had every reason to be wary of entering that war from the East. Further, the J had much reason to suspect that there could ultimately be an inter-mleccha alliance against them, much as the well-known sarvonmatta-samāyoga happens against us — after all, they were a heathen nation with the emperor as the head of the Shinto religion. But the most neglected aspect in all this is the fact the Soviet Rus had an extraordinary capacity to bleed until they attained victory. Even the almost eusocial atiprācya-s realized that this capacity coupled with a brutal dictator as Dzhugashvili meant that they were unlikely to prevail. The Rus had already shown this in the Soviet-J showdown fought at the Mongol-Manchu border-post of Khalkhin Gol. There, the Rus and their Mongol allies led by the famous Zhukov smashed a powerful Japanese army just before the Stalin-Hitler pact was concluded. Through much of the battle, the Japanese and their Manchu allies showed tactical brilliance and fought resolutely to inflict heavy losses on the Rus side in terms of men and material. But the Rus remained unshaken by all those losses and kept fighting relentlessly till they were able to outflank and encircle the Japanese killing or imprisoning more than half their men. This and an earlier Manchurian encounter with the Soviet Rus had shown to the Japanese that it was not the best strategy to pursue the war with the former. With this and the śūlapuruṣa-s signing a pact with the śrava-s it was quite natural that the J did not join the Nazis in opening an eastern front against the Rus.”

L: “Indeed, whatever one may think of them, the tale of Rus in WW-2 has something awful and heroic about it. The Anglo-Saxon propaganda, bought by so many, has systematically tried to erase them from the picture, who were the true victors of that war at a staggering human cost. Their deaths at over 25 million dwarfs the costs incurred by the śūlapuruṣa-s, Jews and Poles of about a 6 million each and by the Hindus of about 2.5 million.”

M: “After all much as tyrant Akbar’s shaikh-s celebrated the deaths of the Kaffirs and fired randomly into their midst as Mana Siṃha engaged the valiant Pratāpa Siṃha, the Anglo-Saxons cheered on as the fascist-Soviet clash played out acknowledging that a death on either side was gain for them.”

V: “It was truly a crystallization of the statement from our national epic: varāhasya śunaś ca yudhyatos tayor abhāve śvapacasya lābhaḥ । (In the fight between the hog and the dog, the death of either is the gain of the dog-eating tribesman).”

M: “That indeed was the way the Anglo-Saxon mleccha-s played the game. They delayed opening a western front against their śūlapuruṣa cousins as long as they could, instead choosing only those engagements in the South that would keep English control of India and other vassals intact. Further, they finally opened that front only when they realized the Rus had beaten the śūlapuruṣa-s against all odds. Their claim of victory against the Japanese with the use of nuclear weapons was another such. After all, the Japanese had been taking heavy losses from the Phosphorus bombs as your fellow traveler mentioned for a while and were trying to negotiate a truce. Those had caused more harm than the āṇavāstra-s in toto. They finally did surrender because they had been shredded in the mainland and Sakhalin by the Soviet Rus, who were then poised to take Hokkaido and possibly execute their emperor. To cap it all the āṅglika-duṣṭa-mahāmleccha combine manufactured the tale that they were the good guys fighting evil. Taking a leaf straight out of the ādirākṣasagrantha, they made it appear that as long as they committed genocide it was not genocide at all, much as those sanctioned by the ekarākṣasa.”

S: “The Soviet Rus were undoubtedly the true victors of WW-2 but as we have often seen with the mleccha-s in our own scientific endeavors claiming our discoveries as theirs, the Franco-Anglo-Saxon entente positioned itself conveniently to suffer the least losses among the major belligerents to claim victory for themselves. Indeed, the āṇavāstra-s were hardly the cause of the victory in WW-2 but it was the Anglo-Saxon trump-card for the world that was to unfold. Its use against Japan in WW-2 can simply be attributed to the mleccha perception of other ethnicities as subhuman and the need to send a message to the winners, the Soviet Rus. When WW-2 caused the mantle of mlecchādhipatyam to finally pass from the āṅgalika-duṣṭa-s to the mahāmleccha-s, it was the āṇavāstra-s that made the limp mleccheśa Truman turgid as a Californian sea cucumber. As a mahāmleccha had told the Rus ambassador: ‘I am going to pull out an atomic bomb out of my hip pocket and let you have it’, even as a Texan would have settled a score by drawing a six-shooter from his hip keeping to their famous maxim: ‘shoot first and ask questions later.’ Indeed, the mahāmleccha-s repeatedly wanted to use the āṇavāstra thereafter, not just against the Rus but also on the Cīna-s and in Campāvati but stopped only because the other members in their own circle seemed to have been uneasy with that. Thus, the Soviet Rus realized that they had to concede what the mahāmleccha-s and the āṅgalika-duṣṭa-s demanded in the immediate aftermath of WW-2.”

V: “In the end, as on the Kuru field, winning and losing is often relative and in victory, and as you say with our scientific discoveries, a thief could turn up to steal it even as the mleccha-s stole the victory of the Soviet Rus. But the game of a thief is open for more than one and the Soviet Rus was the next to be play thief to get their own āṇavāstra-s. In defeat, Japan’s heroic performance allowed it to keep the emperor and the Shinto religion relatively intact. The śūlapuruṣa overreach resulted in their becoming a vassal state losing many of their lands to the surrounding states. We, for all the death taken on behalf of the mleccha-s, were dismembered and our millennial civilizational foe was handed the eastern and western wings of our lands. And as if to add insult to injury were saddled with an uncle and a father of a secular socialist nation who led us to a disastrous defeat the hands of the Duṣṭa-cīna-s. But then as Mitrayu had consoled me when we first met, sometimes life in the margins has its own charms. ”

L: “In the end, as the former mleccheśa had himself admitted, in their zeal to fight the Soviet Rus they had handed their own people to a Gestapo like police state who were kept from revolting with an abundance of fructose-laden corn syrup and soy paste sweetening mountain-high scoops of ice-cream and sacks of potato chips. As those bloated the waistlines of the mahāmleccha it laid the foundations of a disease far beyond anything their praṇidhi-s had ever imagined would emerge from their midst. Thus, we await the unfolding of the phantasmagoria the Hindu soldier saw before turning phantom. Unlike the pleasant passing of Danish girl, his culminated in the climax of a roga that seized the Bhārata-s even as a man when ranged against his svābhāvika-vairin-s is seized by a disease from within.”

Posted in History, Life, Politics | Tagged , , , , , , , , , , ,

An arithmetic experiment and an unsolved problem

We realized that a simple arithmetic experiment we had performed in our youth is actually related to an unsolved problem in number theory. It goes thus: consider the sequence of natural numbers n=1, 2, 3, 4 \cdots Then find the distance of n to nearest prime p that is 1) greater than or equal to n or 2) less than or equal to n. Thus, d_1[n]=p-n, d_2[n]=n-p. We then define the arithmetic function f_1[n]=d_1[n] d_2[n]. Since, 1 has no prime before it, we can either have f_1[1] as undefined or assign 0 to it. The corresponding sequence goes thus:

f_1 \rightarrow ?, 0, 0, 1, 0, 1, 0, 3, 4, 3, 0, 1, 0, 3, 4 \cdots

The function as a nice shape with symmetric maxima that remind one of reptilian teeth (Figure 1).

prime_Distance_1Figure 1.

Now, where do the successive local maxima of this function occur? If we leave out the undefined f_1[1] we see that these occur in a sequence which we call f_2:

f_2 \rightarrow 3, 5, 8, 11, 14, 17, 20, 25, 29, 33, 38 \cdots

One can right away intuitively conclude that this sequence captures the occurrences of primes in natural number space by defining some kind of central position between them. Hence, we can more explicitly ask, what is the relation of f_2 to the arithmetic and geometric means of successive primes behave? We find that the above sequence f_2= \lfloor \textrm{GM}(p_n, p_{n+1}) \rfloor and f_2 = \textrm{AM}(p_n, p_{n+1}) - 1 for primes starting with 3 onward. One can see that the local maxima of f_1, i.e. the values of f_1[f_2] (if we count leaving out the undefined first term in f_1), are all square numbers. These have a specific relationship to the prime difference function \textrm{pd}[n]= p_{n+1}-p_n starting from 3 (Figure 2). Given that from 3 onward every prime is odd, the corresponding \textrm{pd}[n] will be even. Then, we have the following relationship to the local maxima in f_1:

f_1[f_2] = \dfrac{\left(\textrm{pd}[n]\right)^2}{4}

prime_Distance_2Figure 2.

Thus, the local maxima of f_1 help define a certain parabolically rescaled version of the prime difference function, which, as we will see below, has utility in understanding aspects of the occurrence of successive primes.

We know that \textrm{AM} \ge \textrm{GM} and \textrm{GM}(p_n, p_{n+1}) will never be a whole number. Thus, we can define the arithmetic function \textrm{pmd}[n] = \textrm{AM}(p_n, p_{n+1}) - \textrm{GM}(p_n, p_{n+1}). One experimentally notes the asymptotic behavior that as n \to \infty, \; \textrm{pmd}[n] \to 0. However, this secular decay is marked local fluctuations. There are two notable features of this (Figure 3): 1) The maximum value of \textrm{pmd}[n] is 0.22503561260788 for n=4 which corresponds to the p_n=7, p_{n+1}=11. Thus, we can conjecture that the difference between the arithmetic and geometric means of successive primes is always less than one fourth, i.e. \textrm{pmd}[n] < \tfrac{1}{4}. 2) The fluctuations of \textrm{pmd}[n] starting from n=2 exactly mirror the fluctuations defined by the local maxima of f_1[n], i.e. f_1[f_2], with the magnitude of the f_1[f_2] peak tracking the magnitude of the peak in \textrm{pmd}[n]. The first time a peak of given magnitude appears in f_1[f_2] it has the largest corresponding effect in \textrm{pmd}[n] with all subsequent appearances of the peak of the same magnitude being increasingly muted.

prime_Distance_3Figure 3.

Thus, we can empirically determine that (Figure 4):

\dfrac{f_1[f_2]}{\textrm{pmd}[n]} \approx 2.3 n \log(n)

prime_Distance_4Figure 4. The green filled curve is \dfrac{f_1[f_2]}{\textrm{pmd}[n]} while the dark red curve is y=2.3 n \log(x)

Whether there is some closed form for the constant 2.3 remains an open question to us. After we posted this note, an acquaintance from Twitter provided a proof for why the constant in the above equation should be should be 2 for large n.

Prime gaps have been intensely studied since at least Legendre who had a conjecture regarding them; several tighter variants of that conjecture have been proposed repeatedly since then. Hence, looked up the literature to see if our conjecture regarding the difference of the arithmetic and geometric means of successive primes might be equivalent to any of those. We learnt recently that it is a version of a conjecture stated by Andrica, in 1986, just about a decade before when we began exploring the function f_1[n]. It goes thus:

\sqrt{p_{n+1}}-\sqrt{p_n} < 1

The form in which we present the conjecture appears to be a nice statement of a strong version of the Andrica conjecture and f_1[f_2] provides a cleaner comparison for the fluctuations in \textrm{pmd}[n] than the simple prime gap function. Remarkably, simple as these conjectures are, they have not been proven to date. Moreover, it seems that even if the Riemann hypothesis were to be true, that by itself will not imply these conjectures. Thus, yet again we have simple arithmetic statements that can be understood or arrived at even by a school kid but are extraordinarily difficult to prove formally. The philosophical implications of these are interesting to us.

Posted in Scientific ramblings | Tagged , , , , , , ,

Matters of religion-4

Pinakasena was visiting Somakhya and Lootika. He was seeking instruction on the Sadyojāta-mantra from his hosts: “O Ātharvaṇa and Śāradvatī, I wish to learn the rahasya-s pertaining to the deployment of the Sadyojāta-mantra to the great god Kumāra, the patron deity of all our clans. We have borne his worship since the time our ancestors were in the now fallen lands of Gandhāra in Uttarāpatha. Then they bore it to the Yaudheya republic and then to Kānyakubja during the reign of emperor of Harṣavardhana and finally to Dakṣiṇāpatha.”

His hosts led him to their fire room and seated him on the western side of their aupāsana-vedi. There, they asked him to purify himself for the ritual with the sprinkling using the mantra-s to divine horse Dadhikrāvan and the waters. Lootika then signaled to him: “Offer the pouring with with incantation to the daughters of Rudra”. Pinakasena did as instructed:

namo rudriyābhyo .adbhyaH svāhā ||

Obeisance to the daughters of Rudra. Hail to the waters.

Somakhya: “Now invoke Prajāpati the lord of the Bhṛgu-s and Aṅgiras-es and offer fire sticks to him”.

He did as instructed:

namaḥ parameṣṭhine prajāpataye svāhā hiraṇyagarbhāya svāhā bhṛgūṇāṃ patye svāhā .aṅgirasāṃ pataye svāhā ||

Then he offered tarpaṇa-s:

vasubhyo namo rudrebhyo nama ādityebhyo marudbhyo namo mārutebhyo namo .aśvibhyāṃ namo vaiśravaṇāya namo dharmāya namaḥ kāmāya namo ṛṣibhyo nama ārṣeyebhyo namo aṅgirobhyo nama āṅgirasebhyo namo atharvebhyo nama ātharvaṇebhyo namaḥ ||

Then he recited after his hosts and made oblations at the appropriate calls:

yaś chandasām ṛṣabho viśvarūpaś chandobhyaś chandāṃsy āviveśa | satāṃ śakyaḥ provācopaniṣad indro jyeṣṭa indriyāya ṛṣibhyo namo devebhyaḥ svadhā pitṛbhyo bhūr-bhuvaḥ-suvaś chanda oṃ ||

He [Indra], who is the bull among the meters, of all forms, emerging from the meters entered into the meters. The great one, Indra, who is possible to be [reached] by the good, proclaimed the upaniṣat to the Ṛṣi-s for attaining powers (full experience). Obeisance to the gods, for the ancestors the good station, the 3 vyāhṛtis, the Veda, OṂ.

Then stressing to Pinakasena that his instruction was a continuation of the tradition imparted by the god Indra to their ancient ancestors, Somakhya began his instruction after placing a fire stick for the repelling of the non-sacrificing foes:

|| ayajvanaḥ sākṣi viśvasmin bharo3m ||

Thereafter he instructed Pinakasena to offer a fire stick after he recited the incantation:

tvaṃ devānām asi rudra śreṣṭha

tavas tavasām ugrabāho |

hṛṇīyasā manasā modamāna

ā babhūvitha rudrasya sūno || + svāhā + idaṃ na mama |

Somakhya: “The ṛṣi of the Sadyojāta-mantra is Gopatha. Its meter is anuṣṭubh, though it has an additional bhakti that is non-metrical and its devatā is Kumāra Bhavodbhava. Alternatively depending on the prayoga there is a variant form where the devatā is Skanda-Viśākhau. The mantra itself goes thus:

sadyojātaṃ prapadyāmi sadyojātāya vai namaḥ |

bhave-bhave nādibhave bhajasva māṃ bhavodbhava ||

I take refuge in the one who had instantly arisen (Sadyojāta). Verily obeisance to him who has arisen instantly. In existence and after existence do not partition my share; provide me a share of the ultimate existence, O one born of Bhava!

This is the core anuṣṭubh to which in regular practice the bhakti: bhavāya namaḥ || is appended at the end.

There are some key points that you must understand in relation to this ṛk: Why is Skanda known as Sadyojāta? It relates to him arising instantly from the semen of Rudra in the Śaravana or Gaṅgā river. The term is also used for Agni, the father of Skanda, in the ancient Āprī of the Bhārgava-s:

sadyo jāto vy amimīta yajñam agnir devānām abhavat purogāḥ |

At birth he instantly measured out the ritual [space]; Agni became the leader of the gods.

This is reminiscent Skanda becoming the leader, i.e. senāni, of the gods. In the old tradition of the Bhāllavi-s, he is said to have emerged from Rudra as the dual of Agni to burn down the piśācī when invoked by Vṛśa Jāna, the badger-like purohita of the Ikṣvāku lord Tryaruṇa. That is why this ṛk is associated with the mūlamantra of Kumāra: OṂ vacadbhuve namaḥ || or that of Bodhāyana: OṂ bhuve namaḥ svāha ||. Indeed, one may do japa of the mantra using the following yantra. In the corners of the central tryaśra-yamala (forming the hexagonal star) one must place the syllables of one of the above ṣaḍakṣari-s. The in a circle around the central star one must place the 32 syllables of the core anuṣṭubh. Then in the central hexagonal hub of the star one must place the bhakti of 5 syllables. The name of Kumāra, Vacadbhū, is evidently an old linguistic fossil and indicates the emergence of the god from the piśacī-repelling incantation.”

PS:“How is the second part of the mantra to be understood?”

Lootika: “The imperative verb bhajasva is to be applied twice once with the negation particle na and once without it. The na is coupled with the locative āmreḍita bhave-bhave. When with the negation particle bhava is understood in more than one way depending on the votary and their interests. For people of the world, like us, a bhava can be each unit of conscious existence — it could be each time we get up from sleep and return to conscious ahaṃ-bhāva. Here the negated verb na bhajasva might be understood as do not divide what is ours. In the positive sense it is applied to ādibhava — the primal state or when the consciousness associated with you is in an identity loop — a state of bliss. For the yati-s, like the practitioners of the pūrvāṃnāya of yore in the Vaṅga or the Karṇāṭa country, it implies not being partitioned into multiple rebirths and being instead given mokṣa (ādibhava).”

Somakhya: “While performing japa you may meditate on the deva mounted on a peacock with a fowl-banner surrounded by the troops Marut-s bearing spears, backed by Agni riding a rhinoceros. He is in the midst of a fierce battle leading the deva-s against the daitya-s and showering arrows on his foes. Thus one must meditate on him for the destruction of brahmadviṣ-es and dasyu-s. Alternatively, you may meditate on him in his dual form as Skanda-Viśākha. In this case one must replace the terminal bhakti with: namo bhavāya ca śarvāya cobhābhyām akaraṃ namaḥ ||. One may also perform the prayoga where Skanda is visualized as surrounded by the 12 awful goddesses:

1) Vimocinī; 2) Mohinī; 3) Sunandā; 4) Pūtanā; 5) Āsurī; 6) Revatī; 7) Śakunī; 8) Piśācikā; 9) Pāśinī; 10) Mahāmārī; 11) Kālikā; 12) Bhāminī.

Certain traditions holds them to be Paulastyā-s, the 12 sisters of Rāvaṇa and call it the Rākṣasī-prayoga but we do not subscribe to that. This may be done for pediatric purposes.

One may also deploy the mantra visualizing Kumāra along with Viśākha, Śākha, Nejameṣa and Ṣaṣṭhī along with Kauśikī, the daughter Rudra and Umā. They should be accompanied by the following therocephalic and avicephalic goddesses who accompany Kauśikī:

1) Vāyasī, crow-headed; 2) Upakā, owl-headed; 3) Pracaṇḍā, lion-headed; 4) Ugrā, tiger-headed; 5) Jayā, elephant-headed; 6) Jayantī, peahen-headed; 7) Jayamānā, horse-headed; 8) Prabhā, goose-headed; 9) Śivā, jackal-headed; 10) Saramā, dog-headed; 11) Vijayā, falcon-headed; 12) Prabhāvatī, sheldrake-headed; 13) Mṛtyu, crane-headed; 14) Niyati, grebe-headed; 15) Aśani, fowl-headed; 16) Vṛṣadaṃśā, badger-headed; 17) Śakunī, thrush-headed; 18) Potakī, munia-headed; 19) Vānarī, monkey-headed.

The following devī-s are visualized with human heads:

20) Revatī (some say she is cat-headed); 21) Pūtanā; 22) Kaṭapūtanā; 23) Ālambā; 24) Kiṃnarī; 25) Mukhamaṇḍikā; 26) Alakṣmī or Jyeṣthā; 27) Adhṛtiː 28) Lakṣmī; 29) Spṛhā; 30) Aparājitā.

Along with Ṣaṣṭhī and Kauśikī these constitute the 32 aṛṇa-devī-s of the Sadyojāta-ṛk and should be worshiped with bīja-s derived from each of the syllables of the ṛk. The meditation on Kumāra and his emanations visualized in the midst of these devī-s is the highest sādhanā of his mantra and may be deployed for Ṣaṭkarmāṇi. This sādhanā may be accompanied by the use of the 4 \times 4 pan-diagonal magic square that adds up to 34 comprised of numbers from 1:16 with each representing a pair of the 32 goddesses. Of the sum, 32 are the goddesses that I’ve just mentioned. The remaining 2 are are Skanda and Ṣaṣṭhī.

\begin{tabular}{|c|c|c|c|} \hline 16 &2 & 3 & 13\\ \hline 5 & 11 & 10 & 8\\ \hline 9 & 7 & 6 & 12\\ \hline 4 & 14 & 15 & 1\\ \hline \end{tabular}

Now, Pinaki, what śruti-vākya do these goddesses with an avicephalic emphasis bring to your mind?”

PS:“I’m reminded of the mysterious words of my ancestor Atri Bhauma:

vayaś cana subhva āva yanti

kṣubhā martam anuyataṃ vadhasnaiḥ |

And like mighty birds [the Marut-s] swoop down here, turbulently, to the mortal pursued by deadly weapons.

They are again compared the birds by Gotama:

vayo na sīdan adhi barhishi priye |

Like birds [the Marut-s] sit down on the dear sacred grass.”

Given the genetic connection of Skanda and his retinue to the Marut-s, who are said to descend like birds to the sacrificer, I find this avicephalic emphasis resonant.

Lootika “Good. Indeed, these goddesses accompanying Ṣanmukha are the precedents of the yoginī-s of kula practice. They reveal themselves to the sādhaka usually in the form of certain reptiles and mammals. Thus, you may get the confirmation of your mantra-sādhana with your dūtī Shallaki by the apparition of yoginī-s in the form of certain birds in quiet sylvan spots.”

Somakhyaː “To conclude, Lootika will show you how to prepare the dhatturādi-viṣa for the viṣa-prayoga.”

After Lootika showed him how to prepare the guhyaviṣa from oṣadhi-s, she said: “The details of these plants are not something you might know offhand. You can get them from your brother or my sister. In addition to Skanda, Viśākha, Śākha and Nejameṣa you have to invoke and worship the 32 aṛṇa-devī-s when making the viṣa. When a devadatta is subject this prayoga he is seized by a dreadful graha and with a crooked look on his face he wanders yelling and singing: Rudraḥ Skando Viśākho .aham Indro .aham | even as old Vagbhaṭa has described the graha seizure. Such a seizure could also happen due to other reasons by agents of Kumāra such as the ovine sprite or the caprine sprite or the Āpastamba sprite. It could also happen among the ritually weak when they visit Lankā, Mālādvīpa or Kāshmīra where various piśāca-graha-s naturally reside. The seizure often manifests differently among males and females. In females it might manifest as the state of pinning for ones lover as the old Drāviḍa-s would say. In such cases one may deploy this mantra with the visualization of Kumāra with the Mātṛ-s. One also worships them when goes to the holy spots specially to around Eurasia.”

PS: “Indeed, I heard from Shallaki of such a seizure of one of her relatives when he visited Kāshmīra. My brother informs me of the Kaubera-vrata-s that need to be performed in Kāshmīra to invoke Kaubera-piśāca-s to counter seizures.”

Somakhyaː“Thus, go ahead an practice this mantra. Sleep on the ground, avoid eating sweets and drinking sweet beverages, and sitting on cushions. Now you should conclude by performing the tarpaṇa as promulgated by Bodhāyana”:

OṂ skandaṃ tarpayāmi | Om indraṃ tarpayāmi | OṂ ṣaṣṭhīṃ tarpayāmi | ṣaṇmukhaṃ tarpayāmi | OṂ jayantaṃ tarpayāmi | OṂ viśākhaṃ tarpayāmi | OṂ mahāsenaṃ tarpayāmi | OṂ subrahmaṇyaṃ tarpayāmi | OṂ skanda-pārṣadāṃs tarpayāmi | OṂ skanda-pārṣadīś ca tarpayāmi ||

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Conic conquests: biographical and historical

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Studying mathematics with our father was not exactly an easy-going experience; nevertheless, it was the source of many a spark that inspired fruitful explorations and life-lessons. We recount one such thread here, and reflect on how our personal education matched up to history. When we were a kid, and learning about the area and perimeter of plane figures — we used to do it with a graph paper — our father told us that \pi was not \tfrac{22}{7} as our school lessons claimed but a “never-ending” number. He pointed to us that \tfrac{22}{7} was just like \frac{1}{3} which repeated after a certain run of numbers. This inculcated in us a life-long fascination, to the degree our meager mathematical capacity allowed, for both these types of numbers and their deeper significance. Some time thereafter our parents took us to watch a series of documentaries that were screened at a nearby auditorium on the evolution of man and various intellectual developments in science and mathematics in the Occident. In one of them, the presenter mentioned that the yavanācārya of yore, Archimedes, had arrived at the approximation \tfrac{22}{7} for \pi by inscribing polygons in a circle. This sparked a great a excitement in us for we were then fascinated by construction of regular polygons (\mathcal{P}_n). On returning home from the screening, we quickly got back to that geometric activity realizing that the very first construction we learned in life, that of a regular hexagon, yielded \pi \approx 3 (Figure 1).

Figure 1. Construction of a regular hexagon (polygon \mathcal{P}_6) in a unit circle.

That was hardly anything to write home about, but even as people used to think that “developmental ontology recapitulates phylogeny”, this realization recapitulated the very beginnings of the human knowledge of \pi. This value was used for the crudest of constructions in the Vedic ritual going back to its ancient roots. It has been suggested that this knowledge is encapsulated in a cryptic manner with a peculiar play on the name of the god Trita (meaning 3rd):

indro yad vajrī dhṛṣamāṇo andhasā bhinad valasya paridhīṃr iva tritaḥ |

When the vajra-wielding Indra invigorated by the soma draught, split the [fortification] perimeter of Vala (the dānava’s name is used in a possible play on term for a circle), even as Trita [had done].

The allusion to the god Trita in the simile here is unusual and is evidently an allusion to his breaking out a well (an enclosure with a circular section); this strengthens the idea that a word play on the circular perimeter being split up in 3 by the diameter was exploited by Savya Āṅgirasa. This crude approximation continued to be used in sthūla-vyavahāra by the Hindus and the Nirgrantha-s till the medieval period (e.g. in the nagna text Tiloyasāra: vāso tiguṇo parihi |; the Prakrit corresponds to the Sanskrit vyāso triguṇo paridhiḥ |: P(\bigcirc)= 3d ). We also hear that the Jews used the same value in building a religious structure in their early history. While that is a lot of words to expend on this crudest of approximations, one could say that it at least gets you to 95.5\% of the real thing. Around that time our father had introduced us to the radian measure of an angle and informed us that it was the natural one, for after all the number 360 for the degree measure was an arbitrary one coming from an approximation of the year. This most elementary of constructions, the hexagon, gave us an indelible visual feel for the radian for after all if 3 got us to 95.5\% of \pi then the radian should be roughly 57.3^\circ. More importantly, it informed us that this unit is best understood in multiples of \pi and that the interior angle of the hexagon should be \tfrac{\pi}{3}. In terms of history we had caught up with the emergence of the germ of this concept in Āryabhaṭa.

Figure 2. A regular dodecagon from a square and equilateral triangles

The aftermath of the above apprehension led us to doing a few more constructions and origami folding that led us to a somewhat more interesting realization from an aesthetically pleasing construction of a regular dodecagon which goes thus (Figure 2):

1) Draw a starting square and an equilateral triangle facing inwards on each of its sides (relates to the basic origami construction of an equilateral triangle from a square paper as indicated by Sundara Rao more than 100 years ago).

2) The inward-facing vertices of these equilateral triangles will define a new square orthogonal to the original square.

3) The 4 midpoints of the sides of this new square and the intersections of the equilateral triangles help define the sides of a regular dodecagon — in effect arising from a geometric multiplication of 4 \times 3. Thus, this dodecagon is inscribed in the inner square.

4) Notably, this construction by itself helps define two tiles, a 150^\circ-15^\circ-15^\circ isosceles triangle T_1 (violet) and an equilateral triangle T_2 (green). Using these tiles both the inner square (\mathcal{P}_4) and the dodecagon (\mathcal{P}_{12}) can be completely tiled thus (Figure 2): \mathcal{P}_4=32T_1+16T_2 and \mathcal{P}_{12}=24T_1+12T_2. This means that the area of the inscribed dodecagon is \tfrac{3}{4} the area of the inner square.

5) A corollary to the above is that if a unit circle were inscribed in the inner square then square will have area 4 and the dodecagon will have area 3.

Thus, it indicated that we would need a polygon of twice the number of sides to get the same approximation of \pi via its area as that of an inscribed polygon which gives the same from its perimeter. Hence, perimeter of the inscribed polygon is better than the area to obtain an approximation of \pi. Further, these exercises taught us something notable: If the Yajurvedic tradition had used polygon inscription (likely it did not) then it would have required a decagon to get something close to its values \approx 3.09; 98.4\% the real value. While Baudhāyana or Kātyāyana are not explicit about it, Āpastamba is clear that these \bigcirc \leftrightarrow \square conversions are approximate. Squeezing out additional digits beyond that point is a process of diminishing returns; however, in the ancient world the Maitrāyaṇīya school of the Ādhvaryava tradition and the Egyptians achieved similar success reaching close to 99.4..99.5\% of the real value. If one wanted to achieve such a level of approximation with the polygon method you would need to inscribe \mathcal{P}_{18} to get almost exactly the value of the Maitrāyaṇīya tradition. While old Archimedes is said to have labored with a 96-side polygon to reach his \tfrac{22}{7}, Āryabhaṭa would have needed something like 360-polygon to get his value. This made us suspect that it was unlikely he used polygon inscription and instead had a trick inherited from the non-polygonal methods typical of the old Hindu quadrature of the circle [Footnote 1]. On the other hand Archimedes’ early triumph undoubtedly rode on the quadrature of the circle achieved by the Platonic school.

From our father’s instruction we were reasonably conversant with basic trigonometry before it had been taught in school and he used that as base to introduce us to the basics of calculus in the form of limits. He told us that we could use our polygon inscription to informally understand the limit \lim_{x \to 0} \tfrac{\sin(x)}{x}=1. With this in hand, he told us that we could, if we really understood it, prove the formulae of the perimeter and area of a circle as the limiting case of the \infty-sided polygon. From our earliest education in mensuration we had been puzzled by how that mysterious number \pi appeared in these formulae — we understood quite easily how the formulae of rectilinear figures like rectangles and triangles had been derived but this “correction factor” for the circle had been an open question for us. Hence, we were keen to figure this out using our newly acquired knowledge of limits. Being of only modest mathematical ability it took us a few days until we arrived at the proof with limits but we were then satisfied beyond words by the experience of putting down the below:

Figure 3. The circle as the limit of an inscribed polygon

From Figure 3 we can write the perimeter of an inscribed polygon in a unit circle as:

P(\mathcal{P}_n)=2n\sin\left(\frac{\pi}{n}\right) = 2 \pi \dfrac{\sin\left(\frac{\pi}{n}\right) }{\frac{\pi}{n}}

Similarly we can write its area as:

A(\mathcal{P}_n)=n\sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right) = \pi \dfrac{\sin\left(\frac{\pi}{n}\right) }{\frac{\pi}{n}} \cos\left(\frac{\pi}{n}\right)

n \to \infty \; \tfrac{\pi}{n} \to 0 \therefore \displaystyle \lim_{\tfrac{\pi}{n} \to 0} \dfrac{\sin\left(\tfrac{\pi}{n}\right)}{\tfrac{\pi}{n}} \to 1

By taking the above limit we get P(\mathcal{P}_\infty)=P(\bigcirc)=2\pi; \;A(\mathcal{P}_\infty)=A(\bigcirc)=\pi

While this gave us the formulae for the perimeter and the area of a circle, the actual value of \pi was still a challenge and progress on that front had to wait for other developments. Around the same time, our fascination with the other conics was growing, mainly as an offshoot of our concomitant interest in astronomy. Armed with the high-precision German-made templates we had received from our father we began studying these conics closely. We soon realized that the circle was at one end of the continuum of ellipses and the parabola the end. The hyperbolae lay beyond that end almost as if the ellipse had wrapped around infinity and its two apices had folded back towards each other. It also struck us right away that the method of limits we had used to derive the area and perimeter of a circle could not applied to these other conics. Informally (i.e. by squeezing a circle perpendicular to one of the diameters while preserving area), we could figure out that the area of an ellipse should be \pi a b where a, b are its semimajor and semiminor axes. We also got the idea of “area under a curve” intuitively; however, it was unclear how the formulae for perimeters of these other conics could be derived. We had seen formulae for them in tables of functions we had at home [Footnote 2]. However, the tables stated that the multiple formulae it offered perimeter of the ellipse P(E) were approximate:

P(E) \approx 2 \pi \sqrt{ab}

P(E) \approx \pi (a+b)

P(E) \approx \pi\sqrt{2(a^2+b^2)}

The first two formulae were attributed in the tables to Johannes Kepler, who had reason to calculate this as he studied ellipses in course of his monumental work on planetary orbits. While I have confirmed him as the source of the first formula, it is not clear if he was the first to propose the second one. The third formula was proposed by Leonhard Euler. In Hindu tradition, to our knowledge, the perimeter of an ellipse (āyata-vṛtta) was treated for the first time by Mahāvīra in this Gaṇita-sāra-saṃgraha. He gives a formula that goes thus:

vyasa-krtiḥ ṣadguṇitā dvi-saṅ-guṇāyama-krtiyutā padam paridhiḥ |

vyāsa-catur-bhāga-guṇaś cāyata-vṛttasya sukṣma-phalam ||

Six times the square of the minor axis plus the square of twice its major axis; the root of this gives the perimeter. That multiplied by one fourth of its minor axis is the high precision area of the ellipse.

In modern usage the perimeter will be: P(E) \approx 2\sqrt{4a^2+6b^2}

Figure 4. Ellipse perimeter approximations

Figure 4 shows the approximate perimeters obtained from the various formulae for selected ellipses with semimajor axis a=1 and the variable semiminor axis b = .2, .4, .6, .9. The average method is the second of the above formulae. We realized that each approximation is optimized for a different types of ellipses even before we had achieved the exact value for ourselves. One can see that if the ellipse becomes a circle then Mahāvīra’s formula would become P(\bigcirc)=2\sqrt{10}r; this is keeping with his approximation of \pi as \sqrt{10}, which was a misapprehension also held by Brahmagupta contra Āryabhaṭa. Thus, it appears that he tried to “break up” that \pi between the two axes — this approximation does reasonably well at the extremes and has a maximum error of around 3.25\% for b \approx .438. In the rare instances when ellipses where used in Hindu architecture (e.g. in the maṇḍapa of the temple at Kusuma) they are usually of the proportions a=1, b=\tfrac{1}{\sqrt{2}}; for such ellipses the Hindu formula would have given an error of about 2\%.

Our explorations in this direction had set in our mind a strong desire to obtain the exact formula for the perimeters of the ellipses and parabolic arcs. Hence, like our ancestor Bhṛgu going to the great asura Varuṇa we went to our father seeking the way forward. He informed us that for achieving those objectives we needed to apprehend the further branches of calculus and that those would anyhow come as part of our curriculum in college. But we were not going to wait till then; so, he suggested that we go to the shop and get those simple textbooks a bit in advance and I could attempt to study them over the vacations. Over the next two years we made modest progress and by then we were already in junior college where elementary calculus was to start from the second semester. A prolonged shutdown from a strike gave us exactly what we wanted — the time to explore these matters by ourselves. By then, armed with the basics of the different branches of calculus, we made exciting progress for our low standards: 1) We rediscovered for ourselves the hyperbolic equivalents of the circular trigonometric functions, their derivatives and integrals. 2) We studied and (re)discovered some additional methods for constructing conics. 3) Most importantly, we non-rigorously derived for ourselves the general method of determining arc length of a curve between x=a, b using differential and integral calculus:

\displaystyle L = \int_a^b \sqrt{1+\left( \dfrac{dy}{dx} \right)^2}dx

We got our first opportunity to put it to practice when we explored the following question: Given a semicircle, how would you inscribe a circle in it? What would be the locus of the centers of such inscribed circles? The construction shown in Figure 5 provides a completely self-evident proof that the locus should a parabola with the tangent to the semicircle at its apex as its directrix and the diameter of the semicircle as its latus rectum. Nevertheless we expand it for a naive reader (Figure 5).

1) First draw the line tangent to the apex D of the semicircle.

2) To inscribe a circle that touches a semicircle at point E, join its center C to E.

3) Then draw a line perpendicular to radius \overline{CE} at E; this will be the tangent to the semicircle at E.

4) This line intersects the tangent at D at point F.

5) Drop a perpendicular from F to the diameter of the semicircle. It will cut radius \overline{CE} at G which will be the center of an inscribed circle touching the semicircle at E and its diameter at H.

6) \angle CHG = \angle GEF =\tfrac{\pi}{2} and \angle CGH = \angle EGF. Moreover, \overline{GE}=\overline{GH}. Thus, \triangle CGH \cong EGF by the side-angle-angle test. Hence, \overline{GC}=\overline{GF}.

7) Thus, for every inscribed circle its center G will be equidistant from the tangent to the semicircle at its apex and from its center C. Hence, this locus is a parabola with its focus at C and the above line as its directrix. Accordingly, the diameter of the semicircle would be its latus rectum.

Figure 5. Semicircle-parabola construction

Accordingly, we applied the above integration to this parabola (Figure 5) whose equation would be y=-\tfrac{x^2}{2a}+\tfrac{a}{2}, where a is the radius of the generating semicircle to obtain the arc length of the parabola bounded by its latus rectum:

\dfrac{dy}{dx}=-\dfrac{x}{a}

\therefore \displaystyle \int_{-a}^a \sqrt{1+\dfrac{x^2}{a^2}}dx = \left. \dfrac{x}{2}\sqrt{1+\dfrac{x^2}{a^2}}+\dfrac{a}{2}\textrm{arcsinh}(x/a)\right|_{-a}^a = a\left( \sqrt{2} + \textrm{arcsinh}(1) \right)

With this we realized that the parabola as a unique conic (i.e. fixed eccentricity), just as the circle, has an associated constant comparable to \pi that provides its arc length bounded by the latus rectum in terms of the semi-latus rectum a; hence all parabolas are like just as all circles and differ only in scale. Thus, we had rediscovered the remarkable parabolic constant, the ratio of the arc length of the parabola bounded by its latus rectum to its semi-latus rectum: P= \sqrt{2} + \textrm{arcsinh}(1) \approx 2.295587.

This also brought home to us that, unlike the circle and the parabola, the ellipse (and the hyperbola) will not have a single constant that relates their arc length to a linear dimension. Instead there will be a family of those which would be bounded by \pi and \sqrt{2} + \textrm{arcsinh}(1) latex . Our meager mind was immensely buoyed by the successful conquest of the parabola and believed that the comparable conquest of the ellipse was at hand. But hard as we tried we simply could not solve the comparable integral for the ellipse in terms of all the integration we knew. Later that summer we got to meet our cousin who was reputed to have enormous mathematical capacity but had little interest in conics. With a swagger, he said it should be easy but failed to solve it just as we had. However, he had a computer, and for the first time we could attack it with numerical integration. This gave us some intuition of how the integral specifying the arc length of an ellipse behaves and that there is a likely generalization of the circular trigonometric functions to which they might map. At that point we asked an aunt of ours, who used to teach mathematics, if she had any leads to solving that integral. She flippantly asked if we did not know of elliptic integrals? That word struck cord — not wanting to expose our ignorance further we set out to investigate it further. We went back to our father, who handed us a more “advanced” volume and told us that we were now grown up and could pursue our mathematical fancies on our own. That was indeed the case — like our ancestor Bhṛgu before he realized the Vāruṇī-vidyā. Therein we finally learned that the elliptic integrals, like the one we had battled with, were functions in themselves which could not be expressed in terms of elementary functions — there were special tables that gave their values for ellipses of different eccentricities even as we had circular and rectangular hyperbolic trigonometric functions. But those books had a terrible way of teaching elliptical integrals; hence, we had to chart our own method of presenting them for a person of modest intelligence. Once we did so we felt that these could be easily studied in their basic form along with the regular trigonometric functions.

Thus, we learned that our quest for the perimeter of the ellipse following the course from Mahāvīra through Kepler had reached the dawn of modern mathematics by converging on the famous elliptic integral which has attracted the attention of many of a great mind. The early modern attack on the perimeter of an ellipse began with Newton’s attempt with numerical integration, which we had recapitulated using a computer. In the next phase, the 26 year old Leonhard Euler, who declared it to be one great problems that had mystified geometers of the age, used some basic geometry and remarkable sleights of the hand (or should we say the mind) with the binomial theorem to prove the below series for the perimeter of an ellipse. One could say that the paper in which it appeared (“Specimen de constructione aequationum differentialium sine indeterminatarum separatione”) had a foundational role in modern mathematics:

Let d=\dfrac{a^2}{b^2}-1 then,

P(E) = 2 \pi b \left( 1+ \dfrac{1 \cdot d}{2 \cdot 2} - \dfrac{1 \cdot 1 \cdot 3 \cdot d^2}{2 \cdot 2 \cdot 4 \cdot 4} + \dfrac{1 \cdot 1 \cdot 3 \cdot 3 \cdot 5 d^3}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6} -\dfrac{1 \cdot 1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 d^4}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdot 8 \cdot 8} \dots \right)

Our numerical experiments showed that the above series yielded values close to the actual perimeter of an ellipse with 10..20 terms when its eccentricity is between 0..\tfrac{1}{\sqrt{2}}. However, below that it starts faring poorly and has increasingly poor convergence. From Euler’s original series a second series which uses the eccentricity e_E can be derived with somewhat better convergence:

Let m=e_E^2 then,

\displaystyle P(E) = 2 \pi a \left(1 - \sum_{j=1}^\infty \dfrac{1}{2j-1} \left( \dfrac{(2j)!}{(2^j j!)^2} \right)^2 m^j \right)

20 terms of this series gives values close to the real perimeter for ellipses with eccentricities in the range (0,0.95) and poorer approximations at higher eccentricities like e_E=.995. This showed to us that the problem we had struggled with was a truly worthy one and even with Euler’s attack getting precise values throughout the eccentricity range was not an easy one. This line of investigation was brought to a closure by the great Carl Gauss who in his twenties had worked out a series with superior convergence for the perimeter of the ellipse that was related to his discovery of the hypergeometric function with a profound impact on modern mathematics. This series is defined thus:

Let \displaystyle C(n) = \dfrac{\displaystyle \prod_{j=1}^n \left(\dfrac{1}{2}-j+1\right)}{n!}

Let h=\left(\dfrac{a-b}{a+b}\right)^2 then,

\displaystyle P(E) = \pi (a+b) \left(1+ \sum_{j=1}^{\infty} (C(j))^2 \cdot h^j \right)

This series gives accurate perimeters within 20 terms for e_E \in (0,0.99995). Another man who also took a similar path, although almost entirely in the isolation, was Ramanujan of Kumbhaghoṇa but that story is beyond the scope of this note. The above series was hardly the only achievement of Gauss in this direction. He had figured out an algorithm that put the final nail into this problem. Before we get to that, we shall take a detour to define the different original elliptic integrals and take a brief look at the other places we encountered them.

The ellipse \tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1 can be divided into 4 quadrants due its symmetry and in the first quadrant its equation can we written as

y=b\sqrt{1-\dfrac{x^2}{a^2}} and by the chain rule \dfrac{dy}{dx}=-\dfrac{bx}{a^2\sqrt{1-\dfrac{x^2}{a^2}}}

We make the substitution t=\tfrac{x}{a}; hence,

\dfrac{dy}{dx}=-\dfrac{bt}{a\sqrt{1-t^2}}

\therefore \sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\sqrt{1+\dfrac{b^2t^2}{a^2-a^2t^2}} =\sqrt{\dfrac{a^2 \left(1-\left(1-\frac{b^2}{a^2}\right)t^2\right)}{a^2(1-t^2)}}

We observe that e_E^2=1-\tfrac{b^2}{a^2} is the square of the eccentricity of the ellipse. Keeping with the commonly used elliptic integral convention we use e_E = k. Thus, the above expression becomes:

\sqrt{\dfrac{1-k^2t^2}{1-t^2}}

From the above substitution dx=a \cdot dt; when x=0, t=0 and when x=a, t=1. Thus, taking into account all 4 quadrants our arc length integral for the perimeter of the ellipse becomes:

\displaystyle P(E) = 4a \int_0^1 \sqrt{\dfrac{1-k^2t^2}{1-t^2}}dt

More generally, it may be expressed using the substitution t=\sin(\theta) \; \therefore dt=\cos(\theta) d\theta with which a version of the integral for an elliptic arc defined by angle \phi becomes:

\displaystyle E(k,\phi) = \int_0^\phi \sqrt{1-k^2\sin^2(\theta)} d\theta

This was the very form of the integral we had battled with in our youth before realizing that it was the definition of the elliptic integral of the second kind. One can see that for a quadrant of the ellipse with e_E=k the above integral is from 0 to \phi=\tfrac{\pi}{2}. This is then called the complete elliptic integral of the second kind and it may be simply written as E(k). Limits between other angles will give the corresponding lengths of elliptical arcs and the general integral E(k,\phi) is thus the incomplete elliptic integral of the second kind.

If this is the elliptic integral of the second kind then what is the first kind? Incomplete elliptic integral of the first kind is defined as:

\displaystyle F(k,\phi) = \int_0^\phi \dfrac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}

If the limits of this integral are taken from 0 to \phi=\tfrac{\pi}{2} then we get the complete elliptic integral of the first kind which is confusingly denoted by K(k).

The relationship between the complete integrals of the 2 kinds and \pi was discovered by the noted French mathematician Adrien-Marie Legendre, who greatly expanded their study from what Euler had done. As we shall see below, this key relationship led to the Gauss algorithm for calculating \pi most efficiently. If the eccentricity of an ellipse is k then the ellipse of complementary eccentricity k' is defined thus: k^2+k'^2=1. Then we get the Legendre relationship between the 2 kinds of complete elliptic integrals:

K(k') \cdot E(k)+E(k') \cdot K(k) - K(k') \cdot K(k) = \dfrac{\pi}{2}

We had already had our brush with K(k) earlier in our youth in course of the auto-discovery of conic-associated pedal and envelop curves recapitulating some deep history in this direction. For an ellipse or a hyperbola we can define the pedal curve as the locus of the feet of the perpendiculars dropped from the center of the conic to its tangents. We can define a second curve as the envelope of the circles whose centers lie on an ellipse or a hyperbola and which pass through the center of the said conic. We found that these two curves differ only in scale with the later being double the former in a given dimension. If a, b are the semi- major and minor axes of the parent conic, these curves have the polar equations:

\rho^2 = a^2\cos^2(\theta)+b^2\sin^2(\theta) the pedal curve for an ellipse

\rho^2 = 4\left(a^2\cos^2(\theta)+b^2\sin^2(\theta)\right) the envelope curve for an ellipse

\rho^2 = a^2\cos^2(\theta)-b^2\sin^2(\theta) the pedal curve for a hyperbola

\rho^2 = 4\left(a^2\cos^2(\theta)-b^2\sin^2(\theta)\right) the envelope curve for a hyperbola

Figure 6A. The hippopedes of an ellipse

We subsequently learned that these elliptical version of the curves was termed the hippopede by the great Proclus who investigated them (Figure 6A). He was consciously one of the last in the line of great yavana investigators of curves starting from the discovery of the conics at the Platonic academy and this died with the destruction of the Greek tradition by the “Typhonic winds” of the second Abrahamism. One observes that if a=b then the hippopedes become circles. If b=0 then again we get a pair of circles. The hyperbolic versions in contrast specify \infty-shaped curves that cross at origin. If the hyperbola is rectangular (i.e. a=b) then we get the pedal version to be: \rho^2 = a^2\left(\cos^2(\theta)-\sin^2(\theta)\right)= a^2\cos(2\theta). This curve was first studied by Jakob Bernoulli, the eldest brother of the first famous generation of the Bernoulli clan. It has an interesting property: if the ellipse is the locus of points the sum of whose distances from the two foci is a constant, the lemniscate is the locus of points the product of whose distances from the two foci is constant.

Figure 6B. The lemniscates of a hyperbola

We realized that unlike the other above curves whose arc lengths pose some terrible integrals that for the lemniscate can be reduced to a form comparable to what we got for the ellipse. Given a lemniscate with the polar equation: \rho^2 = a^2\cos(2\theta),

\dfrac{d\rho}{d\theta}=-\dfrac{a\sin(2\theta)}{\sqrt{\cos(2 \theta)}}

The arc length formula in polar coordinates is:

\displaystyle \int_a^b \sqrt{\rho^2 + \left(\dfrac{d\rho}{d\theta}\right)^2}d \theta

Thus, the arc length of the first quadrant of the lemniscate is:

\displaystyle \int_0^{\pi/4} \sqrt{a^2 \cos(2\theta)+a^2 \dfrac{\sin^2(2\theta)}{\cos(2\theta)}}d\theta= a\int_0^{\pi/4}\sqrt{\dfrac{\sin^2(2\theta)+\cos^2(2\theta)}{\cos(2\theta)}}d\theta

\displaystyle = a \int_0^{\pi/4} \dfrac{d\theta}{\sqrt{\cos(2\theta)}}

In the above we make the substitution \cos(2\theta)=\cos^2(\phi) \; \therefore -\sin(2\theta)d\theta=-2\sin(\phi)\cos(\phi)d\phi

d\theta = \dfrac{\sin(\phi)\cos(\phi)d\phi}{\sin(2\theta)}= \dfrac{\sin(\phi)\cos(\phi)d\phi}{\sqrt{1-\cos^4(\phi)}} =\dfrac{\sin(\phi)\cos(\phi)d\phi}{\sqrt{(1-\cos^2(\phi))(1+\cos^2(\phi))}}=\dfrac{\cos(\phi)d\phi}{\sqrt{1+\cos^2(\phi)}}

=\dfrac{\cos(\phi)d\phi}{\sqrt{2(1-\frac{1}{2}\sin^2(\phi))}}

This substitution results in the limits of the arc length integral changing to 0..\tfrac{\pi}{2}. Thus, it becomes:

\displaystyle = \dfrac{a}{\sqrt{2}} \int_0^{\pi/2} \dfrac{\cos(\phi)d\phi}{\cos(\phi)\sqrt{(1-\frac{1}{2}\sin^2(\phi))}}=\dfrac{a}{\sqrt{2}} \int_0^{\pi/2} \dfrac{d\phi}{\sqrt{(1-\frac{1}{2}\sin^2(\phi))}}

Thus, the perimeter of this lemniscate is:

\displaystyle P(L)=\dfrac{4a}{\sqrt{2}} \int_0^{\pi/2} \dfrac{d\phi}{\sqrt{(1-\frac{1}{2}\sin^2(\phi))}}

We see that the integral specifying the perimeter of a lemniscate is a complete elliptic integral of the first kind with as \phi=\tfrac{\pi}{2} and k=\tfrac{1}{\sqrt{2}}, i.e. K\left(\tfrac{1}{\sqrt{2}}\right). This was one of the integrals studied by Count Fagnano, a self-taught early pioneer in the calculus of elliptical and lemniscate arcs — we were amused and somewhat consoled to learn that, like us, he has initially tried to solve this integrals in terms of elementary functions and failed. Moreover, like the circle and the parabola the lemniscate is a unique curve, such that the ratio of its perimeter to its horizontal semi-axis a is a constant (the lemniscate constant) mirroring \pi and the parabolic constant P:

L= 2\sqrt{2}K\left(\tfrac{1}{\sqrt{2}}\right) \approx 5.244116

This is also a special value where we have the below relationship which can be used to compute \pi efficiently (corollary to Legendre’s identity):

2K\left(\tfrac{1}{\sqrt{2}}\right)\left(2E\left(\tfrac{1}{\sqrt{2}}\right)-K\left(\tfrac{1}{\sqrt{2}}\right)\right)=\pi

Around the time we acquired a grasp of these elliptic integrals we also learned of another practical appearance of K(k). In elementary physics one learns of simple oscillators like the pendulum and derives its period T using the basic circular trigonometric differential equation:

T \approx 2\pi\sqrt{\dfrac{l}{g}}

Here, l is the length of the pendulum and g \approx 9.8 m/s^2 is the gravitational acceleration. This emerges from an approximation for small angle oscillations where \sin(\theta) \approx \theta and corresponds to the period relationship discovered by Galileo and the apparent failed attempt of Gaṇeśa Daivajña. We had already realized that integrating the differential equation for a larger amplitude presented an integral that we had failed to solve using elementary functions. However, it can be solved with the elliptical integral of the first kind to give the accurate value for period as:

T=4\sqrt{\dfrac{l}{g}}K\left(\frac{\theta_0}{2}\right)

Here \theta_0 represents the initial angle at which the pendulum is released. One can see that if \theta_0 \to 0 then K(k) \to \tfrac{\pi}{2} giving us the low amplitude formula. Taking the standard value of g=9.80665 m/s^2 given in physics textbooks we get the period of a meter pendulum with a low amplitude displacement as T= 2.006409 s. If we instead give it a 60^\circ release then we get T=2.153242 s with the elliptic integral K\left(\sin\left(\tfrac{\pi}{6}\right)\right). Hence, one can see that the Galilean linear approximation is not a bad one for typical low angle releases.

This finally leads to what was a burning question for us in our youth: How do we effectively compute these elliptic integrals? In our opinion, this should be taught first to students and that would go some way in making the elliptics trivial as trigonometric functions. We saw the various series methods of Euler and Gauss. While the latter does quite well it is still a multi-term affair, that takes longer to converge higher the eccentricity. But the 22 year old Gauss solved this problem with a remarkable algorithm that rapidly gives you the values of these integrals — something, which in our early days, we had even done with a hand calculator while teaching it to a physics student. Right then, Gauss realized that it “opens an entirely new field of analysis” as he wrote in his notes accompanying the discovery. This is the famous arithmetic-geometric mean M algorithm which goes thus:

Given 2 starting numbers x_0, y_0, apply the map: x_{n+1}=\dfrac{x_n+y_n}{2}, \; y_{n+1}=\sqrt{x_n \cdot y_n}.

The map converges usually within 5 iterations for typical double precision values to the arithmetic-geometric mean M(x_0,y_0). Let k be the eccentricity value for which we wish to compute the complete elliptic integral of the first kind. k'=\sqrt{1-k^2} then we have,

K(k)=\dfrac{\pi}{2M(1,k')}

For E(k) we used to originally use a Gaussian algorithm (see below) have now rewritten the function using Semjon Adlaj’s more compact presentation of the same:

Given 2 starting numbers x_0, y_0, define z_0=0. Then apply the map: x_{n+1}=\dfrac{x_n+y_n}{2}, \; y_{n+1}=z_n+\sqrt{(x_n-z_n)(y_n-z_n)}, \; z_{n+1}=z_n-\sqrt{(x_n-z_n)(y_n-z_n)}

When x_n=y_n within the limits of your precision stop the process (within 5..6 iterations for double precision). The number they have converged to is the variant arithmetic-geometric mean N(x_0, y_0). If k is the eccentricity and k'=\sqrt{1-k^2} then we have,

E(k) = \dfrac{\pi N(1,k'^2)}{2M(1,k')}

Thus, with the Gaussian algorithm the complete elliptic integrals or perimeter of the ellipse to any desired accuracy is as easy as that. Further, by way of the Legendre identity this also yields the extremely efficient Gaussian algorithm for calculating the value of \pi:

\pi = \dfrac{2M(k)M(k')}{N(k^2)+N(k'^2)-1}

By putting any eccentricity and its complement one can now compute \pi from it — every reader should try it out to see its sheer efficiency.

With the complete integrals in place, we were next keen apprehend the Gaussian algorithm for the incomplete integrals. After some effort with the geometric interpretation of the arithmetic-geometric mean, we realized that it was not ideal for the hand calculator and we had to use to a computer, which was not yet available at home. Nevertheless, we wrote down the algorithm and rushed to the “public computer” input it as soon as we could. It goes thus; We have as our input k the eccentricity parameter and \phi the angle defining the partial elliptical arc. We then initialize with:

x_n=1; \; y_n=\sqrt{1-k^2}

\phi_n=\phi; \; c_n=k

s_n=1-\dfrac{c_n^2}{2}; \; s'_n=0; \; t_n=1

We then iterate the below process for a desired n number of steps. For most values double precision values can be achieved within 5..6 iterations:

d_n=\arctan\left(\dfrac{(x_n-y_n)\tan(\phi_n)}{x_n+y_n\tan^2(\phi_n)}\right)

\phi_{n+1}=2\phi_n-d_n

x_{n+1}=\dfrac{x_n+y_n}{2}

c_{n+1}=\dfrac{x_n-y_n}{2}

y_{n+1}=\sqrt{x_n y_n}

s_{n+1}=s_n-t_nc_{n+1}^2

s'_{n+1}=s'_n+c_{n+1}\sin(\phi_{n+1})

t_{n+1}=2t_n

Finally, upon completing iteration n we compose the solutions for the incomplete integrals as below:

F(k, \phi)= \dfrac{\phi_{n+1}}{2^{n+1} x_{n+1}}

E(k, \phi)= s_{n+1} F(k, \phi)+s'_{n+1}

Of course one can see that this algorithm also yields the corresponding complete integrals:

K(k)=\dfrac{\pi}{2x_{n+1}}=\dfrac{\pi}{2y_{n+1}}

E(k)=s_{n+1}K(k)

It was this method by which we originally computed E(k) in our youth as Adlaj’s algorithm was published in English only later. In any case the Gauss algorithm made a profound impression on us for more than one reason. First, the connection between the convergent M(x,y) and the elliptic integrals was remarkable in itself. Second, Gauss devised this algorithm in 1799 CE when no computers were around. Being a great mental computer (a trait Gauss passed on to one of his sons) it was no issue for him; however, this method was eminently suited for computer age that was lay far in the future. Indeed, in a general sense, it reminded one iterative algorithms of the Hindus like the square root method of Chajaka-putra, the famed Cakravāla or the sine algorithm of Nityānanda. Third, as we learnt for the first time of the Gauss algorithms for the elliptic integrals, we were also exploring and discovering various iterative maps with different types of convergences: fixed points of note, fixed oscillations and strange attractors. This hinted to us the iterative algorithms were an innate feature of computational process that emerge in systems independently of the hardware (though some hardware might be better suited than others to execute them). A corollary was that various numbers underlying attractors could play a direct role in the patterns observed in structures generated by natural computational processes.

That brings us to the final part of this story, namely the relationship between the elliptic integrals and the circular trigonometric functions. As mentioned above, even in course of our futile struggle to solve the elliptic integrals in terms of elementary functions, it hit us that underlying them were elliptical equivalents of trigonometric functions. Hence, when we finally learned of these functions in our father’s book we realized that our geometric intuition about their form was informal but correct. That is shown using the Eulerian form of the ellipse in Figure 7.

Figure 7. The elliptic generalization of circular trigonometric functions

Thus, given an ellipse with semi-minor axis b=1, semi-major axis a>1 its eccentricity is k=\sqrt{1-\tfrac{1}{a^2}}. For a point A on this ellipse determined by the radial vector r (vector connecting it to origin O) and position angle \phi, we can define the following elliptic analogs of the circular trigonometric functions:

\textrm{cn}(u,k)=\dfrac{x}{a}

\textrm{sn}(u,k)=y

\textrm{dn}(u,k)=\dfrac{r}{a}

Here, the variable u is not the position angle \phi itself but is related to \phi via the integral:

u=\displaystyle \int_B^A r \cdot d\phi

When the ellipse becomes a circle, r=a=b and the above integral resolves to \phi with \textrm{cn}(\phi, 0)=\cos(\phi), \textrm{sn}(\phi,0)=\sin(\phi) and \textrm{dn}(\phi,0)=1. Further, one can see that these functions have an inverse relationship with the lemniscate arc elliptic integral F(k, \phi). We have already seen that by definition:

\displaystyle F(k,\phi) = \int_0^\phi \dfrac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}, then:

\textrm{cn}(u,k) =\cos(\phi); \; \textrm{sn}(u,k)= \sin(\phi)

The complete elliptical integral K(k) determines the period of these elliptic functions and provides the equivalent of \tfrac{\pi}{2} in circular trigonometric functions for these elliptic functions. Thus, the values of \textrm{sn}(u,k); \textrm{cn}(u,k); \textrm{dn}(u,k) will repeat at u+4nK(k), where n=1, 2, 3 \dots. Moreover,

\textrm{sn}(0,k) =0; \textrm{cn}(0,k) =1; \textrm{dn}(0,k) =1

\textrm{sn}(K(k),k) =1; \textrm{cn}(K(k),k) =0; \textrm{dn}(K(k),k) =1

Further, the geometric interpretation (Figure 7) also allows one to understand the elliptical equivalents of the fundamental trigonometric relationships:

\textrm{sn}^2(u,k)+\textrm{cn}^2(u,k)=1 \rightarrow a consequence of the definition of an ellipse.

\textrm{dn}^2(u,k) +k^2\textrm{sn}^2(u,k)=1

\textrm{dn}^2(u,k)+k^2=1+k^2\textrm{cn}^2(u,k)\; \therefore \textrm{dn}^2(u,k)= k'^2+k^2\textrm{cn}^2(u,k)

Again parallel to the circular and hyperbolic trigonometric functions, the derivatives of the elliptic functions also have parallel expressions:

\dfrac{\partial \textrm{sn}(u,k)}{\partial u}=\textrm{cn}(u,k) \textrm{dn}(u,k)

\dfrac{\partial \textrm{cn}(u,k)}{\partial u}=-\textrm{sn}(u,k) \textrm{dn}(u,k)

\dfrac{\partial \textrm{dn}(u,k)}{\partial u}=-k^2\textrm{sn}(u,k) \textrm{cn}(u,k)

At this point we will pause to make a few remarks on early history of these elliptic functions that has a romantic touch to it. While still in his early 20s, Carl Gauss studied the elliptic integrals of the first type in the context of the lemniscate arc length problem leading to the celebrated arithmetic-geometric mean algorithm that we saw above. In course of this study, he discovered that the inverse of this integral led to general versions of the circular trigonometric functions like sine and cosine. He had already discovered their basic properties, such as those stated above, and made several higher discoveries based on them. He had already realized that these were doubly periodic when considered in the complex plane. However, as was typical of him (and the luxuries of science publication in the 1700-1800s) he did not publish them formally. Almost 25 years later, the brilliant young Norwegian Niels Abel, rising like a comet in the firmament, rediscovered these results of Gauss and took them forward establishing the foundations of their modern study. One striking point was how Abel’s notation closely paralleled that of Gauss despite their independent discovery. When we learnt of this and reflected at our own limited attempt in this direction, it reinforced to us the idea that such mathematics is not created but merely discovered by tapping into a deep “Platonic” realm. Abel submitted an initial version of his work on elliptic integrals at the French National Academy; however, it seems to have been lost due to Augustin-Louis Cauchy discarding it unread among his papers. The subsequent year Abel published a more elaborate work which rediscovered Gauss’s findings.

Around the same time, the brilliant mathematician Carl Jacobi also rediscovered the same results and extended them further. This sparked a rivalry between him and Abel with a flurry of publications each bettering the other. Consequently, Legendre, the earlier pioneer of the elliptic integrals, remarked that as a result they were producing results at such a pace that it was hard for his old head to keep up with them. But this competition was to soon end with Abel slipping into deep debt from his European travels and dying shortly thereafter from tuberculosis. The Frenchman Évariste Galois, who paralleled the research of his contemporary Niels Abel in so many ways, wrote down numerous mathematical discoveries in his last letter just before his death in a duel at the age of 20. In those were found studies on the elliptic functions including rediscoveries of Abel’s work and generalizations that Jacobi was to arrive at only a little later. Ironically, in that letter he stated to his friend: “Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.” With Abel and Galois dead, the field was open to Jacobi. While he did not live much longer either, he had enough time to take their investigation to the next stage and these generalizations of circular trigonometric continue to be known as Jacobian elliptic functions.

Now again in our youth we were keen write computer functions to that could accurately output the values of these elliptic functions so that we could play with them more easily. In the process, we learned of Ramanujan blazing his own trail through the elliptic functions that led to series for evaluating them. However, the computationally most effective approach to calculate them was the Gaussian arithmetic-geometric mean algorithm which we present below. This algorithm has two parts: first, in the “ascending part” wherein we compute the iterates of the means as in the elliptic integral algorithm. Second, having stored the above iterates we “descend” with them to compute the values of corresponding \phi from which we can extract the Jacobian elliptic through the circular trigonometric functions. As input we have the variable u and the eccentricity k:

x_1=1; \; y_1=\sqrt{1-k^2}; \; c_1=k

Then we carry out a desired n iterations thus:

x_{n+1}=\dfrac{x_n+y_n}{2}

y_{n+1}= \sqrt{x_n y_n}

c_{n+1} = \dfrac{x_n - y_n}{2}

Once this is complete we compute:

\phi_{n+1}=2^{n+1} a_{n+1} u

Then we carry out the “descent” in n till n=1:

d = \arcsin \left( \dfrac{c_{n+1} \sin\left(\phi_{n+1}\right)}{a_{n+1}}\right)

\phi_n = \dfrac{\phi_{n+1}+d}{2}

Once the descent is complete we extract the Jacobian elliptic functions thus:

\textrm{sn}(u,k)=\sin\left(\phi_1\right)

\textrm{cn}(u,k)=\cos\left(\phi_1\right)

\textrm{cd}(u,k)=\cos\left(\phi_2-\phi_1\right)

\textrm{dn}(u,k) =\dfrac{\textrm{cn}(u,k)}{\textrm{cd}(u,k)}

Figure 8. \textrm{sn}(u,k) computed for u \in (0,25), k \in (0,0.99). For k=0 it is equivalent to the sine function

With this we could finally visualize the form of these elliptic functions (Figure 8): with increasing k, \textrm{sn}(u,k) develops from a sine curve to one with increasing flat crests and troughs.

We end this narration of our journey through the most basic facts pertaining to the elliptic functions with how it joined our other long-standing interest, the oval curves, and helped us derive ovals parametrized using Jacobian elliptics. From the above account of the fundamental identities of the elliptic functions and considering the derivative only with respect to u for a constant k we get:

\dfrac{d\textrm{cn}(u) }{du}= -\sqrt{(1-\textrm{cn}^2(u))(1-k^2+k^2\textrm{cn}^2(u))}

\therefore (x')^2=(1-x^2)(1-k^2+k^2x^2)

By differentiating the above again and resolving it we get the differential equation:

x''=-2k^2x^3 + (2k^2-1)x

This differential equation whose solutions take the form of the \textrm{cn}(u,k) function is a generalization of the harmonic differential equation. Having obtained we discovered much to our satisfaction that the curves parametrized by \textrm{cn}(u,k) and its derivative (i.e. position-momentum plots of dynamics defined by this DE) can take the form of ovals (Figure 9), a class of curves we were coevally investigating. These “elliptic ovals” are part of continuum ranging from elliptic hippopede-like curves to elliptic lemniscates, paralleling the continuum of classic Cassinian ovals (Figure 9). It was this intuition that led us to the discovery of the chaotic oval-like curves we had narrated earlier. These curves have an interesting property: k=\tfrac{1}{\sqrt{2}} marks a special transition value. For all k less that the solutions define concentric curves (Figure 9). For all k greater than that we get lemniscates, ovals, and centrally dimpled curves.

Figure 9. Solutions of the above \textrm{cn}(u,k) differential equation for different eccentricities k with different starting x_0 and y_0=0

Footnote 1: Already in Yajurvedic attempt recorded by authors like Baudhāyana we see an alternating pattern of positive and negative fractions of decreasing magnitude to effect convergence. The Yajurvedic formula can be written as \pi \approx 4\left(1-\tfrac{1}{a} + \tfrac{1}{a \cdot 29}- \tfrac{1}{a \cdot 29 \cdot 6}+ \tfrac{1}{a \cdot 29 \cdot 6 \cdot 8}\right); a=8 is used by Baudhāyana in his conversion. However, if we use \tfrac{853}{100} we get \pi \approx 3.1415. There is a history of such correction within the śrauta tradition recorded by ritualists like Dvārakānatha Yajvān who has a correction to Baudhāyana’s root formula giving \pi \approx \left (\tfrac{236}{39(2+\sqrt{2})}\right)^2 \approx 3.141329 indicating that in later practice values much closer to the real value were used.

Footnote 2: Something people used in the era when computers were not household items. The scientific calculator gave you most of the basic ones like the trigonometric triad and logarithms but for the rest you looked up such tables.

vyasa-krtiḥ ṣadguṇitā dvi-saṅ-guṇāyama-krtiyutā padam paridhiḥ |

vyāsa-catur-bhāga-guṇaś cāyata-vṛttasya sukṣma-phalam ||

Six times the square of the minor axis plus the square of twice its major axis; the root of this gives the perimeter. That multiplied by one fourth of its minor axis is the high precision area of the ellipse.

In modern usage the perimeter will be: P(E) \approx 2\sqrt{4a^2+6b^2}

Figure 4. Ellipse perimeter approximations

Figure 4 shows the approximate perimeters obtained from the various formulae for selected ellipses with semimajor axis a=1 and the variable semiminor axis b = .2, .4, .6, .9. The average method is the second of the above formulae. We realized that each approximation is optimized for a different types of ellipses even before we had achieved the exact value for ourselves. One can see that if the ellipse becomes a circle then Mahāvīra’s formula would become P(\bigcirc)=2\sqrt{10}r; this is keeping with his approximation of \pi as \sqrt{10}, which was a misapprehension also held by Brahmagupta contra Āryabhaṭa. Thus, it appears that he tried to “break up” that \pi between the two axes — this approximation does reasonably well at the extremes and has a maximum error of around 3.25\% for b \approx .438. In the rare instances when ellipses where used in Hindu architecture (e.g. in the maṇḍapa of the temple at Kusuma) they are usually of the proportions a=1, b=\tfrac{1}{\sqrt{2}}; for such ellipses the Hindu formula would have given an error of about 2\%.

Our explorations in this direction had set in our mind a strong desire to obtain the exact formula for the perimeters of the ellipses and parabolic arcs. Hence, like our ancestor Bhṛgu going to the great asura Varuṇa we went to our father seeking the way forward. He informed us that for achieving those objectives we needed to apprehend the further branches of calculus and that those would anyhow come as part of our curriculum in college. But we were not going to wait till then; so, he suggested that we go to the shop and get those simple textbooks a bit in advance and I could attempt to study them over the vacations. Over the next two years we made modest progress and by then we were already in junior college where elementary calculus was to start from the second semester. A prolonged shutdown from a strike gave us exactly what we wanted — the time to explore these matters by ourselves. By then, armed with the basics of the different branches of calculus, we made exciting progress for our low standards: 1) We rediscovered for ourselves the hyperbolic equivalents of the circular trigonometric functions, their derivatives and integrals. 2) We studied and (re)discovered some additional methods for constructing conics. 3) Most importantly, we non-rigorously derived for ourselves the general method of determining arc length of a curve between x=a, b using differential and integral calculus:

\displaystyle L = \int_a^b \sqrt{1+\left( \dfrac{dy}{dx} \right)^2}dx

We got our first opportunity to put it to practice when we explored the following question: Given a semicircle, how would you inscribe a circle in it? What would be the locus of the centers of such inscribed circles? The construction shown in Figure 5 provides a completely self-evident proof that the locus should a parabola with the tangent to the semicircle at its apex as its directrix and the diameter of the semicircle as its latus rectum. Nevertheless we expand it for a naive reader (Figure 5).

1) First draw the line tangent to the apex D of the semicircle.

2) To inscribe a circle that touches a semicircle at point E, join its center C to E.

3) Then draw a line perpendicular to radius \overline{CE} at E; this will be the tangent to the semicircle at E.

4) This line intersects the tangent at D at point F.

5) Drop a perpendicular from F to the diameter of the semicircle. It will cut radius \overline{CE} at G which will be the center of an inscribed circle touching the semicircle at E and its diameter at H.

6) \angle CHG = \angle GEF =\tfrac{\pi}{2} and \angle CGH = \angle EGF. Moreover, \overline{GE}=\overline{GH}. Thus, \triangle CGH \cong EGF by the side-angle-angle test. Hence, \overline{GC}=\overline{GF}.

7) Thus, for every inscribed circle its center G will be equidistant from the tangent to the semicircle at its apex and from its center C. Hence, this locus is a parabola with its focus at C and the above line as its directrix. Accordingly, the diameter of the semicircle would be its latus rectum.

Figure 5. Semicircle-parabola construction

Accordingly, we applied the above integration to this parabola (Figure 5) whose equation would be y=-\tfrac{x^2}{2a}+\tfrac{a}{2}, where a is the radius of the generating semicircle to obtain the arc length of the parabola bounded by its latus rectum:

\dfrac{dy}{dx}=-\dfrac{x}{a}

\therefore \displaystyle \int_{-a}^a \sqrt{1+\dfrac{x^2}{a^2}}dx = \left. \dfrac{x}{2}\sqrt{1+\dfrac{x^2}{a^2}}+\dfrac{a}{2}\textrm{arcsinh}(x/a)\right|_{-a}^a = a\left( \sqrt{2} + \textrm{arcsinh}(1) \right)

With this we realized that the parabola as a unique conic (i.e. fixed eccentricity), just as the circle, has an associated constant comparable to \pi that provides its arc length bounded by the latus rectum in terms of the semi-latus rectum a; hence all parabolas are like just as all circles and differ only in scale. Thus, we had rediscovered the remarkable parabolic constant, the ratio of the arc length of the parabola bounded by its latus rectum to its semi-latus rectum: P= \sqrt{2} + \textrm{arcsinh}(1) \approx 2.295587.

This also brought home to us that, unlike the circle and the parabola, the ellipse (and the hyperbola) will not have a single constant that relates their arc length to a linear dimension. Instead there will be a family of those which would be bounded by \pi and \sqrt{2} + \textrm{arcsinh}(1) latex . Our meager mind was immensely buoyed by the successful conquest of the parabola and believed that the comparable conquest of the ellipse was at hand. But hard as we tried we simply could not solve the comparable integral for the ellipse in terms of all the integration we knew. Later that summer we got to meet our cousin who was reputed to have enormous mathematical capacity but had little interest in conics. With a swagger, he said it should be easy but failed to solve it just as we had. However, he had a computer, and for the first time we could attack it with numerical integration. This gave us some intuition of how the integral specifying the arc length of an ellipse behaves and that there is a likely generalization of the circular trigonometric functions to which they might map. At that point we asked an aunt of ours, who used to teach mathematics, if she had any leads to solving that integral. She flippantly asked if we did not know of elliptic integrals? That word struck cord — not wanting to expose our ignorance further we set out to investigate it further. We went back to our father, who handed us a more “advanced” volume and told us that we were now grown up and could pursue our mathematical fancies on our own. That was indeed the case — like our ancestor Bhṛgu before he realized the Vāruṇī-vidyā. Therein we finally learned that the elliptic integrals, like the one we had battled with, were functions in themselves which could not be expressed in terms of elementary functions — there were special tables that gave their values for ellipses of different eccentricities even as we had circular and rectangular hyperbolic trigonometric functions. But those books had a terrible way of teaching elliptical integrals; hence, we had to chart our own method of presenting them for a person of modest intelligence. Once we did so we felt that these could be easily studied in their basic form along with the regular trigonometric functions.

Thus, we learned that our quest for the perimeter of the ellipse following the course from Mahāvīra through Kepler had reached the dawn of modern mathematics by converging on the famous elliptic integral which has attracted the attention of many of a great mind. The early modern attack on the perimeter of an ellipse began with Newton’s attempt with numerical integration, which we had recapitulated using a computer. In the next phase, the 26 year old Leonhard Euler, who declared it to be one great problems that had mystified geometers of the age, used some basic geometry and remarkable sleights of the hand (or should we say the mind) with the binomial theorem to prove the below series for the perimeter of an ellipse. One could say that the paper in which it appeared (“Specimen de constructione aequationum differentialium sine indeterminatarum separatione”) had a foundational role in modern mathematics:

Let d=\dfrac{a^2}{b^2}-1 then,

P(E) = 2 \pi b \left( 1+ \dfrac{1 \cdot d}{2 \cdot 2} - \dfrac{1 \cdot 1 \cdot 3 \cdot d^2}{2 \cdot 2 \cdot 4 \cdot 4} + \dfrac{1 \cdot 1 \cdot 3 \cdot 3 \cdot 5 d^3}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6} -\dfrac{1 \cdot 1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 d^4}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdot 8 \cdot 8} \dots \right)

Our numerical experiments showed that the above series yielded values close to the actual perimeter of an ellipse with 10..20 terms when its eccentricity is between 0..\tfrac{1}{\sqrt{2}}. However, below that it starts faring poorly and has increasingly poor convergence. From Euler’s original series a second series which uses the eccentricity e_E can be derived with somewhat better convergence:

Let m=e_E^2 then,

\displaystyle P(E) = 2 \pi a \left(1 - \sum_{j=1}^\infty \dfrac{1}{2j-1} \left( \dfrac{(2j)!}{(2^j j!)^2} \right)^2 m^j \right)

20 terms of this series gives values close to the real perimeter for ellipses with eccentricities in the range (0,0.95) and poorer approximations at higher eccentricities like e_E=.995. This showed to us that the problem we had struggled with was a truly worthy one and even with Euler’s attack getting precise values throughout the eccentricity range was not an easy one. This line of investigation was brought to a closure by the great Carl Gauss who in his twenties had worked out a series with superior convergence for the perimeter of the ellipse that was related to his discovery of the hypergeometric function with a profound impact on modern mathematics. This series is defined thus:

Let \displaystyle C(n) = \dfrac{\displaystyle \prod_{j=1}^n \left(\dfrac{1}{2}-j+1\right)}{n!}

Let h=\left(\dfrac{a-b}{a+b}\right)^2 then,

\displaystyle P(E) = \pi (a+b) \left(1+ \sum_{j=1}^{\infty} (C(j))^2 \cdot h^j \right)

This series gives accurate perimeters within 20 terms for e_E \in (0,0.99995). Another man who also took a similar path, although almost entirely in the isolation, was Ramanujan of Kumbhaghoṇa but that story is beyond the scope of this note. The above series was hardly the only achievement of Gauss in this direction. He had figured out an algorithm that put the final nail into this problem. Before we get to that, we shall take a detour to define the different original elliptic integrals and take a brief look at the other places we encountered them.

The ellipse \tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1 can be divided into 4 quadrants due its symmetry and in the first quadrant its equation can we written as

y=b\sqrt{1-\dfrac{x^2}{a^2}} and by the chain rule \dfrac{dy}{dx}=-\dfrac{bx}{a^2\sqrt{1-\dfrac{x^2}{a^2}}}

We make the substitution t=\tfrac{x}{a}; hence,

\dfrac{dy}{dx}=-\dfrac{bt}{a\sqrt{1-t^2}}

\therefore \sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\sqrt{1+\dfrac{b^2t^2}{a^2-a^2t^2}} =\sqrt{\dfrac{a^2 \left(1-\left(1-\frac{b^2}{a^2}\right)t^2\right)}{a^2(1-t^2)}}

We observe that e_E^2=1-\tfrac{b^2}{a^2} is the square of the eccentricity of the ellipse. Keeping with the commonly used elliptic integral convention we use e_E = k. Thus, the above expression becomes:

\sqrt{\dfrac{1-k^2t^2}{1-t^2}}

From the above substitution dx=a \cdot dt; when x=0, t=0 and when x=a, t=1. Thus, taking into account all 4 quadrants our arc length integral for the perimeter of the ellipse becomes:

\displaystyle P(E) = 4a \int_0^1 \sqrt{\dfrac{1-k^2t^2}{1-t^2}}dt

More generally, it may be expressed using the substitution t=\sin(\theta) \; \therefore dt=\cos(\theta) d\theta with which a version of the integral for an elliptic arc defined by angle \phi becomes:

\displaystyle E(k,\phi) = \int_0^\phi \sqrt{1-k^2\sin^2(\theta)} d\theta

This was the very form of the integral we had battled with in our youth before realizing that it was the definition of the elliptic integral of the second kind. One can see that for a quadrant of the ellipse with e_E=k the above integral is from 0 to \phi=\tfrac{\pi}{2}. This is then called the complete elliptic integral of the second kind and it may be simply written as E(k). Limits between other angles will give the corresponding lengths of elliptical arcs and the general integral E(k,\phi) is thus the incomplete elliptic integral of the second kind.

If this is the elliptic integral of the second kind then what is the first kind? Incomplete elliptic integral of the first kind is defined as:

\displaystyle F(k,\phi) = \int_0^\phi \dfrac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}

If the limits of this integral are taken from 0 to \phi=\tfrac{\pi}{2} then we get the complete elliptic integral of the first kind which is confusingly denoted by K(k).

The relationship between the complete integrals of the 2 kinds and \pi was discovered by the noted French mathematician Adrien-Marie Legendre, who greatly expanded their study from what Euler had done. As we shall see below, this key relationship led to the Gauss algorithm for calculating \pi most efficiently. If the eccentricity of an ellipse is k then the ellipse of complementary eccentricity k' is defined thus: k^2+k'^2=1. Then we get the Legendre relationship between the 2 kinds of complete elliptic integrals:

K(k') \cdot E(k)+E(k') \cdot K(k) - K(k') \cdot K(k) = \dfrac{\pi}{2}

We had already had our brush with K(k) earlier in our youth in course of the auto-discovery of conic-associated pedal and envelop curves recapitulating some deep history in this direction. For an ellipse or a hyperbola we can define the pedal curve as the locus of the feet of the perpendiculars dropped from the center of the conic to its tangents. We can define a second curve as the envelope of the circles whose centers lie on an ellipse or a hyperbola and which pass through the center of the said conic. We found that these two curves differ only in scale with the later being double the former in a given dimension. If a, b are the semi- major and minor axes of the parent conic, these curves have the polar equations:

\rho^2 = a^2\cos^2(\theta)+b^2\sin^2(\theta) the pedal curve for an ellipse

\rho^2 = 4\left(a^2\cos^2(\theta)+b^2\sin^2(\theta)\right) the envelope curve for an ellipse

\rho^2 = a^2\cos^2(\theta)-b^2\sin^2(\theta) the pedal curve for a hyperbola

\rho^2 = 4\left(a^2\cos^2(\theta)-b^2\sin^2(\theta)\right) the envelope curve for a hyperbola

Figure 6A. The hippopedes of an ellipse

We subsequently learned that these elliptical version of the curves was termed the hippopede by the great Proclus who investigated them (Figure 6A). He was consciously one of the last in the line of great yavana investigators of curves starting from the discovery of the conics at the Platonic academy and this died with the destruction of the Greek tradition by the “Typhonic winds” of the second Abrahamism. One observes that if a=b then the hippopedes become circles. If b=0 then again we get a pair of circles. The hyperbolic versions in contrast specify \infty-shaped curves that cross at origin. If the hyperbola is rectangular (i.e. a=b) then we get the pedal version to be: \rho^2 = a^2\left(\cos^2(\theta)-\sin^2(\theta)\right)= a^2\cos(2\theta). This curve was first studied by Jakob Bernoulli, the eldest brother of the first famous generation of the Bernoulli clan. It has an interesting property: if the ellipse is the locus of points the sum of whose distances from the two foci is a constant, the lemniscate is the locus of points the product of whose distances from the two foci is constant.

Figure 6B. The lemniscates of a hyperbola

We realized that unlike the other above curves whose arc lengths pose some terrible integrals that for the lemniscate can be reduced to a form comparable to what we got for the ellipse. Given a lemniscate with the polar equation: \rho^2 = a^2\cos(2\theta),

\dfrac{d\rho}{d\theta}=-\dfrac{a\sin(2\theta)}{\sqrt{\cos(2 \theta)}}

The arc length formula in polar coordinates is:

\displaystyle \int_a^b \sqrt{\rho^2 + \left(\dfrac{d\rho}{d\theta}\right)^2}d \theta

Thus, the arc length of the first quadrant of the lemniscate is:

\displaystyle \int_0^{\pi/4} \sqrt{a^2 \cos(2\theta)+a^2 \dfrac{\sin^2(2\theta)}{\cos(2\theta)}}d\theta= a\int_0^{\pi/4}\sqrt{\dfrac{\sin^2(2\theta)+\cos^2(2\theta)}{\cos(2\theta)}}d\theta

\displaystyle = a \int_0^{\pi/4} \dfrac{d\theta}{\sqrt{\cos(2\theta)}}

In the above we make the substitution \cos(2\theta)=\cos^2(\phi) \; \therefore -\sin(2\theta)d\theta=-2\sin(\phi)\cos(\phi)d\phi

d\theta = \dfrac{\sin(\phi)\cos(\phi)d\phi}{\sin(2\theta)}= \dfrac{\sin(\phi)\cos(\phi)d\phi}{\sqrt{1-\cos^4(\phi)}} =\dfrac{\sin(\phi)\cos(\phi)d\phi}{\sqrt{(1-\cos^2(\phi))(1+\cos^2(\phi))}}=\dfrac{\cos(\phi)d\phi}{\sqrt{1+\cos^2(\phi)}}

=\dfrac{\cos(\phi)d\phi}{\sqrt{2(1-\frac{1}{2}\sin^2(\phi))}}

This substitution results in the limits of the arc length integral changing to 0..\tfrac{\pi}{2}. Thus, it becomes:

\displaystyle = \dfrac{a}{\sqrt{2}} \int_0^{\pi/2} \dfrac{\cos(\phi)d\phi}{\cos(\phi)\sqrt{(1-\frac{1}{2}\sin^2(\phi))}}=\dfrac{a}{\sqrt{2}} \int_0^{\pi/2} \dfrac{d\phi}{\sqrt{(1-\frac{1}{2}\sin^2(\phi))}}

Thus, the perimeter of this lemniscate is:

\displaystyle P(L)=\dfrac{4a}{\sqrt{2}} \int_0^{\pi/2} \dfrac{d\phi}{\sqrt{(1-\frac{1}{2}\sin^2(\phi))}}

We see that the integral specifying the perimeter of a lemniscate is a complete elliptic integral of the first kind with as \phi=\tfrac{\pi}{2} and k=\tfrac{1}{\sqrt{2}}, i.e. K\left(\tfrac{1}{\sqrt{2}}\right). This was one of the integrals studied by Count Fagnano, a self-taught early pioneer in the calculus of elliptical and lemniscate arcs — we were amused and somewhat consoled to learn that, like us, he has initially tried to solve this integrals in terms of elementary functions and failed. Moreover, like the circle and the parabola the lemniscate is a unique curve, such that the ratio of its perimeter to its horizontal semi-axis a is a constant (the lemniscate constant) mirroring \pi and the parabolic constant P:

L= 2\sqrt{2}K\left(\tfrac{1}{\sqrt{2}}\right) \approx 5.244116

This is also a special value where we have the below relationship which can be used to compute \pi efficiently (corollary to Legendre’s identity):

2K\left(\tfrac{1}{\sqrt{2}}\right)\left(2E\left(\tfrac{1}{\sqrt{2}}\right)-K\left(\tfrac{1}{\sqrt{2}}\right)\right)=\pi

Around the time we acquired a grasp of these elliptic integrals we also learned of another practical appearance of K(k). In elementary physics one learns of simple oscillators like the pendulum and derives its period T using the basic circular trigonometric differential equation:

T \approx 2\pi\sqrt{\dfrac{l}{g}}

Here, l is the length of the pendulum and g \approx 9.8 m/s^2 is the gravitational acceleration. This emerges from an approximation for small angle oscillations where \sin(\theta) \approx \theta and corresponds to the period relationship discovered by Galileo and the apparent failed attempt of Gaṇeśa Daivajña. We had already realized that integrating the differential equation for a larger amplitude presented an integral that we had failed to solve using elementary functions. However, it can be solved with the elliptical integral of the first kind to give the accurate value for period as:

T=4\sqrt{\dfrac{l}{g}}K\left(\frac{\theta_0}{2}\right)

Here \theta_0 represents the initial angle at which the pendulum is released. One can see that if \theta_0 \to 0 then K(k) \to \tfrac{\pi}{2} giving us the low amplitude formula. Taking the standard value of g=9.80665 m/s^2 given in physics textbooks we get the period of a meter pendulum with a low amplitude displacement as T= 2.006409 s. If we instead give it a 60^\circ release then we get T=2.153242 s with the elliptic integral K\left(\sin\left(\tfrac{\pi}{6}\right)\right). Hence, one can see that the Galilean linear approximation is not a bad one for typical low angle releases.

This finally leads to what was a burning question for us in our youth: How do we effectively compute these elliptic integrals? In our opinion, this should be taught first to students and that would go some way in making the elliptics trivial as trigonometric functions. We saw the various series methods of Euler and Gauss. While the latter does quite well it is still a multi-term affair, that takes longer to converge higher the eccentricity. But the 22 year old Gauss solved this problem with a remarkable algorithm that rapidly gives you the values of these integrals — something, which in our early days, we had even done with a hand calculator while teaching it to a physics student. Right then, Gauss realized that it “opens an entirely new field of analysis” as he wrote in his notes accompanying the discovery. This is the famous arithmetic-geometric mean M algorithm which goes thus:

Given 2 starting numbers x_0, y_0, apply the map: x_{n+1}=\dfrac{x_n+y_n}{2}, \; y_{n+1}=\sqrt{x_n \cdot y_n}.

The map converges usually within 5 iterations for typical double precision values to the arithmetic-geometric mean M(x_0,y_0). Let k be the eccentricity value for which we wish to compute the complete elliptic integral of the first kind. k'=\sqrt{1-k^2} then we have,

K(k)=\dfrac{\pi}{2M(1,k')}

For E(k) we used to originally use a Gaussian algorithm (see below) have now rewritten the function using Semjon Adlaj’s more compact presentation of the same:

Given 2 starting numbers x_0, y_0, define z_0=0. Then apply the map: x_{n+1}=\dfrac{x_n+y_n}{2}, \; y_{n+1}=z_n+\sqrt{(x_n-z_n)(y_n-z_n)}, \; z_{n+1}=z_n-\sqrt{(x_n-z_n)(y_n-z_n)}

When x_n=y_n within the limits of your precision stop the process (within 5..6 iterations for double precision). The number they have converged to is the variant arithmetic-geometric mean N(x_0, y_0). If k is the eccentricity and k'=\sqrt{1-k^2} then we have,

E(k) = \dfrac{\pi N(1,k'^2)}{2M(1,k')}

Thus, with the Gaussian algorithm the complete elliptic integrals or perimeter of the ellipse to any desired accuracy is as easy as that. Further, by way of the Legendre identity this also yields the extremely efficient Gaussian algorithm for calculating the value of \pi:

\pi = \dfrac{2M(k)M(k')}{N(k^2)+N(k'^2)-1}

By putting any eccentricity and its complement one can now compute \pi from it — every reader should try it out to see its sheer efficiency.

With the complete integrals in place, we were next keen apprehend the Gaussian algorithm for the incomplete integrals. After some effort with the geometric interpretation of the arithmetic-geometric mean, we realized that it was not ideal for the hand calculator and we had to use to a computer, which was not yet available at home. Nevertheless, we wrote down the algorithm and rushed to the “public computer” input it as soon as we could. It goes thus; We have as our input k the eccentricity parameter and \phi the angle defining the partial elliptical arc. We then initialize with:

x_n=1; \; y_n=\sqrt{1-k^2}

\phi_n=\phi; \; c_n=k

s_n=1-\dfrac{c_n^2}{2}; \; s'_n=0; \; t_n=1

We then iterate the below process for a desired n number of steps. For most values double precision values can be achieved within 5..6 iterations:

d_n=\arctan\left(\dfrac{(x_n-y_n)\tan(\phi_n)}{x_n+y_n\tan^2(\phi_n)}\right)

\phi_{n+1}=2\phi_n-d_n

x_{n+1}=\dfrac{x_n+y_n}{2}

c_{n+1}=\dfrac{x_n-y_n}{2}

y_{n+1}=\sqrt{x_n y_n}

s_{n+1}=s_n-t_nc_{n+1}^2

s'_{n+1}=s'_n+c_{n+1}\sin(\phi_{n+1})

t_{n+1}=2t_n

Finally, upon completing iteration n we compose the solutions for the incomplete integrals as below:

F(k, \phi)= \dfrac{\phi_{n+1}}{2^{n+1} x_{n+1}}

E(k, \phi)= s_{n+1} F(k, \phi)+s'_{n+1}

Of course one can see that this algorithm also yields the corresponding complete integrals:

K(k)=\dfrac{\pi}{2x_{n+1}}=\dfrac{\pi}{2y_{n+1}}

E(k)=s_{n+1}K(k)

It was this method by which we originally computed E(k) in our youth as Adlaj’s algorithm was published in English only later. In any case the Gauss algorithm made a profound impression on us for more than one reason. First, the connection between the convergent M(x,y) and the elliptic integrals was remarkable in itself. Second, Gauss devised this algorithm in 1799 CE when no computers were around. Being a great mental computer (a trait Gauss passed on to one of his sons) it was no issue for him; however, this method was eminently suited for computer age that was lay far in the future. Indeed, in a general sense, it reminded one iterative algorithms of the Hindus like the square root method of Chajaka-putra, the famed Cakravāla or the sine algorithm of Nityānanda. Third, as we learnt for the first time of the Gauss algorithms for the elliptic integrals, we were also exploring and discovering various iterative maps with different types of convergences: fixed points of note, fixed oscillations and strange attractors. This hinted to us the iterative algorithms were an innate feature of computational process that emerge in systems independently of the hardware (though some hardware might be better suited than others to execute them). A corollary was that various numbers underlying attractors could play a direct role in the patterns observed in structures generated by natural computational processes.

That brings us to the final part of this story, namely the relationship between the elliptic integrals and the circular trigonometric functions. As mentioned above, even in course of our futile struggle to solve the elliptic integrals in terms of elementary functions, it hit us that underlying them were elliptical equivalents of trigonometric functions. Hence, when we finally learned of these functions in our father’s book we realized that our geometric intuition about their form was informal but correct. That is shown using the Eulerian form of the ellipse in Figure 7.

Figure 7. The elliptic generalization of circular trigonometric functions

Thus, given an ellipse with semi-minor axis b=1, semi-major axis a>1 its eccentricity is k=\sqrt{1-\tfrac{1}{a^2}}. For a point A on this ellipse determined by the radial vector r (vector connecting it to origin O) and position angle \phi, we can define the following elliptic analogs of the circular trigonometric functions:

\textrm{cn}(u,k)=\dfrac{x}{a}

\textrm{sn}(u,k)=y

\textrm{dn}(u,k)=\dfrac{r}{a}

Here, the variable u is not the position angle \phi itself but is related to \phi via the integral:

u=\displaystyle \int_B^A r \cdot d\phi

When the ellipse becomes a circle, r=a=b and the above integral resolves to \phi with \textrm{cn}(\phi, 0)=\cos(\phi), \textrm{sn}(\phi,0)=\sin(\phi) and \textrm{dn}(\phi,0)=1. Further, one can see that these functions have an inverse relationship with the lemniscate arc elliptic integral F(k, \phi). We have already seen that by definition:

\displaystyle F(k,\phi) = \int_0^\phi \dfrac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}, then:

\textrm{cn}(u,k) =\cos(\phi); \; \textrm{sn}(u,k)= \sin(\phi)

The complete elliptical integral K(k) determines the period of these elliptic functions and provides the equivalent of \tfrac{\pi}{2} in circular trigonometric functions for these elliptic functions. Thus, the values of \textrm{sn}(u,k); \textrm{cn}(u,k); \textrm{dn}(u,k) will repeat at u+4nK(k), where n=1, 2, 3 \dots. Moreover,

\textrm{sn}(0,k) =0; \textrm{cn}(0,k) =1; \textrm{dn}(0,k) =1

\textrm{sn}(K(k),k) =1; \textrm{cn}(K(k),k) =0; \textrm{dn}(K(k),k) =1

Further, the geometric interpretation (Figure 7) also allows one to understand the elliptical equivalents of the fundamental trigonometric relationships:

\textrm{sn}^2(u,k)+\textrm{cn}^2(u,k)=1 \rightarrow a consequence of the definition of an ellipse.

\textrm{dn}^2(u,k) +k^2\textrm{sn}^2(u,k)=1

\textrm{dn}^2(u,k)+k^2=1+k^2\textrm{cn}^2(u,k)\; \therefore \textrm{dn}^2(u,k)= k'^2+k^2\textrm{cn}^2(u,k)

Again parallel to the circular and hyperbolic trigonometric functions, the derivatives of the elliptic functions also have parallel expressions:

\dfrac{\partial \textrm{sn}(u,k)}{\partial u}=\textrm{cn}(u,k) \textrm{dn}(u,k)

\dfrac{\partial \textrm{cn}(u,k)}{\partial u}=-\textrm{sn}(u,k) \textrm{dn}(u,k)

\dfrac{\partial \textrm{dn}(u,k)}{\partial u}=-k^2\textrm{sn}(u,k) \textrm{cn}(u,k)

At this point we will pause to make a few remarks on early history of these elliptic functions that has a romantic touch to it. While still in his early 20s, Carl Gauss studied the elliptic integrals of the first type in the context of the lemniscate arc length problem leading to the celebrated arithmetic-geometric mean algorithm that we saw above. In course of this study, he discovered that the inverse of this integral led to general versions of the circular trigonometric functions like sine and cosine. He had already discovered their basic properties, such as those stated above, and made several higher discoveries based on them. He had already realized that these were doubly periodic when considered in the complex plane. However, as was typical of him (and the luxuries of science publication in the 1700-1800s) he did not publish them formally. Almost 25 years later, the brilliant young Norwegian Niels Abel, rising like a comet in the firmament, rediscovered these results of Gauss and took them forward establishing the foundations of their modern study. One striking point was how Abel’s notation closely paralleled that of Gauss despite their independent discovery. When we learnt of this and reflected at our own limited attempt in this direction, it reinforced to us the idea that such mathematics is not created but merely discovered by tapping into a deep “Platonic” realm. Abel submitted an initial version of his work on elliptic integrals at the French National Academy; however, it seems to have been lost due to Augustin-Louis Cauchy discarding it unread among his papers. The subsequent year Abel published a more elaborate work which rediscovered Gauss’s findings.

Around the same time, the brilliant mathematician Carl Jacobi also rediscovered the same results and extended them further. This sparked a rivalry between him and Abel with a flurry of publications each bettering the other. Consequently, Legendre, the earlier pioneer of the elliptic integrals, remarked that as a result they were producing results at such a pace that it was hard for his old head to keep up with them. But this competition was to soon end with Abel slipping into deep debt from his European travels and dying shortly thereafter from tuberculosis. The Frenchman Évariste Galois, who paralleled the research of his contemporary Niels Abel in so many ways, wrote down numerous mathematical discoveries in his last letter just before his death in a duel at the age of 20. In those were found studies on the elliptic functions including rediscoveries of Abel’s work and generalizations that Jacobi was to arrive at only a little later. Ironically, in that letter he stated to his friend: “Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.” With Abel and Galois dead, the field was open to Jacobi. While he did not live much longer either, he had enough time to take their investigation to the next stage and these generalizations of circular trigonometric continue to be known as Jacobian elliptic functions.

Now again in our youth we were keen write computer functions to that could accurately output the values of these elliptic functions so that we could play with them more easily. In the process, we learned of Ramanujan blazing his own trail through the elliptic functions that led to series for evaluating them. However, the computationally most effective approach to calculate them was the Gaussian arithmetic-geometric mean algorithm which we present below. This algorithm has two parts: first, in the “ascending part” wherein we compute the iterates of the means as in the elliptic integral algorithm. Second, having stored the above iterates we “descend” with them to compute the values of corresponding \phi from which we can extract the Jacobian elliptic through the circular trigonometric functions. As input we have the variable u and the eccentricity k:

x_1=1; \; y_1=\sqrt{1-k^2}; \; c_1=k

Then we carry out a desired n iterations thus:

x_{n+1}=\dfrac{x_n+y_n}{2}

y_{n+1}= \sqrt{x_n y_n}

c_{n+1} = \dfrac{x_n - y_n}{2}

Once this is complete we compute:

\phi_{n+1}=2^{n+1} a_{n+1} u

Then we carry out the “descent” in n till n=1:

d = \arcsin \left( \dfrac{c_{n+1} \sin\left(\phi_{n+1}\right)}{a_{n+1}}\right)

\phi_n = \dfrac{\phi_{n+1}+d}{2}

Once the descent is complete we extract the Jacobian elliptic functions thus:

\textrm{sn}(u,k)=\sin\left(\phi_1\right)

\textrm{cn}(u,k)=\cos\left(\phi_1\right)

\textrm{cd}(u,k)=\cos\left(\phi_2-\phi_1\right)

\textrm{dn}(u,k) =\dfrac{\textrm{cn}(u,k)}{\textrm{cd}(u,k)}

Figure 8. \textrm{sn}(u,k) computed for u \in (0,25), k \in (0,0.99). For k=0 it is equivalent to the sine function

With this we could finally visualize the form of these elliptic functions (Figure 8): with increasing k, \textrm{sn}(u,k) develops from a sine curve to one with increasing flat crests and troughs.

We end this narration of our journey through the most basic facts pertaining to the elliptic functions with how it joined our other long-standing interest, the oval curves, and helped us derive ovals parametrized using Jacobian elliptics. From the above account of the fundamental identities of the elliptic functions and considering the derivative only with respect to u for a constant k we get:

\dfrac{d\textrm{cn}(u) }{du}= -\sqrt{(1-\textrm{cn}^2(u))(1-k^2+k^2\textrm{cn}^2(u))}

\therefore (x')^2=(1-x^2)(1-k^2+k^2x^2)

By differentiating the above again and resolving it we get the differential equation:

x''=-2k^2x^3 + (2k^2-1)x

This differential equation whose solutions take the form of the \textrm{cn}(u,k) function is a generalization of the harmonic differential equation. Having obtained we discovered much to our satisfaction that the curves parametrized by \textrm{cn}(u,k) and its derivative (i.e. position-momentum plots of dynamics defined by this DE) can take the form of ovals (Figure 9), a class of curves we were coevally investigating. These “elliptic ovals” are part of continuum ranging from elliptic hippopede-like curves to elliptic lemniscates, paralleling the continuum of classic Cassinian ovals (Figure 9). It was this intuition that led us to the discovery of the chaotic oval-like curves we had narrated earlier. These curves have an interesting property: k=\tfrac{1}{\sqrt{2}} marks a special transition value. For all k less that the solutions define concentric curves (Figure 9). For all k greater than that we get lemniscates, ovals, and centrally dimpled curves.

Figure 9. Solutions of the above \textrm{cn}(u,k) differential equation for different eccentricities k with different starting x_0 and y_0=0

Footnote 1: Already in the Yajurvedic attempt recorded by authors like Baudhāyana we see an alternating pattern of positive and negative fractions of decreasing magnitude to effect convergence. The Yajurvedic formula can be written as \pi \approx 4\left(1-\tfrac{1}{a} + \tfrac{1}{a \cdot 29}- \tfrac{1}{a \cdot 29 \cdot 6}+ \tfrac{1}{a \cdot 29 \cdot 6 \cdot 8}\right); a=8 is used by Baudhāyana in his conversion. However, if we use \tfrac{853}{100} we get \pi \approx 3.1415. There is a history of such correction within the śrauta tradition recorded by ritualists like Dvārakānatha Yajvān who has a correction to Baudhāyana’s root formula giving \pi \approx \left (\tfrac{236}{39(2+\sqrt{2})}\right)^2 \approx 3.141329 indicating that in later practice values much closer to the real value were used.

Footnote 2: Something people used in the era when computers were not household items. The scientific calculator gave you most of the basic ones like the trigonometric triad and logarithms but for the rest you looked up such tables.

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Generalizations of the prime sieve and Pi

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Eratosthenes, the preeminent yavana philosopher of early Ptolemaic Egypt [footnote 1], composed a hymn to the god Hermes of which only some fragments have come down to us. This connection to Hermes is evidently related to his Egyptian locus, where the old ritual-experts saw all manner of clever inventions and ritual ordinances as being set down by their god Thoth, who was syncretized with Hermes of the yavana conquerors of Egypt. In this hymn, Eratosthenes describes Hermes as looking down on universe from the highest sphere of heaven (\sim Ārya parame vyoman [footnote 2]. As he did so, Hermes is mentioned as perceiving the harmony of the spheres of the planets and the world axis passing through the earth in the center. Eratosthenes represented these harmonies in the form of the lyre that was invented by the god Hermes and gifted to the god Apollo. Eratosthenes, held that while surveying the universe from the highest sphere, Hermes saw that the “harmony of the spheres” was the same as the harmony of his lyre. This equivalence of the harmonies was seen by these yavanācarya-s as illustrating the desmos (equivalent of Ārya saṃbandha: the common thread, bindings, equivalences) that runs across different branches of mathematics (noted by Friedrich Solmsen who brought to light the religious background of Eratosthenes, something that is ignored by those who wish to paint him in the image of a modern “scientist” of the Occident ). Indeed, similar “harmonies” were perceived by the yavana sage Pythagoras and before him in the form of the numerical sambandha-s of the Ārya-s of the Yajurveda — such mysterious numerical patterns and conjunctions bring together apparent disparate branches of mathematics.

Keeping with the god with whom he had a special connection, Eratosthenes was the inventive kind, who, while a Platonist (made clear by the sage Iamblichus), was somewhat unlike those of the pure geometric school — he described to king Ptolemaios a method of doubling the Delian altar of Apollo with a machine rather than the geometric constructions of Eudoxus (believed to be divinely inspired) using the curve known as the kampyle (y^{2}=\tfrac{x^{4}}{a^{2}}-x^{2}) or the conics used by Plato’s associates Menaechmus and Dinostratus. Likewise, he was more like the Ārya-s in his algorithmic methods pertaining to numbers — perhaps most famously the sieve for prime numbers is attributed to him. We had the experience of such a sambandha while exploring the prime sieve and its generalizations.

The prime sieve is a simple but powerful algorithm for extracting the sequence of prime numbers p that one might have learned in elementary school. While a product of the ancient world, its power is best appreciated in the modern computer age and is recommended to students as one of the first computer programs to write. In its modern form it goes thus (the ancient yavana concept of numbers was geometric and not exactly the same as we might take it be):

• Write out the sequence of integers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 \dots

• Move the first integer in this sequence to sequence p and kill all its multiples from the above sequence (circled):

• This yields a new sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 \dots Again move the first term of this sequence to sequence p and kill its multiples in this sequence (circled):

• Repeat this procedure with 5, 7, 11, 13, 17, 19 \dots for as many cycles as required. Thus, you get the sequence of prime numbers p: 2, 3, 5, 7, 11, 13, 17, 19 \dots

Millennia after Eratosthenes, in the 1950s, Jabotinsky generalized this sieve algorithm to generate other notable sequences. The first of these goes thus:

• Write out the sequence of integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 \dots

• Move the first integer in this sequence to sequence a and kill all integers by jumping by a skip of size equal to that first integer, starting from that first integer of the above sequence (circled):

• This yields a new sequence: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \dots Again move the first term of this sequence to sequence a and kill all integers by jumping with a skip equal to the new first integer starting from it (circled):

• This yields a new sequence: 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40 \dots. We illustrate the next few steps below:

• Thus, the above sieve yields the sequence

a=1, 2, 4, 6, 10, 12, 16, 22, 24, 28, 36, 40, 42, 46, 52, 60, 66, 70, 76, 82 \dots

The form of the sequence originally published by Erdős and Jabotinsky is the above sequence plus 1 (Sometimes called Ludic numbers). i.e.:

2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53 \dots

This sequence can also be generated by applying a sieve just as above starting with an initial sequence 2, 3, 4, 5, 6 \dots and killing integers with a skip of size equal to first term of the intermediate sequence minus 1. As one can see, this version of the sequence includes certain primes and certain odd numbers which are not primes (e.g. 25, 77, 91, 115, 119, 121, 143, 161, 175 …).

We know that prime number sequence p has a scaling which can be asymptotically represented by p[n] \sim n\log(n). Now we can ask how does this sequence a generated by the generalized sieve scale (Figure 1)?

Figure 1. Scaling of sequence a.

We see that a grows more rapidly than p (Figure 1, black line) though it must be asymptotic to the latter. Thus, there must be additional terms to get a better asymptotic fit than n\log(n) (Figure 1, gray line). Figure 1 shows at till the term 10000 a lies between:

n\log(n)+\tfrac{1}{2}n(\log(\log(n)))^2 (brown dotted line) and

n\log(n)+\tfrac{1}{2}n(\log(\log(n)))^2 + (2-\gamma)n\log(\log(n)) (brown solid line),

where \gamma = -\textrm{di}\gamma(1) \approx 0.5772157 is Euler’s constant.

Erdős and Jabotinsky have shown that for large n the growth of sequence a exceeds the above higher bound by a further correction term.

We can generate another related sequence by a similar sieve where we kill the terms by skipping over 2 \times the value first term. This results in the sequence

b: 1, 2, 3, 5, 6, 8, 11, 12, 14, 18, 20, 21, 23, 26, 30, 33, 35, 38, 41, 44 \dots

Unlike the sequence a it includes both odd and even numbers and one notices that b[n]=\tfrac{a[n]}{2} starting from the term a[2].

As one can see it scales at half rate of the sequence a. Thus, this type of generalized sieve reveals that sequences arising from a process, where the skip is dependent on the first term of the intermediate sequence, are asymptotic with a function whose base term is of the form a_n \sim k n\log(n), where k is a constant. One may say that of these the one which yields the primes is some kind of an optimal sieve that lets neither too few nor two many numbers pass through it. Thus, intuitively, the primes can be seen as an inherent optimal path through the number world.

The next type of sieve of Jabotinsky is operated thus:

• Write out the sequence of integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 \dots

• Move the first integer in this sequence to sequence f and kill all integers by jumping by a skip of size 1 starting from the first integer of the above sequence (circled):

• This yields a new sequence: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 \dots Now we operate on this sequence by sending the first term to sequence f and starting from it killing all terms by jump with a skip of 2 (circled):

• This yields a new sequence: 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54 \dots We repeat the above procedure, killing terms in each iteration with skips of 3, 4, 5… As a result we get the sequence:

f: 1, 2, 4, 6, 10, 12, 18, 22, 30, 34, 42, 48 \dots

This sequence relates to a function on positive integers \textrm{jab}(n); we illustrate this function by the example of \textrm{jab}(8):

• We take 8 and reduce it by 1 to get 7.

• We then get the lowest multiple of 7 that is greater than 8 \rightarrow 14.

• We then reduce 7 by 1 \rightarrow 6 and obtain lowest multiple of 6 that is greater than 14 \rightarrow 18.

• We continue this procedure till the successive reduction ends in 1. The corresponding sequence of their multiples is:

8 \rightarrow 14 \rightarrow 18 \rightarrow 20 \rightarrow 20 \rightarrow 21 \rightarrow 22 \rightarrow 22

• Thus, we get \textrm{jab}(8)=22

Remarkably, it turns out that f is the sequence of \textrm{jab}(n).

Figure 2. Scaling of sequence f

Even more remarkably, f[n] = \tfrac{n^2}{\pi}+c; where c is a correction term. Erdős and Jabotinsky showed that c=\mathcal{O}(x) in the big \mathcal{O} notation. For relatively small n we can safely write f[n] \approx \tfrac{n^2}{\pi}, i.e. a parabolic scaling with \tfrac{1}{\pi} as the coefficient (brown line in Figure 2). Thus, the generalized sieving procedure yields a sequence that provides a mysterious “desmos” to \pi yet again linking disparate branches of mathematics. More generally, it points that beneath these branches lie relatively simply computational processes such as the sieve, whose original form was attributed to Eratosthenes, which yields a variety of entities organically, like prime numbers or \pi.

Finally, this parabolic scaling with \pi as the constant also brings to mind an interesting iterative generator of \pi and e, apparently first discovered by Cloitre:

• Initiate two maps with the terms x_1=y_1=0 and x_2=y_2=1

• Iterate the maps as x_n=x_{n-1}+\tfrac{x_{n-2}}{n-2} and y_n=\tfrac{y_{n-1}}{n-2}+y_{n-2}

x_n scales as \tfrac{n}{e} and y_n scales as \sqrt{\tfrac{2n}{\pi}}

Figure 3. Maps generating e and pi.

Here, while the map generates e via a linear relationship it generated \pi via a parabolic relationship.


Footnote 1: Sometimes he is called the new or second Plato; Archimedes addresses him as philosophías proestō̃ta

Footnote 2: This evidently derives from an old IE tradition as one can also compare it with the phrase used for the cognate deity Pūṣaṇ: saṃcakṣāṇo bhuvanā deva īyate by the Aṅgiras, Bharadvāja Bārhaspatya, in RV 6.58.2. In Greek world, there were two evolutes descending from the Proto-Indo-European cognate of Pūṣaṇ: Hermes and Pan

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Pandemic days: the fizz is out of the bottle

Just this morning our brother remarked that the fear of the virus has inverted this month with respect to the actual infection. We literally hear this: in the past few months, while home-bound, we at least had aural quietness for most of the day, something we generally prefer. Since the lock-down was lifted in our regions over the past two days we hear the loud blaring of music from the restaurant next door even as diners flock to it in throngs. We can only see those seated outside and do not know if any are seated inside. If it is only the former, it is a lesser evil, but the music is any case an annoyance for the still home-bound residents. A walk to the deśi grocery store revealed a similar situation on the street and with other restaurants. Masks, social distancing and the like are mostly breaking down. Most diners and ramblers are young individuals who clearly know that they are at lower risk but the danger they have inflicted on the rest of society, if any, will be known over this month. However, there are much bigger things than that are happening.

Humans are social animals and we have been wondering how the lock-down and social distancing was going to spill over into the psychological domain. Right from the early stages of the pandemic when it was still mostly confined to the Cīna-s, we heard reports of the psychological impact of being cooped up at home. The Americans, in particular, are a very outdoor people, especially with summer making its advent. Hence, the spring lock-down has clearly had a negative effect on their psyche and over the last two weeks it has exploded in the form of an extensive jana-kopa. With the jana-kopa unfolding, the political players seeking to take advantage of it have decided to throw social distancing to the winds and we are now witnessing massive congregations of people. Earlier, the politicians of the opposite pakṣa were mainly the ones against lock-downs. Ironically, the politicians cheering for the current congregations are the very ones who were in favor of social distancing and strong lock-downs until two weeks ago. The virus, of course, does not care for the whims of either pakṣa, and what is unfolding might end up teaching us some important epidemiological lessons: how much does mass social aggregation outdoors really contribute to viral spread? However, that might be conflated with other factors like the end of the lock-down and consequent social aggregation we mentioned above.

The flash-point for this jana-kopa were a set of incidents relating to the well-known and long-standing racial conflict which is baked into the foundation of the USA. However, that is not the primary topic of our discussion here. We primarily intend to touch upon how an outbreak of a memetic viral pandemic rapidly overtook the original cause for the flash-point of the jana-kopa. Here, we are seeing a pratyakṣa of how a real viral infection interacts with a memetic pathogen, much like how a viral infection could facilitate in a secondary bacterial infection in biological pathology.

Before we look at that, we need to have quick survey of the lay of land that facilitates the spread of the latest strain of the memetic infection. As we have noted before, the world has come a long way from the sārthavāha-s of yore who even when wide-ranging did not disrupt the local systems. For instance, in say the Marahaṭṭā country, where we formerly lived, there was the system of the bārā balutedāra-s which has perhaps lasted in some form from the pre-Āryan days of the Harappan civilization. This system comprised of 12 jāti-s who offered essential services for a village. These spanned the entire range from the jāti-s with a low social rank to those with a large component of steppe ancestry and old Āryan descent who were high-ranked. Irrespective of the real or imagined the local issues rising from inter-jāti conflicts, these still for most part worked as a rather cohesive unit for probably at least 1400 years. The sārthavāha operated on top of this system and bore produce across the country and beyond. However, such vaṇij-s did not do anything to interfere disruptively with the system in the villages. The modern sārthavāha built by the stock market or the internet is mostly a contrasting species. It is clear they have greater reach and rapacity than the old sārthavāha-s but they are also prone to more evil. There is an entire spectrum of the balance of evil they cause to the utility they provide to society. The self-enriching mithyā-kraya-kartṛ Sora has absolutely nothing useful to offer to the world and a purely evil player. The vyāpārin Bejha offers several useful services for the jana-samudāya but has leveraged the wealth he has accumulated in the process to do evil. Among the corporations, karṇa-piśācī guggulu might offer some small services of use to the populace but has used its capture of an important resource to further evil. A similar situation is seen with the duṣṭa, Jāka the dāraka: he offers a platform for online advertisement and conversation but has used it to further evil causes and contribute to the demise of freedom of expression. Probably, Mukhagiri also falls in the same category as Jāka the dāraka.

Why are these modern mleccha and mūlavātūla sārthavāha-s such duṣṭa-s? We posit that it has two parts to it: 1) They are to a degree conscious that their rapacious actions cause a measure of harm to individuals and the emergent entity known as society which they comprise. Moreover, in the past the mleccha and mūla-rugṇa sārthavāha-s used to practice what came to be known as capitalism in an unbridled way and they saw that it acquired a bad name. So there has been some urge in the midst of at least some of them to perform a bit of parihāra. Like all humans they too subliminally subscribe to the view that they can gain compensatory puṇya by performing acts in support of religion. In the past, the sārthavāha-s in our midst too, keen to acquire puṇya, used to make donations to various deva-prāsāda-s or build new ones themselves (or pāṣaṇdālaya-s). 2) The original ekarākṣasonmāda-s had a single rakṣas as the object of their worship. They took the jabberings of their unmatta-s — much like one seized by a rākṣasa-graha — to be the very words of this rakṣas and as the basis of their impious systems. Closer to our times, starting with durātman-s Marx and Engels, there arose among the mleccha-s and the ādipraiṣhaka-s a mutation of the old unmāda-s where the rakṣas count was reduced from 1 to 0. However, as we have noted before, the rest of the unmāda, such as the promise of a utopia on earth or in Lalaland remained intact. This version of the unmāda is now the ascendant religion among the employees of or of these very sārthavāha-s themselves. Thus, what has happened is that they seek to earn puṇya by granting endowments for this new abrahma-ruj.

Even if one were not infected fully by this secular mutation of the old unmāda-s, among the lay mleccha-s there is a certain naive econolatory: they worship the faceless market running by its own vrata-s and attribute all human abhyudaya to these magical market forces. They see it as self-correcting and the panacea for all evil — whereas in the past they used see the prathama and dvitīya unmāda-s as the foundation of their well-being they now either additionally or exclusively genuflect towards the miraculous market. In this worship of the market, especially given that it is faceless, its purveyors, the vaṇij, naturally become important idols in society. Now in itself this might have not been specially negative with the vaṇij were dharma-kartṛ-s. However, as we saw above a significant fraction of the mleccha and ādi-rugṇa vaṇij-s have been infected by the caturthonmāda. As a consequence, these idols of society are not purveying the neutral capitalism that the mleccha elite worship but a new unmāda. This, along with their global reach, has allowed that unmāda to attain pandemic proportions.

One may ask had it not already done so already in the heydays of the Soviets and Mao, leaking over to India as the Naxal rudhironmāda? Yes, it had but there are important distinctions with respect to the current strain. First, the old strain was very much dependent on the Soviet continuation of the old Rus empire. Once that fell, it was like a chariot that had lost a wheel. While it briefly infected the Cīna-s, what people do not get is that it was domesticated to the ultimate advantage of the Cīna-s as it encountered an even more potent endogenous coopter in the form Wei Yang Lord Shang’s legalism. Thus, the old strain was mostly a failure. However, a lineage of it survived in mleccha academia where it was incubated in the Ivy League soft departments and their more pedestrian arborizations as a local disease, much like a virus jumping from animal to animal in a Cīna “wet-market”. Like SARS-1 it has had localized outbreaks and was mostly used a weapon by mleccha-s against others like the Hindus — e.g. to reinvigorate the Naxals in the form of their urban variant (“the urban Naxals” as a Bollywood producer terms it). Its local outbreaks were mostly seen in the mleccha academia and news media. A few years ago we heard of a left-leaning professor couple being forced out of a mleccha liberal arts college by a particularly sustained outbreak of this disease. Some other, otherwise rather neutral and harmless, mleccha physicians/professors were also seriously targeted but being part of the Ivy elite themselves they suffered little from the attacks. Similarly, there was a mlecchikā who used to originally think of the rākṣasonmāda as a great blessing on humanity. However, as it happens with that disease, when you get too close to it you get the full blast of its stench. Thus, she suffered an incident of rākṣasa-juṣṭi and suddenly saw light the truly hard way. That also opened her eyes to the caturthonmāda and immediately brought her under attack from those infected by it. Thus, those who were observant and discerning knew that this extremely dangerous mānasika-ruj was just waiting to jump out into the wider population like a viral pandemic.

Rather notably, the jana-kopa triggered by the kṛṣṇa-mleccha-saṃgrāma in big mleccha-land along with the psychological and economic fallout of the Wuhan disease finally allowed this unmāda to break out as a pandemic itself – indeed we see it catching fire across the mleccha world. Like the Wuhan corruption it might eventually hit Bhārata too. Already it was used by the duṣṭa Sora against Bhārata in conjunction with the marūnmatta-s in the recent past. A part of the urban Bhārata elite are nothing other than mleccha-s with an asita-tvak; hence, they would get similarly afflicted by the disease. Like all the prior abrahma-ruj-s, this latest strain retains its inherent hate for dharma and theoclasm. Hence, the Hindu elite will need to become aware of this dangerous disease and not be fooled by the subterfuge it practices of hiding beneath other causes of jana-kopa. One thing that will show its face to Hindus will be its collaboration with marūnmāda and its śanti-dūta-s and pretonmāda and its prema-dūta-s in Bhārata.

While abrahma-ruj-s are of the mānasika kind they have some of the same behaviors and consequences as a jivaruj. One might think that one is cured of it but it might have merely has mutated in the infected person’s mind and persist as a different version of the disease even as persistent bacterial infection. As an example, we have a TSPian ex-marūnmattikā, who is popular in mleccha secular circles and even supported by Hindus, who exhibits the same hate for H as a full-blown marūnmatta. Dick Dawkins is another example of the same. Here, the disease has assumed a new persistent form while superficially appearing as though cured. Even if one is truly cured of an abrahma-ruj one can have a lifelong chronic inflammatory condition — this is seen among some mleccha atheists and neo-heathens. That inflammatory condition can be seen in the form of destructive dur-vāsana-s persisting among them. Given this background, our general prognosis is that we are not going to see the mleccha-s especially the mahā-mleccha come out of this. It might not be this incident per say but this might be one in a series of many that will eventually result in the decline and perhaps even demise of the mahā-mleccha. The only lucky thing for them is that they have no challengers right now who could exploit it well enough to give them the coup de grace.

So what might play out? Given that the current strain is still an unmāda it will compete vigorously with other unmāda-s for its host. Since, none of these unmāda-s kill the host the battle is not going to lead to attenuation in anyway. Finally, the second amendment of the mahāmleccha constitution and those who know how to make use of it will be the last line of defense for the pretonmāda. So these and the those infected by the new strain will clash. The former have the advantage of fecundity while the latter of propaganda and the economic backing from the converted vyāpārin-s. The biological virus is still not done and could still stir the mix. At this stage even though the new strain will clearly win in the urban centers it is not going to yet take the rural lands. We do not know if that failure would allow it to gain a total victory. In any case we can say that already a covert war between two unmāda-s, like the crusade and the jihād, is under way among the maha-mleccha. In Europe the fourth strain will win and in turn it would likely face defeat at the hands of the more vigorous marūnmāda. Marūnmāda’s foothold in mahāmleccha is still weak; hence, while they will take the side of the 4th strain the basic resolution of the war remains unclear. Whoever wins, the mahāmleccha will be weakened as a whole. But from a Hindu perspective, the pakṣas locked in yuddha among the mlecchas will just be like being offered a choice between the evil Qarmathians and the monstrous Ghaznavids.

Posted in History, Life, Politics | Tagged , , , , , , ,

A simple second order differential equation, ovals and chaos

In our youth as a consequence of our undying fascination with ovals we explored many means of generating them. In course of those explorations we experimentally arrived at a simple second order differential equation that generated oval patterns. It also taught us lessons on chaotic systems emerging from differential equations even before we actually explored the famous Lorenz and Rossler attractors for ourselves. The inspiration came from the very well-known harmonic oscillator which is one of the first differential equations you might study as a layperson: \tfrac{d^2x}{dt^2}=-ax, where a is a positive constant that has some direct meaning in physics as the ratio of the restoration constant to the mass of the oscillator. With this foundation, we wondered what might happen if we used a cubic term instead which was in turn coupled to a periodic forcing from a regular harmonic oscillator. Thus, we arrived at the below equation with 3 parameters a, b, c:

\dfrac{d^2x}{dt^2}=-abx^3+a\cos\left(\dfrac{2\pi t}{c}\right) \; \; \; \S 1

It became obvious to us this equation has no solution in terms of elementary functions but we could semi-intuitively figure out that solutions are likely to generate oval-like figures. To test this out we had actually solve it numerically. Just before this foray, a visiting relative, who was a college teacher, mentioned to us that the mark of man was to solve ordinary differential equations using the method of Rangā and Kuṭṭi on hand-held calculator. We had learned Euler’s method as a part of our education in basic numerical calculus but the this new method, apparently discovered by these south Indian savants, was something we had never heard of. Accordingly, we looked up a book featuring numerical analysis, which our father often recommended to us, and realized that the relative was actually referring to a generalization of the Eulerian method by the two śulapuruṣa-s Runge and Kutta. This differential equation offered us an opportunity to apply it to something interesting and we spent sometime writing a program to do the same. With that in hand, we could solve the above 2nd order DE by writing it as two linked 1st order ordinary DEs.

\dfrac{dx}{dt}=ay

\dfrac{dy}{dt}=-bx^3+\cos\left(\dfrac{2\pi t}{c}\right)

Figure 1. Some solutions of \S 1 for different values of the parameters a, b, c with x_0=y_0=0 (click to magnify).

Consequently, we had an unfolding of the solutions for it which presented an interesting picture (Figure 1): We saw before our eyes the entire range of oscillations with simple periodicity, quasi-periodicity, superimposed beats of different frequencies and various types of chaotic oscillations. Thus, we saw this simple DE recapitulate all manner of oscillatory phenomena we had encountered in nature: the changes is numbers of molecules in during the cell cycle, populations of organisms in the ecosystem, climatic patterns and the light-curves of variable stars.

Figure 2. x, \dot{x} for the same values of the parameters a, b, c as in Figure 1 with x_0=y_0=0 (click to magnify).

If we plotted x, \dot{x} we got our desired oval-like shapes. Notably, the cases where the solution is clearly chaotic we get a space-filling entangling of near oval paths in this plot.

Figure 3. Plots of solutions with the b parameter changing while keeping a, c constant in each row (click to magnify).
Figure 4. x, \dot{x} plots of the above solutions (click to magnify).

From Figure 4 we can see that the amplitude parameter a and the wavelength parameter c of the sinusoidal term of \S 1 are the primary determinants of “shape” of the solution, while the parameter b associated with the cubic terms interacts with them to determine where chaos occurs.

Posted in Scientific ramblings | Tagged , , , , , , , , , ,

Viṣṇu, the Marut-s and Rudra

This note might be read as a continuation of the these two earlier ones:
1) The Aśvin-s and Rudra
2) The roots of Vaiṣṇavam: a view from the numerology of Vedic texts

In the Ṛgveda (RV), the Marut-s are seen associating with Viṣṇu on several occasions. This often occurs in the context of the epithets evaya or eṣa (meaning swift or ardent) being applied to them. This association is not per say out of place or surprising because it might be seen in the context of Viṣṇu accompanying Indra in the battle against Vṛtra, where the Marut-s too accompany Indra as his troops (gaṇa-s). Thus, they might be seen as surrounding Viṣṇu, one of the leaders of the deva-s in this battle. However, this association is not purely a mythic one — in the context of the RV it extends to the Marut-s and Viṣṇu being invoked together for receiving offerings in a specific ritual. For example, this is clearly stated by Gṛtsamada Śaunahotra:

tān vo maho maruta evayāvno
viṣṇor eṣasya prabhṛthe havāmahe ।
hiraṇyavarṇān kakuhān yatasruco
brahmaṇyantaḥ śaṃsyaṃ rādha īmahe ॥ RV2.34.11

Verily those great Maruts, speeding along
we invoke in the ritual offering of the swift Viṣṇu
with extended sruc-s, the golden eminent ones,
composing incantations, we implore them of praiseworthy liberality.

This is again presented in the context of the soma offering by Gotama Rāhūgaṇa:

te ‘vardhanta svatavaso mahitvanā
nākaṃ tasthur uru cakrire sadaḥ ।
viṣṇur yad dhāvad vṛṣaṇam madacyutaṃ
vayo na sīdann adhi barhiṣi priye ॥ RV 1.85.7

Those growing in their greatness, the self-powerful ones
They stood in the high heaven and made [themselves] a broad seat.
When Viṣṇu washed the bull dripping with exhilaration (soma)
like birds they sat down on the dear ritual grass.

Unlike the Marut-s, in the RV, their father Rudra is not commonly paired with Viṣṇu except in the general context of the incantations invoking multiple deities. This is keeping with the absence of any evidence for his participation in the Vedic narration of the Vṛtra-hatyā. However, there is one exception where Rudra is mentioned along with Viṣṇu in a ritual context similar to the Marut-s by Vasiṣṭha Maitravaruṇi:

asya devasya mīḻhuṣo (‘)vayā
viṣṇor eṣasya prabhṛthe havirbhiḥ ।
vide hi rudro rudriyam mahitvaṃ
yāsiṣṭaṃ vartir aśvināv irāvat ॥ RV 7.40.5

The appeasement of the god who is bountiful (Rudra)
[is done] in the ritual of the swift Viṣṇu with oblations;
for Rudra knows his Rudrian might.
May you Aśvin-s drive on your food-bearing orbit.

This is notable in two ways: 1) there is a specific mention of Rudra being appeased in the ritual of Viṣṇu. This exactly parallels the offering to the Marut-s in the ritual of Viṣṇu. 2) There is also a reference to Aśvin-s being called for the ritual. They are said to come on their food-bearing orbit (vartis), which reminds one of their epithet Rudravartanī, i.e. they who follow on the track of Rudra.

What is the ritual being referred to here? The answer to this comes from the deployment in the somayāga of another sūkta emphasizing the link of the Marut-s and Viṣṇu — the famous Evayamarut-sūkta of the Atri-s. This sūkta is deployed in a key recitation of the hotṛ-s in the somayāga known as the Śilpa-śastra-s. While it is described in all the RV brāhmaṇa-s, the most detailed account is given the Aitareya-brāhmaṇa:

śilpāni śaṃsanti । devaśilpāny eteṣām vai śilpānām anukṛtīha śilpam adhigamyate । hastī kaṃso vāso hiraṇyam aśvatarī-rathaḥ śilpaṃ; śilpaṃ hāsminn adhigamyate ya evaṃ veda; yad eva śilpānī3n ātma-saṃskṛtir vāva śilpāni; chandomayaṃ vā etair yajamāna ātmānaṃ saṃskurute ।

They recite the Śilpa-s. These are divine art-works; by imitating these [divine] art-works a [human] work of art is achieved here. An elephant (evidently image of one), metal-work, weaving, gold-work, mule-cart-making are [such human] craft-works. A work of art is accomplished by him who knows thus. Regarding those known as the Śilpa-s, the Śilpa-s are a perfection of the self; indeed by them the ritualist perfects himself imbued with the meters.

In this introduction to the Śilpa-śastra-s, the brāhmaṇa teaches the Hindu “Platonic” principle that all human craft-works are imitations of the works of the deva-s (also c.f. the Ratu-s of the Iranians). It is in this spirit the ritualist engages in this śastra recitation so that he might become saṃskṛta or perfect even as the metrical chants — their composers saw them as similar to crafts, sometimes using the phrase that they composed the ṛk-s much like a craftsman making a chariot. Now, the Śilpa-śastra-s consist of a long series of chants: 1) the two sūkta-s of the ancient Nābhānediṣṭha, the descendant of Manu Sāvarṇi (his name is also recorded in Iranian tradition). The first of these prominently features the gods Rudra and the Aśvin-s (explicitly termed sons of Rudra-s in this sūkta) and is recited by the hotṛ. 2) The long aindra-sūkta-s of the Kāṇva-s known as the Vālakhilyā-s in the bṛhatī and satobṛhatī meters. These are recited by maitravaruṇa by intricate separations of the pada-s and half verses. 3) The sūkta of the great descendant of Dīrghatamas, Sukīrti Kākṣīvata (RV 10.131), which is central to the offering of beer in sautrāmaṇi ritual, featuring the Aśvin-s as physicians primarily along with Indra and Sarasvatī again in her medicinal form. Then the enigmatic Vṛṣākapi-sūkta is deployed which presents the banter between Indra and Indrāṇi. As this sūkta is recited the ritualist identifies its verses with the constitution of his body from hair, skin, fleshy organs, bones and marrow. This is recited by the brāhmaṇacchaṅsin. 4) The recitation of the śastra that includes the Rudra-dhāyyā and the Evayāmarut-sūkta to Viṣṇu and the Marut-s (see below). In extant tradition the Rudra-dhāyyā is a single ṛk RV 1.43.6 of Kaṇva Ghaura. Between the two is inserted the sūkta of Bharadvāja starting with “dyaur na ya …” (RV 6.20). This śastra is explicitly recited with the insertion of the “o” vowels, i.e. nyūṅkha style of recitation by the acāvāka, the fourth of the hotraka-s. The Aitareya-brāhmaṇa records an interesting old tale regarding this śastra, which suggests that it was redacted to attain its current structure:

sa ha bulila āśvatara āśvir vaiśvajito hotāsann īkṣāṃ cakra: eṣāṃ vā eṣāṃ śilpānāṃ viśvajiti sāṃvatsarike dve madhyaṃdinam abhi pratyetor hantāham ittham evayāmarutaṃ śaṃsayānīti । tad dha tathā śaṃsayāṃ cakāra; tad dha tathā śasyāmane gauśla ājagāma; sa hovāca hotaḥ kathā te śastraṃ vicakram plavata iti; kiṃ hy abhūd ity ? evayāmarud ayam uttarataḥ śasyata iti; sa hovācaindro vai madhyaṃdinaḥ,
kathendram madhyaṃdinān ninīṣasīti; nendram madhyaṃdinān ninīṣāmīti hovāca; chandas tv idam amadhyaṃdinasācy ayaṃ jāgato vātijāgato vā। sarvaṃ vā idaṃ jāgataṃ vātijāgataṃ vā । sa u māruto maiva śaṃsiṣṭeti; sa hovācāramāchāvakety; atha hāsminn anuśāsanam īṣe; sa hovācaindram eṣa viṣṇunyaṅgaṃ śaṃsatv; atha tvam etaṃ hotar upariṣṭād raudryai dhāyyāyai purastān mārutasyāpyasyāthā iti; tad dha tathā śaṃsayāṃ cakāra । tad idam apy etarhi tathaiva śasyate ॥

That Bulila Āśvatara Āśvi was the hotṛ in the viśvajit ritual; he observed: of these Śilpa-s in the year-long Viśvajit ritual these two (the recitations of the maitravaruṇa and brāhmaṇacchaṅsin) are added to the midday recitation. Well, let me have the Evayāmarut recited [by the acāvāka]. He then made that to be recited. When it was being thus recited Gauśla came up; he said: “O hotṛ why is your śastra sinking without a wheel?” B.A.A: “What happened? The Evayāmarut is being recited to the north of the altar.” He (G) said: “Indra is verily the midday. Why do you seek lead Indra away from from the midday?” “I do not seek to lead Indra away from the midday”, he replied. G: “The meter is also not that for the midday, jagati or atijagati. All these [incantations] (Evayāmarut) are either jagati or atijagati. It is also of the Marut-s; do not recite it.” He (B.A.A) said: “Stop O acāvāka”. He (B.A.A) then sought an instruction on this. He (G) said: “He may recite the Indra hymn with the mark of Viṣṇu. Now, O Hotṛ this is inserted between the preceding Rudra Dhāyyā and the following Māruta (i.e. Evayāmarut). Then he (B.A.A) made it be be thus recited. Even now that is how it is recited.”

The tale hints that originally or in certain traditions this Śilpa-śastra consisted of just the invocation of Rudra, Viṣṇu and the Marut-s in an offering centered on Viṣṇu and the Marut-s. However, as stated, it was emended to include Indra to maintain the connection of the midday rite with Indra. Nevertheless, even in this, the sūkta indicated to Indra was chosen such that the original connection of the offering with Viṣṇu was retained. The second ṛk of the inserted sūkta goes thus:

divo na tubhyam anv indra satra
asuryaṃ devebhir dhāyi viśvam ।
ahiṃ yad vṛtram apo vavrivāṃsaṃ
hann ṛjīṣin viṣṇunā sacānaḥ ॥ RV 6.20.2

Just as that of Dyaus, to you, O Indra, the power
of the Asura-s was entirely ceded by the deva-s,
when you, O drinker of silvery juice, accompanied by Viṣṇu
smote the snake Vṛtra blocking the waters.

This is the mark of Viṣṇu mentioned in the brāhmaṇa. Given that the immediate juxtaposition of Rudra, Viṣṇu and the Marut-s in a ritual all together is a distinctive one, it is apparent that one of the form encapsulated in the final Śilpa-śastra was something widely known to the early RV composers and specifically alluded to. This is further supported by the observation that the above Vāsiṣṭha ṛk also mentions the Aśvin-s. As can be seen in the Śilpa-śastra-s, another prominent deity of the recitations are the Aśvin-s who are explicitly coupled with Rudra in the first Nābhānediṣtha sūkta.

It may also be noted that the juxtaposition of Rudra, the Marut-s and Viṣṇu also has a further echo in the śrauta ritual. In the piling of the Agnicayana altar, after the fifth and final layer of bricks has been laid it is said to ghora or terrible and if the adhvaryu steps on it he is said to die. This is because the newly laid altar is said to be possessed by the fierce manifestation of Agni as Rudra. Hence, Rudra has to be pacified by offerings of goat milk with a milkweed leaf on a specific brick of the altar (brick 189 in the standard Agnicayana eagle altar) with the recitation of the Śatarudrīya by the adhvaryu. Once that is over, the ritualist goes clockwise around the altar impersonating Rudra by holding a bow and 3 arrows. Stopping at the vertices of a pentagon in course of that circuit, at each stop he recites the incantation to Rudra of the 5 years of the saṃvatsara cycle as the lord of the wind. The adhvaryu gives a pitcher of water to the pratiprasthātṛ and asks him to make 3 circuits pouring it out in a continuous stream. As he does so, the adhvaryu and the ritualist recite the incantation to the Marut-s (e.g. Taittirīya-saṃhitā 4.6.1.1) who are called upon to provide the energy residing in stones, mountains, wind, rain, the fury of Varuṇa, water bodies, herbs and trees as the strength of food. Once the pacification of Rudra is complete, and the fire is installed on it, the altar is said to be śānta or peaceful and to give the yajamana a great bounty that is asked for in the camaka incantations. These accompany the vasor dhārā offerings wherein a continuous stream of ghee is poured into the fire with a special furrowed log-guide known as the praseka that is as tall as the ritualist. Finally, after the offering is done, the ghee-soaked praseka itself is offered in the fire. This ritual begins with the following gāyatrī incantation:

agnāvīṣṇū sajoṣasemā vardhantu vāṃ giraḥ । dyumnair vajebhir āgatam ॥

Agni and Viṣṇu, may these chants glorify you together. Come with radiance and vigor!

Regarding these oblations the śruti of the Taittirīya-s offers the following brāhmaṇa:

brahmavādino vadanti: yan na devatāyai juhvaty atha kiṃdevatyā vasor dhāreti ? agnir vasus tasyaiṣā dhārā; viṣṇur vasus tasyaiṣā dhārā āgnāvaiṣṇavy arcā vasor dhārāṃ juhoti; bhāgadheyenaivainau sam ardhayati; atho etām evāhutim āyatanavatīṃ karoti; yatkāma enāṃ juhoti tad evāva runddhe; rudro vā eṣa yad agnis; tasyaite tanuvau ghorānyā śivānayā; yac chatarudrīyaṃ juhoti yaivāsya ghorā tanūs tāṃ tena śamayati; yad vasor dhārāṃ juhoti yaivāsya śivā tanūs tāṃ tena prīṇāti; yo vai vasor dhārayai pratiṣṭhāṃ veda praty eva tiṣṭhati ॥ in TS 5.7.3

The brahmavādin-s say: “Given that they do not offer to any deity (i.e svāhā-s are uttered without naming the deity), which deity has the vasor dhārā oblation? Wealth is Agni (or Agni is a Vasu); this stream is his. Wealth is Viṣṇu; this stream is his. With the verse addressed to Agni and Viṣṇu (the above gāyatrī) he offers the stream of wealth; verily he unites them with their proper portions. He also makes this offering in order to have an abode. He wins that desire for which he makes this offering. The fire is Rudra; now two are his bodies, one is dreadful, the other is auspicious. That in which he offers the Śatarudrīya is its dreadful one. He pacifies it with that [Śatarudrīya offering]. That in which he offers the vasor dhārā is the auspicious one. He delights it with that [vasor dhārā offering]. He, who knows the foundation of the vasor dhārā indeed stands well-founded.”

Thus, Rudra and the Marut-s on one hand and Viṣṇu on the other are identified with the opposing but juxtaposed characteristics of Agni and invoked as deities in the two key rituals associated with completed Agnicayana altar. That this juxtaposition is not just incidental but a deeper feature of the traditions of the Indo-Aryan world and in a more general form the greater Indo-European world is hinted by the tendencies expressed in the itihāsa-s: In the Rāmāyaṇa, Rudra offered great favors the malefic Rāvaṇa who is opposed to the Indra-Viṣṇu duo humanized as Rāma and Lakṣmaṇa. More explicitly, in the Mahābhārata, the humanized manifestation of Rudra, Aśvatthāman is the malevolent force opposed to the humanized Indra-Viṣṇu-Agni manifestations in the form of Arjuna, Kṛṣṇa and Dhṛṣṭadyumna and the other daiva forces. Similarly, Rudra backs Jayadratha allowing him to overcome the Pāṇḍu-s on the fateful day of the slaying of Abhimanyu. Yet again, in the same epic a malefic Rudra-backed figure Śiśupāla (born with 3 eyes) is also presented in opposition to the humanized manifestation of Viṣṇu. A comparable form of opposition also extends the Greek world, where, in their national epic, the Rudrian deities Apollo and Ares prominently back the Trojans against the Greeks who are backed by Athena.

Based on the inferred prominence of Rudra and Viṣṇu in the para-Vedic and “greater” Vedic horizons (i.e. the root of the ādhvaryava tradition), we can say that this juxtaposition of them was a reflection in the “standard aindra religion” of tendencies which were more pronounced outside it: i.e. the cults centered on Rudra and Viṣṇu. This is hinted by the fact that right in the Kauṣītaki-brāhmaṇa, in the corresponding account of the Śilpa-śastra-s with the Rudra-dhāyyā and the Evayāmarut with the nyūñkhā “o” insertions, we encounter the below statement:

atho rudro vai jyeṣṭhaś ca śreṣṭhaś ca devānām ।

Now Rudra is indeed the eldest and the best of the gods.

This indicates that the early “śaiva” view was already impinging on the “standard aindra religion”. As we have noted before, the Aitareya-brāhmaṇa correspondingly, provides an early “vaiṣṇava” viewpoint hinting the primacy of Viṣṇu. That such tendencies were ancient is indicated by the fact that they are not restricted to branches of the Ārya-s who eventually conquered India — indeed they are hinted by parallels seen among the Iranians and the Germanic peoples. Interestingly, it was that stream of the religion that was to eventually dominate the Ārya religious traditions in India in the form of the Śaiva and Vaiṣnava cults.

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Pandemic days-4: viruses get new hosts

That we have come to be in these pandemic days evokes some wonderment or even disbelief among laypeople. The general thinking of a large section of the populace is that this event is something completely unexpected or out of the way. Hence, some of them are quite prone to invoke different kinds of outre explanations, the most common being: “It must be a Cīna bioweapon (I hear from Cīna-s that in their midst it is common to think of it as a mleccha bioweapon).” In the least many of them might say it is something “unnatural”. However, for those more familiar with the natural history of these matters it is more of an expected thing that was almost waiting to happen and events specifically like this have probably happened going back some time into the past. The only thing we could not say is when exactly it would happen. In this note we shall rehash these matters in the language of an educated layperson. In our earlier writings on this we sort of took it for granted that this is probably clear to everyone but apparently it is not and in any case there are some interesting points to place out there for the reader.

First, it is not that pandemics are a distant memory; they happen quite frequently with the the negative-strand RNA viruses of the influenza genus: many people might remember H1N1 influenza and some may have even gotten it. Older people will remember how the retrovirus HIV-1 caused the AIDS pandemic. Of course none of these were anywhere as crippling as the current Wuhan disease but these at least give us a feel for the potentialities of such things and that they are not a matter of distant folktale. Second, apart from pandemics there have been several smaller viral outbreaks like Ebolavirus and Henipavirus, both also negative-strand RNA viruses. Third, there is also the regular vector-borne pestilence of the positive-strand RNA viruses from which you or somebody in your family might have suffered or died: the flaviviruses, like the Yellow fever virus, Japanese encephalitis virus, West Nile virus, Dengue virus and Zika virus and their more distant relative the togavirus, Chikungunya virus. Not be left behind we even have the occasional Nucleo-cytoplasmic Large DNA virus like the Monkeypox virus give a smallpox look-alike to its victims. Thus, infectious viral disease is very much part of our existence and it does not take much imagination to see one of these will emerge to deliver a punch more to the extreme right end of the distribution of effects.

Of course when this is pointed out someone would say: “Come on, did any of those put us in a state like what we are in right now? This coronavirus is special !” There is a reason I did not mention coronaviruses in the above list — they are indeed special in a way to deserve separate consideration but what we are experiencing is also quite expected given what we know of the natural history of these the coronaviruses. To apprehend this distinction let us first look at some examples of other viruses acquired by humans from non-human animals. There have been numerous crossovers of viruses between different species in course of evolution. For example, the positive-strand RNA hepeliviruses, which include the likes of the animal Rubella and Hepatitis E viruses on one hand and the plant Beet necrotic yellow vein viruses on the other represent an extreme crossover between plants and animals. Thus, this process is an unavoidable part of life. There many more cases of recent crossovers of viruses from non-human animals to humans, some of which are well-studied. However, the mode in which these crossovers get established in humans makes a big difference: the vector-borne flaviviruses and togaviruses are easy to establish in humans from the bite of an insect like the ubiquitous mosquito but to keep transmitting they need more of those bites in the critical phase of viremia (when the virus is in the blood). This is in principle preventable to quite a degree by relatively simple means like the use of insect repellents (already mentioned in the Atharvaveda) or mosquito nets (a luxury that even the normally morose tathāgata allowed for his saṃgha around 2500 years ago). Indeed, some countries have done quite well with with various insect-borne viruses by relatively simple but rigorous prevention programs.
HIV1Figure 1. A cryo-electron micrographic image of the capsid of the HIV-1 virus: a beautiful object.

Another very well-studied example of crossover from non-human animals to human is AIDS, which is caused by two distinct but related viruses, HIV-1 and HIV-2. Of these, HIV-1 was transmitted from chimpanzees and gorillas to humans in west central Africa on at least 4 distinct occasions (2 times from chimps, 1 time from gorilla and 1 time from either). Only one of these (HIV-1 M) after festering in Africa for nearly 30-50 years radiated out of the Kinshasa region to establish a global pandemic. Chimps in turn acquired the chimp precursor of HIV-1, SIVcpz, from the SIV infecting Cercopithecus and Cercocebus monkeys which they prey upon. The monkeys infected with SIV are unaffected by the virus and lead a mostly normal life. However, in chimps it is sexually transmitted with roughly the same probability per heterosexual coitus (0.0008–0.0015 ) as in humans (0.0011) and greatly increases the mortality of the infected ape. Its dispersion through the chimp populations appears to have been primarily driven by the mobility of infected females. Gorillas appear to have acquired SIVgor from chimps. Since gorillas do not hunt chimps or vice versa but both live in overlapping ranges, it raises the possibility of rare gorilla-chimp matings during which the infection was transmitted. The acquisition of HIV-2 by humans was from a Cercocebus monkey (sooty mangabey) precursor, SIVsmm. Given that in the monkey community the highest infection by SIVsmm is seen in high-ranked females, it is evidently harmless to the monkeys. In humans too only a minority of the infected individuals proceed to developing AIDS and it is limited to West Africa. While human-chimp/gorilla matings might have occurred on rare occasions, the relatively low and similar probability of sexual transmission in humans and chimps, and the HIV-2 crossover from monkeys suggest that both HIVs primarily originated from mucosal contacts or blood during “bush-meat” hunting — thus humans and chimps got AIDS in a similar way from their prey. However, after crossover from monkeys its transmission in all three great apes (Homo included) is primarily sexual. At least in humans, this is still a relatively low probability event per coitus and quite preventable by behavioral means. Thus, even the famous AIDS pandemic took a long time to break out and only one out 5 independent non-human to human crossovers resulted in a pandemic. Not surprisingly, it was eventually managed quite effectively in the general population except for the locations with exaggerated sexual promiscuity.

Coming to coronaviruses proper, apart from being in the hall of fame for having the largest single-segmented RNA genomes, they are specialists of transmission by the respiratory and the orofecal route. Since, you cannot avoid getting air, food or water into your body these viruses are much harder to manage than the rest once they jump to humans. Moreover, primates being very “facially” oriented creatures, have particular risk to infection by these modes. It is this distinction that makes them one of the most likely viral agents to pack a big punch if they establish a pandemic. In evolutionary terms, the crown clade of coronaviruses consists of 4 major subclades: the alphacoronaviruses (alpha-CoV) and betacoronaviruses (beta-CoV) which primarily infect mammals form one higher order group. Basal to them are the deltacoronaviruses and gammacoronaviruses primarily infect birds (Figure 2; but see below for exceptions). Outside of these lie the more basal Gull Coronavirus and the lizard-infecting Guangdong Chinese water skink coronavirus. This suggests that the original radiation of the coronaviruses was likely in the late Paleozoic-Mesozoic where they emerged in reptiles and probably infected both the great branches of reptiles lepidosaurs (including lizards) and archosaurs (including dinosaurs). With the close of the Mesozoic they lingered on in the surviving dinosaurs, i.e. the birds, as the delta-CoV and gamma-CoV lineages. From birds it is likely that they made at least two major jumps to mammals probably facilitated by these vertebrates sharing a warm-blooded physiology with body temperatures in the same general range. One was to bats, which was probably via shared nesting sites and this founded the alpha-CoV and beta-CoV lineages within bats. The next primary transmission, probably due to predation of birds by dolphins transmitted the gamma-CoVs to dolphins/whales. The alpha-CoV and beta-CoV radiated extensively in bats alongside numerous other viruses such as the negative-strand RNA filoviruses (Ebola-like), henipaviruses, and lyssaviruses (rabies) for which bats play great hosts. Further, from bats and birds coronaviruses appear to have episodically invaded a wide range of placental mammals.

coronaviruses

Figure 2. Modified from original figure published in “Discovery of a Novel Coronavirus, China Rattus Coronavirus HKU24, from Norway Rats Supports the Murine Origin of Betacoronavirus 1 and Has Implications for the Ancestor of Betacoronavirus Lineage A” by Susanna K. P. Lau et al.

In the past 20 years we have been witness to several such invasions of humans and domesticated mammals (Figure 2):
1) SARS-CoV of the Beta-CoV clade, the agent of the SARS outbreak which began in November 2002 in Foshan City, Guangdong, China, definitely started from bats but reached humans via civets, which are eaten by the Cīna-s. In August 2003 a virology student in Singapore and in April 2004 two laboratory personnel at the Chinese Institute of Virology in Beijing were independently infected by laboratory SARS-CoV due to poor virological technique.
2) In 2012 MERS outbreak started in Arabia with the transmission of a distinct beta-CoV from dromedaries to humans. Dromedaries are often imported to Arabia from Africa where a closely related virus has been found in bats suggesting that the camels first acquired it from bats in Africa and then transmitted it humans.
3) Again in 2012, the coronavirus HKU15 was detected in pigs in Hong Kong. This delta-CoV appears to have jumped from birds to mammals probably in the unhygienic live markets of China. In 2014 it caused outbreaks of a diarrheal disease in several states of the USA. In Asia it seems to have further spread from pigs to wild cats.
4) In 2017, not far from the place of original SARS outbreak, in Guangdong, China, a novel alpha-CoV caused a major outbreak of acute swine diarrhea syndrome (SADS-CoV) decimating a large number of piglets, which are an important food item of the Cīna-s. It was transmitted from bats which are infected by a related HKU2r-CoV. Outbreaks of this virus have continued in Chinese pig populations till as of an year back.
5) In November of 2019 a repeat performance of the SARS event happened in Wuhan, China, with the related SARS-CoV-2 jumping from bats to humans directly or via an intermediate which could have been something like cats that are consumed by the Cīna-s. This has become the agent of the current pandemic.

One thing we have learned from the intense scrutiny of the few proteins encoded by the HIVs and SIVs it that they have evolved 3 independent mechanisms (via the proteins Vpu in HIV-1 M, via Nef in SIV of chimps, gorilla and monkeys, and Env in HIV-2) of countering a key general purpose host immunity mechanism against enveloped viruses, namely inhibition of the surface protein BST-2 (tetherin), which blocks the budding of virions (viral particles). Interestingly, the SARS coronavirus has evolved its own independent mechanism to do the same and we believe a similar mechanism is used by its cousin SARS-CoV-2, the causative agent of the Wuhan disease, and more generally by both alphacoronaviruses and betacoronaviruses. We shall not dilate on that here (as it will touched upon in a more formal venue) but shall simply state that such adaptations that allow disabling of this general immunity mechanism appear to be one general convergent feature common to distant viruses that might facilitate the jump to humans and closely related apes.

Cīna authors themselves have stated in no unclear terms that the Cīna love for “live (i.e. slaughtered on the pan) meat”, which is held to be more nutritious in traditional Cīna medicine, provides huge opportunities for such outbreaks to happen. Given that 4 different coronavirus outbreaks have happened from crossovers with connections to bush-meat or unhygienic live markets prior to 2019, from a natural history standpoint the current pandemic was just a matter of time. Further, the lab accidents resulting in infections and human-to-human transmission in the Chinese case indicate that those too in principle could be further sources of infection, especially in the Cīna context. Notably, the way MERS reached the other end of Asia in the form of the Korean outbreak and killed tens of people there showed how globalization and Galtonism would drive local epidemics to pandemics. Given all this, the current pandemic is not unexpected since one of these outbreaks was eventually going to hit the “sweet spot” like SARS-CoV-2, especially, as noted above, coronaviruses are imminently suited for something like this. In light of this, the utter failure in the response of several governments all over the world shows that certain forms of predictive knowledge, especially in the mathematical or biological domain, remain rather privileged and are not easily grasped by the elite.

Given that we have had 5 coronaviral outbreaks in humans and livestock in the past 20 years, one question that comes to mind is whether there have been such coronaviral outbreaks/pandemics in the past? As we noted before, pandemics from globalization is not a new thing; so, there is no strong reason why there should not have been past coronaviral outbreaks. Before SARS, coronaviruses were hardly seen as threatening to humans and little effort was expended on the two human coronaviruses discovered in the 1960s (see below). After SARS more attention was paid to the apparently milder coronaviruses that infect humans yielding a wealth of data. This growing interest led the discovery of a new alpha-CoV, HCoV NL63, in 2004 as an agent of human respiratory disease. Subsequent studies have shown that it is responsible for croup in children (a condition already described in old Hindu medicine as caused by the Śvagraha, an agent of the god Kumāra) and sometimes more serious lower respiratory track involvement in both children and adults. Like SARS-CoV-2, it appears to trigger rare instances of the Kawasaki disease in children. Despite its recent discovery, HCoV NL63 does not represent a recent crossover from non-human animals because at the time of its discovery it was already a well-established pandemic. Nevertheless, its closest relatives are viruses infecting bats which suggest that it might have invaded humans ultimately from bats about 1000-500 years ago. Another related alpha-CoV, HCoV 229E, which was discovered in 1966 and has subsequently been shown to be a notable cause respiratory infections throughout the world with a preponderance in immunocompromised individuals. Interestingly, an early serological study in the 1960s showed that HCoV 229E antibodies were detected primarily in adults as opposed to near absence in children — something which reminds one of the higher severity of SARS-CoV-2 in adults as opposed to children. The closest relatives of HCoV 229E are again found in bats and suggest a crossover perhaps in the last 2000 years.

Another comparable pair of human coronaviruses are HCoV OC43 and HCoV HKU1 that belong to the so-called “lineage A” of Beta-CoV. The HCoV OC43, the first human coronavirus to be discovered, was reported in 1965 as “a novel type of common-cold virus.” Subsequently, it has been widely reported as major cause of upper respiratory tract infections (perhaps 5-30% of such infections) and a more severe lower respiratory track involvement as a pneumonia in elderly people. Interestingly, it has been reported to also cause rare instances of fatal encephalitis and Kawasaki’s disease, both of which have also been seen with SARS-CoV-2. Notably, in one survey up to 57% of the patients with HCoV OC43 infections reported enteric tract manifestations. Indeed, early studies in the 1980s associated strains of HCoV OC43 with human gastroenteritis. This is again rather reminiscent of the enteric involvement suggested for SARS-CoV, MERS-CoV and SARS-CoV-2. Following the renewed interest in these viruses in the post-SARS era, a related virus HCoV HKU1 was reported in 2004 from a 71-year-old man with pneumonia who had just returned to Hong Kong from Shenzhen, China. A subsequent survey showed that it was also widely established in humans across the world and primarily caused a “cold-like” URTI though in some cases it might worsen to a pneumonia. However, it causes a significantly higher incidence of febrile seizures than other respiratory tract viruses. These two beta-CoVs have no particularly close relatives among bat viruses. Instead, HCoV OC43 belongs to a complex of closely viruses in “lineage A” including the bovine coronavirus (BCoV) and the equine coronavirus (ECoV) both of which cause episodic outbreaks of enteric disease, sometimes with respiratory manifestations in livestock. Of the two, HCoV OC43 is closer to the the BCoV. In 1994, a recent crossover of BCoV to humans was reported resulting in a case of acute diarrhea in a human patient (HECV-4408). This suggests that HCoV OC43 represents another such earlier crossover from cattle that got established in humans as a pandemic. It is not clear when exactly this happened but likely it happened sometime after the domestication of cattle by humans. Some have proposed that OC43 was the agent of the “Russian flu” in 1889-1890 CE. However, we believe that this is based on erroneous molecular clocks estimates. However, it cannot be ruled out that the “Russian flu” was another viral cross over. In contrast, HCoV HKU1 defines a distinct subclade within “lineage A” but is nested among rodent coronaviruses such as the Murine Hepatitis virus, the Rat-CoV and the China Rat HKU24. This suggests that it might have crossed over from rodents which are widely consumed by humans in East Asia (Figure 2).

There are some interesting common features of above-discussed four human coronaviruses: today they cause relatively mild URTIs in healthy individuals but have the potential for causing more serious conditions including fatal pneumonia or neural complications. This is reminiscent of SARS-CoV-2, which is relatively mild in majority of individuals but causes a far more severe infection in the rest (of course at a much higher rate than the above-mentioned CoVs). This raises the possibility that they were once virulent viruses comparable to SARS-CoV-2 that crossed over directly or indirectly from bats, cattle and rodents in the past and have now evolved to a mild state due to selection on the host and virus. Thus, in these milder human CoVs we might be seeing remnants of a past outbreaks that might have begun as severe infections in some ways comparable to the current one. What might be the scenarios in which they might have begun? At least HCoV OC43 may have started early with cattle domestication. In contrast, HCoV NL63 and HCoV 229E from bats and HCoV HKU1 from rodents probably originated in China or elsewhere East Asia where consumption of such animals is prevalent. Africa is another possibility, though the relatively low connectivity of Africa to the rest of the world until not long ago makes it less likely. The 2007, the camelids known as alpacas (llamas) were found to be infected by a novel coronavirus closely related to HCoV 229E resulting in the  alpaca respiratory disease. This points to a recent crossover from another animal source — this evidently parallels the original crossover of the related virus to humans. These pre-modern events probably spread more slowly than today’s SARS-CoV-2 and could have even involved a relatively severe local epidemic followed by attenuation of the virus before a wider spread. This also suggests that, barring successful intervention, such a trajectory is probably the one by which SARS-CoV-2 will become less threatening.

This course of attenuation seems to follow a full circle: Both SIV/HIV and coronaviruses are asymptomatic or mild respectively in monkeys and bats. Once they crossover to apes the disease is way more severe and fitness reducing. Then eventually they might reconfigure to a milder state in the new host. Indeed, in traditionally bat-eating tribes of Africa, India and East Asia there might have been past epidemics which were generally limited due to the isolation of these tribe and over time attenuated forms of the virus probably emerged giving them higher immunity against such viruses. Those viruses that established a severe infection in humans probably limited themselves in the pre-modern era as they rapidly destroyed the populations they invaded before they could access another population. Thus, the drive tends to be towards a more attenuated state over time unless other factors are in play. However, this is a complex process as the virus first undergoes selection for better establishment in the new host which might not necessarily attenuate it in the early stages and then probably it undergoes more attenuating changes. The first set of such changes happened in both the SIV/HIV and also in parallel in SARS-CoV and SARS-CoV-2. In SARS-CoV it is primarily in the form of the disruption or loss of a viral protein called ORF8 whose mystery we have finally solved and will be presented in a more formal venue. This happened independently in Singaporean isolates of SARS-CoV-2 suggesting that the two viruses are likely under similar selection from human immunity.

Studies on SIV/HIV infectivity has shown that Old World monkeys are resistant to HIV-1 because they likely have multiple defenses against it. The best studied is the TRIM5 protein which inactivates the virus by interfering with the proper uncoating of the viral capsid inside the cell. Old World Monkeys also produce a cyclic peptide retrocyclin-2, which is a broad-spectrum anti-microbial killing both bacteria and HIV-1. Neither does the human TRIM5 bind HIV efficiently nor do humans produce retrocyclin-2. Such changes illustrate things happening on the host side due to natural selection by episodes of viral outbreaks. One may ask why lose such potent defenses? First, TRIM5 is under selection depending on the episodes of retroviral epidemics a population goes through; thus, it may evolve away from one specificity towards what is the latest danger. So a past epidemic evidently drew TRIM5 specificity in great apes of our lineage away from HIV-1 binding because that was a more demanding pressure. Thus, when HIV-1 entered human populations they were in a state of adaptation that was likely still directed at a now gone virus. Second, the case of SARS-CoV-2 illustrates that much of the human morbidity appears to arise from self-damage caused by an unregulated immune response in some individuals. Thus, viral pressure can also select against some immune weaponry that can cause such self-damage. This can result in evolution of a more attenuated interaction of the host with the virus. However, this can also come at the expense of once effective weaponry which is lost under such selection — it is possible that we lost retrocyclin-2 due to some past pathogen pressure acting at the base of the great ape lineage. This also reminds one the actions of the Cīna researcher who created Crispr Ding and Crispr Dong by editing out the gene for the CCR5 7-transmembrane receptor — the receptor used by HIV-1 to get into cells. However, CCR5 is a receptor protein for at least 5 signaling proteins called chemokines in the immune system. While it might provide resistance to HIV-1 and in the past might have also interacted with the Smallpox Virus, it is rather useful for immunity against other viruses like the West Nile virus. Hence, docking it might actually increase susceptibility in a different direction.

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Rāsabha-nyāya-śikṣā

Vrishchika had been seeing several kids of patients affected by the chemical leak that had happened sometime ago. While she saw some purely for routine clinical practice, she was also particularly interested in the several cases exhibiting heterotaxy and had amassed a collection of such. With Indrasena she had identified a collection of mutations in these patients and those from elsewhere. Eventually, with Lootika and Somakhya they had translated it into some interesting and ancient biochemistry concerning membrane protein stability that was widespread in eukaryotes. Now they were sitting down at Somakhya and Lootika’s house to work on a manuscript on this matter. As they did so, they asked their sons to get some lessons from Jhilleeka.

To get her nephews started Jhilleeka asked them: “So what did you two learn from Tigmanika’s grandfather little ones?”
Guhasena: “He taught us something very important which the very basis of the apprehension of the space of our everyday life: the gardabha-nyāya.”
J: “Could you tell me what that is?”
Tigmanika: “He said that even though in the early days when our ancestors roamed on the steppes they saw the braying ass as something alien and evil, they came around to see it as an animal that should be respected. He felt that was because it taught us the gardabha-nyāya: gardabho ‘pi tryaśre hrasiṣṭam patham ādhatte |
Guhasena: “It is the Donkey’s theorem: No side a triangle is greater than the sum of the other two.”
J: “What more did his tell you about this?”
G: “He also taught us the area of a triangle and the bhujā-koṭi-karṇa-nyāya.”

J: “So, nephews, what is the sūtra for the area of a triangle given its three sides?”
T: “bhujayogārdhasya bhūjayogārdha-pratyeka-bhujona-ghātāt padam phalam iti |
J: “That formula is known from at least the work of the old scientist Brahmagupta who had a mix of good and bad ideas. Guha write it down on the board.”

G: A = \dfrac{1}{4}\sqrt{(a+b+c)(a+b-c)(a+c-b)(b+c-a)}

J: “Boys do you realize from that is self-evident from this that if a triangle exists the Donkey’s theorem must be true?”
T and G: “Yes! If any side of a triangle were greater than the sum of the other two then by this formula the triangle cannot have an area because what is under the square root will be negative. So the existence of a triangle predicates the gardadbha-nyāya.”

J: “That is good. Knowing your grandfather, I am sure he gave you some problems — he used to entertain me with such whenever we met. Were you able to solve them?”
T: “Yes, he gave us three. We solved one of them easily and we believe we solved the second one but mom said our solution was not good enough and that our grandfather might call us idiots if we show it to him.”
J: “OK. What were they?”

DonkeyFigure 1

G: “The first goes thus: Let a, b, c be the 3 sides of a triangle from shortest to longest side. In what interval will the ratio of the sum of the short and long side to the middle side fall? We solved it thus: At one extreme will be a degenerate triangle whose sides will be a=0, b=1, c=1. At the other extreme we have another degenerate triangle with sides a=1, b=1, c=2. If we reduce the triangles, all others will have sides in between these. Hence, 1 \le \tfrac{a+c}{b} \le 3.”

J: “Not bad. That’s indeed the case. It is not for nothing our intellectual predecessors would say: ‘trai-rāśikenaiva yad etad uktaṃ vyāptaṃ sva-bhedair hariṇeva viśvam |‘ All that has been stated is permeated by the principle of proportions even as Viṣṇu pervades all with his own [manifold] forms. Those forms of Viṣṇu are the 1000s which exist in the different adhvan-s as laid out in the teachings of the Vaiṣṇava mode of tāntrikopāsana. Alright, so what was the next problem whose solution was found wanting by Lootika?”

T: “Let a, b, c be the 3 sides of a triangle from shortest to longest side. What will be the interval in which the maximal proportion of the short to the middle side or middle to the long side fall?”
J: “So, what did you do?”
T: “One can see that when we have and equilateral triangle or degenerate triangles such that a=0, b=1, c=1, or a=1, b=1, c=2 then \tfrac{a}{b}=1; \tfrac{b}{c}=1 or \tfrac{a}{b}=0; \tfrac{b}{c}=1 or \tfrac{a}{b}=1; \tfrac{b}{c}=\tfrac{1}{2}; So, \max\left(\tfrac{a}{b}, \tfrac{b}{c}\right) \le 1. Thus, we have one end of the interval but the other end of the interval needed some experiments to determine.”
G: “We can see that it should be a degenerate triangle such that from \tfrac{a}{b}, \tfrac{1-\tfrac{a}{b}}{1} we should get the maximum. By writing out different proportions for degenerate triangles we realized it is some number close to 0.618.”
J: “That’s not bad nephews. You are close to the answer. What you have to do is to get the actual number with some formal algebra. You were right in reasoning that it should be a degenerate triangle where a+b=c=1. If \tfrac{a}{b} were to decrease from 1 such that it remains greater than b then you come closer and closer to the lower bound of your desired interval. But if you cross that bound then b becomes greater than \tfrac{a}{b}. So your bound will be where the two are equal. Thus, you can write: \tfrac{a}{b}= b; a=b^2. But since it is a degenerate triangle a=1-b; by plugging that we get the quadratic equation b^2+b-1=0. Now, I have taught you how to solve those; so write the answer on the board”
G and T: b=\tfrac{\sqrt{5}-1}{2}
J: “As you can see that number is close to what your experiment yielded. It is known as \phi' or the conjugate Golden Ratio. Thus, as a consequence of the gardabha-nyāya the maximal proportion of the successive sides of a triangle always lies in the interval \phi' \le \max\left(\tfrac{a}{b}, \tfrac{b}{c}\right) \le 1. The more advance in age the more you will realized of the ubiquity of the former bound like Indra pervading existence. Let us move on; how did you fare with the 3rd problem?”

G and T: “Given an angle \theta, construct a triangle such that its sides a, b, c will be in an arithmetic progression. We have been able to obtain a few cases like 1, 1, 1 or 3, 4, 5 but we have been breaking our head over how to solve it generally?”
J: “Fine. First I’ll give you the construction and then explain the rationale. Do this construction on your tablets as I tell you:

Angle_arithmetic_prog_TriFigure 2

1) Draw \angle{\theta} with point A as its apex and one ray in the horizontal direction. On the other ray mark out a point Q. From Q drop a perpendicular to meet horizontal line at point P.
2) Along the horizontal ray of \angle{\theta} mark point R such that \overline{AR}=2\overline{AQ}.
3) Trisect \overline{PR} to get \overline{PC'}=\tfrac{\overline{PR}}{3}.
4) Mark point C on the horizontal ray of \angle{\theta} such that \overline{AC}=\overline{PC'}
5) Mark point B on ray \overrightarrow{AQ} of \angle{\theta} such that \overline{AB}=\tfrac{\overline{AQ}}{2}.
6) \triangle ABC is the desired triangle with sides in an arithmetic progression.

arith_trigs1Figure 3

Now check out a few examples of such triangles with this construction and verify if we are getting what we want.”

G and T: “Aunt, that is almost like magic: how did you know this would work?”
J: “It is simple. You have to first bring to mind an important generalization of the bhujā-koṭi-karṇa-nyāya known as the koṭijyā-nyāya which applies to any triangle:

c^2=a^2+b^2-2ab\cos(C)

Kids, you can see that if the sides are in arithmetic progression we can write them as a=x-k, b=x; c=x+k. Moreover, constant of the progression k=b-a=c-b. Now we apply the koṭijyā-nyāya for such a triangle:
c^2+b^2-2bc\cos(A)=a^2
(x+k)^2+x^2-2x(x+k)\cos(A)=(x-k)^2
x^2+2kx+k^2+x^2-2x(x+k)\cos(A)=x^2-2kx+k^2
2k+x-2(x+k)\cos(A)=-2k
4k+x=2(x+k)\cos(A)
\cos(A)=2-\tfrac{3b}{2c}

That is the formula which we have expressed in the construction I just gave you. When we are done you can figure that out and how the koṭijyā-nyāya implies the gardabha-nyāya.”

Jhilleeka next decided to use the gardabha-nyāya to introduce her nephews to a fundamental understanding of the conics. J: “I’m sure Tigma’s grandfather has introduced you to the conics, the ellipse, the parabola and the hyperbola those most important curves a person ought have a proper understanding of. Such was a signal sent to the yavanācārya-s by their god Apollo, who is a cognate of our Rudra.”
T and G: “Yes we have learned to draw them and have even taught our sister Prithika who seems to have taken a fancy for drawing ellipses.”
J: “That is indeed an auspicious sign. Now, if I tell you that the existence of conics are a natural consequence of the truth of the gardabha-nyāya (triangle inequality), then would it surprise you two?”
T and G: “That sounds very interesting. Could you please tell us more.”

J: “OK. Now consider the following problem: Given 2 points F_1 and F_2 on the same side of a line l, when will the sum of the distances from a point P on l to F_1 and F_2 reach a minimum. I advise you to follow along reproducing my construction and to think about this deeply by yourself for sometime after I’m done.”

ellipse_donkeyTFigure 4

J: “As you can see, we have point P on l and F_1 and F_2.
1) Reflect F_1 on l to get F'_1. Because it is a mirror image \overline{F_1P}=\overline{F'_1P}.
2) We thus get \triangle{F'_1PF_2} for which the Donkey’s theorem applies thus: \overline{F'_1P}+\overline{F_2P} \ge \overline{F'_1F_2}
3) Hence, as P moves along l we will get \min(\overline{F'_1P}+\overline{F_2P}) only then the triangle become degenerate, i.e. point P lies on the segment F'_1F_2. We can call that version of P point P_0. Thus, we get \min(\overline{F'_1P}+\overline{F_2P})=\overline{F'_1P_0}+\overline{F_2P_0}=\overline{F_1P_0}+\overline{F_2P_0}

You can see that for any given line l there will be only one such point P_0 with respect to the points F_1, F_2 and the sum \overline{F_1P_0}+\overline{F_2P_0} will be some constant k, which will have to be greater than the distance between F_1, F_2. Now, there will be other lines like l on which a P_0 will yield the same constant with respect its distances from F_1, F_2. So what will the total set of such points amount to?”

T: “Since we define an ellipse as the curve on which every point has the same sum of distances from its 2 foci, these P_0 should define an ellipse. Now I see how the mere existence of an ocean of lines and points in the fabric of space automatically specify ellipses!”
J: “Good. So what will that collection of lines be?”
G: “These should all be tangents to the ellipse since only one point on the line P_0 also lies on the ellipse.”
J: “Good. Also observe in the figure I’ve drawn that the angle made by \overline{F_1P_0} and l on one side is congruent to the angle made by \overline{F_2P_0} and l on the other side. As you would have learned the angles are congruent in this way when you reflect a light ray of a mirror. Thus, you can see that if you make a mirror with the curvature of an ellipse and place a light source at one focus then the rays will be reflected and converge at the other focus.

Now the two of you work out a similar case when the points F_1 and F_2 are on opposite side of the line l. However, in this case we instead seek when the difference of the distances of a point P on line l from F_1 and F_2 reaches a maximum rather than a minimum”

hyperbola_donkeyTFigure 5

T: “We learned of another way writing of the Donkey’s theorem: as a+b \ge c we have a \ge c-b. If b \ge c we write the absolute value; thus, a \ge|c-b|. If the absolute value of the difference were to be bigger than a it would violate the Donkey’s theorem. Therefore, a \ge |c-b| and this is the form of the theorem we have to use to find the solution.”
G: “I can intuitively feel that we are headed to a hyperbola.”
J: “That’s right but now work it out fully.”

The two worked it out with their aunt helping and after sometime obtained the following solution on their tablets (Figure 5):
1) Reflect Point F_1 on line l to get F'_1
2) Due to the reflection, \overline{F_1P}=\overline{F'_1P}. Further, since F_1, F_2 are fixed F'_1F_2 is of fixed length.
3) Apply the Donkey’s theorem to \triangle{F'_1PF_2}: \overline{F'_1F_2} \ge |F'_1P-F_2P|.
4) The difference |F'_1P-F_2P| being less than the distance between F'_1,F_2 will attain a maximum only when \triangle{F'_1PF_2} becomes degenerate and P, F'_1, F_2 lie on the same line — let us call this unique P as P_0 and here \overline{F'_1F_2} = |F'_1P_0-F_2P_0|
5) By definition a hyperbola is the curve such that points on it have a constant difference of their distances from the 2 focii. Thus, all P_0 that will yield a certain constant difference k of their distances from F_1, F_2 will define a hyperbola and the corresponding lines will be tangents of that hyperbola.

Jhilleeka remarked: “Here too you will notice that, as in the ellipse, the line l bisects the \angle{F_1P_0F_2}.”
G: “But aunt Jhilli what about the parabola?”
J: “That is a good question. You may know this thing about the conics: There are many flavors of ellipses and hyperbolas but ultimately all parabolas and circles are the same. Imagine an ellipse and a hyperbola with a common fixed focus and passing through a fixed point. Let the second focus of each of them move in opposite directions towards infinity. When the two foci of the ellipse coincide we get a circle. As the second focus moves towards infinity the ellipse converges to a parabola from within. As the second focus of the hyperbola moves to infinity it converges to the parabola from without. Thus, the parabola is a limit for both the ellipse and the hyperbola and is unique. Because of it emerging from one of the foci going to infinity it is endlessly open on one side and has only one axis of symmetry unlike the other two. Hence, we need to take a slightly more complicated approach to understand its emergence from the Donkey’s theorem. Let me show you that.”

parabola_Donkey_bkkFigure 6

J: “For this we have to consider the following setup: Let there be two lines l and l' and a fixed point F. For a point P on l, when will the signed difference of the distances of P to l' and to F reach a maximum? To work this out do this construction:

1) Reflect point F in l to get F'. Hence, \overline{FP}=\overline{F'P}.
2) From F, F' choose that point which is closer to l'. In our case (Figure 6) it is F'. Since F is fixed so is F' and the distance of F' to l' is a positive constant k.
3) From the construction we can see that the distance of P from l' will be k+y when P is above point C, i.e. the point of intersection of l and d, a line parallel to l' through F'. When P is below C then its distance from l' will be k-y.
4) Thus, the signed difference between the distances of P to l' and F' (or F) will be k \pm y- \overline{F'P}.
5) We see that \triangle{F'BP} is a right triangle. A specific consequence of the Donkey’s theorem for such a triangle is that the karṇa (hypotenuse) will always be longer than the two legs. Hence, k \pm y -\overline{F'P} < k unless +y=\overline{F'P}. In this instance, k + y -\overline{F'P} = k and it will be the maximum signed value that it can ever attain. This will happen only when F' lies on the perpendicular dropped from P to l'. We can call this special P as P_0.

You can see that the collection of all such P_0 on lines l defined by the maximal signed difference of the distances of P from l' and F being k will amount to a parabola. The parabola as you all know is defined as curve such that every point on it shows the same distance from the focus and a fixed line. You can see that the fixed line is d the directrix of the parabola.”

T and G: “It is amazing that the conics are a natural consequence of boundary conditions of the gardabha-nyāya.”
J: “Study and think about this simple demonstration more deeply. The old thinkers thought that certain very simple principles lay at the heart of diverse manifestations in mathematics: For example, the soma-drinking Nilakaṇṭha of Cerapada thought that trairāśika-nyāya and the bhujā-koṭi-karṇa-nyāya pervades all astronomical calculations (bhujā-koṭi-karṇa-nyāyayena trairāśika-nyāyena cobhābhyāṃ sakalaṃ graha-gaṇitaṃ vyāptam |). Bhāskara-II expanding on what he mentioned regarding the deva Viṣṇu states further that:

yathā bhagavatā śrī-nārāyaṇena janana-maraṇa-kleśa-apahāriṇā nikhila-jagat-janana+eka-bījena sakala-bhuvana-bhāvanena giri-sarit-sura-nara-asura+ādibhiḥ sva-bhedair idaṃ jagat vyāptaṃ tathā +idam akhilam gaṇita-jātam trairāśikena vyāptam ||

Just as the lord Śrī-nārāyaṇa, who takes away the afflictions of birth and death, who is the one seed that gave rise to the entire universe, causing the manifestation of all the worlds by pervading this universe within his own forms as mountains, rivers, gods, men, demons and the like so also all these branches of mathematics are pervaded by the trairāśika (i.e. the principle of proportions).

You may wonder why so much ado over something as simple as the principle of proportions and its exaltation by comparison to Viṣṇu’s universal manifestation taught by the pāñcarātrika-s. Nephews, I’d say that it is not mere hyperbole nor a peculiar analogy but an intertwined metaphor of how the origin of the universe from the root cause termed Śrī-Nārāyaṇa according to the pāñcarātrika-s, i.e. the conjugate of Lakṣmī and Viṣṇu is an expression of elementary mathematical principles. The principle of proportions governs the measurement of space and existence of scale that is the basic foundation of this universe. Likewise, the simple gardabha-nyāya we have dwelt upon is a pervasive principle linked to the generalization of the bhujā-koṭi-karṇa-nyāya that “selects” for particular genera of curves in space on which the foundations of physics rest. ity alaṃ vistareṇa |

Posted in Heathen thought, Life, Scientific ramblings | Tagged , , , , , , , , , , , ,

Chaotic behavior of some floor-squared maps

Consider the one dimensional maps of the form:

x_{n+1}=\dfrac{\left(\lfloor x_n \rfloor \right)^2 + \{x_n\}^2 }{ax_n}, where \{x_n\}=x-\lfloor x \rfloor is the fractional part of x

What will be evolution of a x_0 under this map when a=2 or a=3? We can see that for x_0>0 it will tend converge. However, the behavior is far more interesting for x_0<0: It turns out that in these cases the trajectory of x_n exhibits chaotic behavior (Figure 1, 2, 3, 4).

floor_square01a2_evFigure 1

Here, a=2 and we use x_0=-1.464; of course all other x_0<0 show comparable behavior (but the choice of this for illustration x_0 will become clear below). The evolution is plotted after discarding the first few values of x_n. While the evolution of x_0 under the map is chaotic it is not entirely random. There are preferred zones and which x_n inhabits. This can be better visualized by plotting a histogram of all the values of x_n in the evolution under the map for 20000 iterations.

floor_square01a2_histFigure 2

We can see that for any value of x_0<0 the iterates will be quickly pushed below -1. Further, we can also see that once a value is -2.5 \le x \le -1 it will remain orbiting within these bounds. Thus, it settles into an attractor in this interval. However, we observe that there is are 2 zones of exclusion in this interval. Even though we initiate the mapping in the middle of the larger zone of exclusion (which is why we chose x_0=-1.464), we observe that x_n moves away from that zone and mostly keeps away from it. As it oscillates between -1 and -2.5 x_n repeatedly approaches the second zone of exclusion from either side but gets repelled by it. How can we precisely determine these repellors of the map? Those can be determined by solving the equation:

2x^2=\left(\lfloor x \rfloor \right)^2+\left(x-\lfloor x \rfloor \right)^2

We have to solve such equations piecemeal due to the discontinuity of the \lfloor x \rfloor function. Because of the interval within which the attractor lies we have to only consider its solutions where \lfloor x \rfloor=-2 and \lfloor x \rfloor=-3. By substituting these two values of the floors in the above floor equation we get the two quadratics x^2-4x-8=0 and x^2-6x-18=0, whose roots will yield the repellors. The solutions of the first are 2 \pm 2\sqrt{3}. Since only the negative root is within interval of for our attractor, we have one repellor as r_1=2 - 2\sqrt{3}. Similarly, from the second equation we get the second repellor to be r_2=3 - 3\sqrt{3}. These are shown as green lines in the above figures. One can see we initiated the map close to r_1 and saw how it was repelled. However, if x_0=2 - 2\sqrt{3}, x_0=3 - 3\sqrt{3} then x_n remains fixed. Thus, while r_1, r_2 repel x_n in their vicinity they are fixed points on the map (green points).

floor_square01a3_evFigure 3

Here, a=3 and we initiate the mapping with x_0=-0.618. We can see that in this case for all x_0 < 0 the iterates with be pushed below -\tfrac{1}{3}. Further, once -\tfrac{5}{3} \le x_n \le -\tfrac{1}{3} we can see that x_n will be trapped in an orbit within this interval. As in the above case, in the example illustrated in Figure 3, after briefly oscillating close to x_0, x_n gets repelled away from it and that region is an exclusion zone for x_0<0. However, unlike in the above case we do not have a clear second exclusion zone here (Figure 4).

floor_square01a3_histFigure 4

Instead, we see that there are repeated attempts to come close to a certain line followed by repulsion away from it to flanking bands. As a result we do not get the second exclusion zone but have a saddle-like distribution around the line that is repeatedly approached. As in the above case to identify the primary repellor and the secondary value that behaves like a pseudo-attractor and also a repellor we need to solve the floor equation:

3x^2=\left(\lfloor x \rfloor \right)^2+\left(x-\lfloor x \rfloor \right)^2

Again we solve it piecemeal. This time given the interval of our attractor [-\tfrac{5}{3}, -\tfrac{1}{3}] we have to consider only \lfloor x \rfloor= -1 and \lfloor x \rfloor= -2. By substituting the first floor value in the above equation we get the quadratic x^2-x-1=0 whose solutions are \phi, -\tfrac{1}{\phi}, where \phi is the Golden Ratio. Taking only the negative value that is in the interval which matters for us, we get the repellor of this map to be r_1=\tfrac{-1}{\phi}. With the next floor we get the quadratic x^2-2x-4=0 with roots 2 \phi, -\tfrac{2}{\phi}. The second of these give us r_2=-\tfrac{2}{\phi} the pseudo-attractor-repellor. This pseudo-attractor-repellor is like the superficially alluring woman who draws you but is a repellor you when you get too close. As in the above case r_1, r_2 are also fixed points of the map.

Finally, we can investigate the evolution of closely separated x_0 to see how closely they parallel each other (Figure 5).

floor_square01a2_evcompFigure 5

Here we compare the evolution of x_0=-0.1464 and x_0=-0.1465 under the first map. We observe that while they are statistically the same in behavior the actual trajectories rapidly diverge. This is a hall mark of chaotic behavior.

Posted in Scientific ramblings | Tagged , , , , , , ,

Notes on miscellaneous brāhmaṇa passages from the Yajurveda

The upasthāna ritual is performed to let the sacrificial fire remain in residence after the primary oblations are complete. In the triple-fire śrauta rite this is done at the āhavanīya altar with several incantations specified in the saṃhitā-s of the Yajurveda, e.g. Taittirīya-Saṃhitā 1.5.5. Among those incantations the following is recited:

indhānās tvā śataṃ himā dyumantaḥ samidhīmahi ।
vayasvanto vasyaskṛtaṃ yaśasvanto yaśaskṛtaṃ ॥
suvīrāso adābhyam agne sapatna-dambhanaṃ varṣiṣṭhe adhi nāke ॥

Kindling you, may we kindle [you] in luster through a hundred snows [winters].
The strong one the maker of strength, the famous one the maker of fame.
With good heroes, the undeceived, O Agni, the deceiver of foes in the highest heaven.

The Taittirīya-Saṃhitā furnishes the following brāhmaṇa passage for the above mantra-s (In TS 1.5.7):

indhānās tvā śataṃ himā ity āha । śatāyuḥ puruṣaḥ śatendriya āyuṣyevendriye prati tiṣṭhati । eṣā vai sūrmī karṇakāvatī । etayā ha sma vai devā asurāṇāṃ śatatarhāṃ stṛṃhanti । yad etayā samidham ā dadhāti vajram evaitac chataghīṃ yajamāno bhrātṛvyāya pra harati stṛtyā achaṃbaṭkāram ।

He recites: “may we kindle [you] through a hundred snows [winters]”. He lives a hundred years and has a hundred senses. Verily he is stable in life and senses. This is a red-hot tube with cutting edges. By this indeed the deva-s have struck hundreds of piercings on the asura-s. When he takes up the fire-stick with this [incantation], the ritualist hurls the śataghnī like the vajra for his enemy’s overthrow without fail.

The text is notable in being, what to our knowledge, the earliest mention and description of the weapon frequently encountered in the itihāsa-s and purāṇa-s and also found in the Arthaśāstra of Viṣṇugupta Cāṇakya. The weapon is described as being a sūrmī, which tradition holds to be a red-hot hollow cylinder. It is described as being karṇakāvati, i.e. with prongs or cutting edges. As its name goes, śataghnī, and from the description of the attack by the deva-s on the asura-s in this brāhmaṇa, it is clear that it was intended to be a weapon of “mass destruction”. Thus, it is reminiscent of what the braḥmaṇa-s are described as having deployed on the Egyptian Herakles by Flavius Philostratus in his biography of Apollonius of Tyna. It is in this context that he says that if Alexander had penetrated beyond the Vipāśā river he might have not been able to take the fort of Indians even if he had 10000 Achilles-es and 30000 Ajax-es with him. Given that the itihāsa-s mention it as being used as a mechanical device for the defense of forts it is likely that it was indeed some form a siege engine of the early Arya-s.

For the same incantation the Maitrāyaṇī-Saṃhitā (in MS 1.5.2.8) gives a rather different explanation:

manor ha vai daśa jāyā āsan । daśaputrā । navaputrā । aṣṭaputrā । saptaputrā । ṣaṭputrā । pañcaputrā । catuṣputrā । triputrā । dviputrā । ekaputrā । ye nava āsan tān eka upasamakrāmat । ye aṣṭau tān dvau । ye sapta tān trayaḥ । ye ṣaṭ tān catvāraḥ । atha vai pañca eva pañca āsan । tā imāḥ pañca । daśata imān pañca nirabhajan । yad eva kiṃ ca manoḥ svam āsīt । tasmāt te vai manum eva upādhāvan । manau anāthanta । tebhya etāḥ samidhaḥ prāyacchat । tābhir vai te tān nirādahan । tābhir enān parābhāvayan । parā pāpmānaṃ bhātṛvyaṃ bhāvayati ya evaṃ vidvān etāḥ samidha ādadhāti ।

Manu indeed had 10 wives. [Respectively,] with 10 sons, with 9 sons, with 8 sons, with 7 sons, with 6 sons, with 5 sons, with 4 sons, with 3 sons, with 2 sons, with 1 son. The 1 son [of the wife with 1] joined with the 9 sons [of the wife with 9]; Those of the she with 2 sons with those of she with 8 sons; 3 sons with 7 sons; 4 sons with 6 sons. The 5 [of she with] 5 remained that. They were just 5. The 10s (i.e. 50) dispossessed these 5. Now Manu had some [possession] of his own. Therefore, those [5] indeed ran to Manu himself. The sought protection in Manu. He gave each these fire-sticks. With those [fire-sticks] the [5 sons] indeed burnt up those [50]. They defeated those [50]. He who knows this and offers the fire-sticks defeats his evil enemies.

This brāhmaṇa is notable for more than one thing: We know that the Hindus were aware of the formula for the sum of natural numbers 1..n from the famous citation of the sage Śākapūṇi in the Bṛhaddevatā regarding the hidden Āgneya ṛk-s associated with the “jātavedase sunavāma…” sūkta (Ṛgveda 1.99). There he gives the sum of 1+2+3…+1000=500500. This brāhmaṇa gives the derivation of the formula for the sum of such a series cryptically and is reminiscent of how Carl Gauss said he computed such a sum as kid. Both brāhmaṇa-s from the two YV saṃhitā-s importantly play on the power of the word śata (100) in the incantation. Once Manu gives his 5 dispossessed sons the samidh-s they are said to have beaten their 50 rival half-brothers. How does this come about? Again it seems to rely on an arithmetic symbolism. When he gave them the samidh-s, as the TS states that samidh makes you śatendriya. Thus, it seems to imply that the 5 become 5\times 100 > 50 thereby beating their half brothers.

That apart it also provides a clue regarding the origin of the story formula of the Mahābharata. We have remarked several times before that when old history was composed it was poured in the bottles of preexisting myth and thereby took their shape. Thus, when we encounter formulaic numbers like 100 (Kaurava-s) and 5 (Pāṇḍava-s) one is immediately alerted to the use of a preexisting mythic frame. Indeed, such a formula related to the count and the differentiation of the Pāṇḍu-s extends beyond the Indo-Iranian horizon. In the Secret History of the Chingizid Mongols we hear the tale of their legendary ancestress Alan Qo’a bearing 5 sons. Notably, there too 2 of her sons are of different father from the from other 3. A birth through a divine non-human father is implied for 3 of them (all 5 in the case of the Pāṇḍu-s). Further, Alan Qo’a specifically instructs her 5 sons to be united despite their being half-brothers. So there is a hint of the need of unity in face of conflict fraternal conflict. These motifs echo what are seen far away in time and space in the Mahābhārata of the Hindus. Hence, we were curious as to whether we might find earlier echoes of the Mahābhārata story formula in earlier Indo-Āryan tradition. We believe that the above brāhmaṇa is indeed the “essentialized” motif of fraternal conflict featuring the number of 5 against 50 that became 5 versus 50 \times 2 in the Mahābhārata where the party of 5 prevailed in the end despite being initially dispossessed as in this case.

Now we move on to an entirely different theme from a brāhmaṇa passage from the Kapiṣṭhala-kaṭha-Saṃhita 35.8.

sā eṣā anuṣṭup । tasyāḥ saptākṣaram ekam pādam aṣṭākṣarāṇi trīṇi । teṣāṃ saptānāṃ yāni trīṇi tāny aṣṭāv upayanti । tany ekādaśa । sā triṣṭup । yāni catvāri tāny aṣṭāv upayanti । tāni dvādaśa । sā jagatī । yāny aṣṭau sā gāyatrī ।parā pāpmānaṃ bhātṛvyaṃ bhāvayati ya evaṃ vidvān etāḥ samidha ādadhāti ।

There was this Anuṣṭubh. 1 of its feet had 7 syllables, the [remaining] 3 had 8. Of the [foot] which had 7, 3 of them went to the 8. That made 11. This was the Triṣṭubh. The [remaining] which were 4 [of that foot] went to the 8. That made 12. This was the Jagatī. The [remaining] feet which had 8 [syllables became] the Gāyatrī (Kapiṣṭala-kaṭha-saṃhitā 35.8).

The model being proposed here is something comparable to what we see in evolution of nucleic acid and protein sequences. The original Anuṣṭubh is conceived thus:
--------|--------
--------|-------

There is fission and fusion with deletion shown by []:
--------|---[----] \; \rightarrow\; ----------- (1 foot of Triṣṭubh) or
--------|[---]---- \; \rightarrow\; ------------ (1 foot of Jagatī)

What remains is:
--------|--------
-------- (Gāyatrī)

Of course, no surviving regular Anuṣṭubh from the earliest Vedic tradition has a 7-syllabled final foot. So is this merely a case of brāhmaṇa numerology or does it reflect some early development in prosody? Now the existence of a basic 8-syllabled unit is something mentioned early in Vedic tradition e.g. by Kurusuti Kāṇva:

vācam aṣṭāpadīm ahaṃ navasraktim ṛtaspṛśam ।
indrāt pari tanvam mame ॥ (RV8.076.12)

An eight-[syllable]-footed, nine-cornered utterance, touching Ṛta
I have measured out its body from Indra.

Being a Gāyatrī with 3 feet of 8 syllables each, the ṛk is testimony for what it states. Further, what is the “nonagonal” form of the utterance? The sūkta in question is composed as sets of tṛca-s of Gāyatrī-s. Thus each tṛca has 3 \times 3 8-syllabled feet giving us the navasrakti structure the ṛṣi mentions. Thus, we can be sure that 8 was definitely a recognized unit of the foot from early one. Further, the very term pāda as a metrical foot has a Iranian cognate pa\deltaa used in the same sense. Further, πούς (poús) as used by Aristophanes suggests that this was indeed the originally the Greek cognate of the Vedic pāda. Thus a comparable metric foot was recognized early in Indo-European. By comparing Sanskrit, Avestan and Greek, Martin West proposed that there are reconstructible proto-feet of Indo-European prosody with 8 syllables and other versions of them with 6 and 7 syllables. If this were true then the brāhmaṇa might be recalling or reconstructing an origin mechanism wherein the shortened 7-syllabled element fragments and merges with adjacent 8-syllabled feet to give rise to the Triṣṭubh and Jagatī.

In any case this irregular hypothetical Anuṣṭubh is not the standard Anuṣṭubh of Sanskrit which since the earliest Vedic record came to be of 4 \times 8 syllables. Indeed, this form of the Anuṣṭubh was the basis of a child’s mathematical problem presented by Bhāskara-II in his Līlāvatīː

samānām ardha-tulyānām viṣamāṇām pṛthak pṛthak ।
vṛttānām vada me saṃkhyām anuṣṭubh-chandasi drutam ॥

Tell me quickly what are the counts of the variations of Anuṣṭubh meter with:
(i) same [configuration of feet]
(ii) half-equivalent [feet]
(iii) dissimilar [feet]

Each syllable can take a short (0) or a long state (1) . If every foot has the same configuration of 0,1 then we only have to look for combinations of one 8-footed element. If it has half-equivalent feet the two 2 \times 8 syllable configurations are the same. But we have remove all instances where all 4 feet are same for a strict definition. Now if all feet are dissimilar then we have 32 syllables available but we have to remove all those where either all or a pair of feet are the same. Thus we get the answers as:
(i) 2^8=256
(ii) 2^{16}-2^8= 65536-256=65280
(iii) 2^{32}-2^{16}=4294967296-65536=4294901760

Interestingly, these numbers make an appearance in modern computing e.g. as in 16-bit and 32 bit computing and the largest 32 bit number.

Posted in Heathen thought, History | Tagged , , , , , , , , ,

The culmination of Galtonism or pandemic days-3

We saw a list of famous elderly people whom the virus has already placed in the abode of Vaivasvata at the time of writing. We recognized at least two names: (1) Robert Carroll the paleontologist, whose hard to find book on vertebrate evolution had influenced us as a youth. (2) John Conway, the creative mathematician, whose work influenced us in a similar way as Hofstader, his rival in the realm of sequences.

But are we seeing the first glimmer of light at the end of tunnel? In this note we shall look at the turning of the curve in parts of the world beyond the root-source, the neo-empire of the “sons” of Chin Shi Huang.

covid19.npluscSun.Apr.12.2020

Figure 1: This is the daily infection curve after the first 35 cases to date for selected countries in the \log_{10} scale. One can see that the worst hit countries in Europe, Italy and Spain, are turning the curve suggesting that this round of the infection has run its course and is flattening. France is on a similar trajectory as the other Romance European nations but earlier in the game; however, its curve has also probably already begun to turn. Germany too, which was hit by a large number of cases but has had fewer deaths, has turned the curve suggesting that they have a reasonable chance of emerging looking better than the rest. England has done much worse than its continental cousin and old enemy Germany. We cannot make much of its case load because they are not testing as much. But if what we see is a reasonably random sample they could be near a turn. England’s bigger cousin across the pond, the US, is also showing signs of having begun turning. India, while keeping the infection load relatively low is still in the growth phase and it is not clear when the turn might happen. South Korea may be seen as the control case where the curve has been successfully flattened.

covid19.nplusrSun.Apr.12.2020

Figure 2: This shows the rate of infection averaged over 3 days since the first 30 cases. South Korea is the only country that has brought the rate down successfully. India has kept a lower rate than the rest but as of now it still seems to be growing. The European states seem to generally cluster together and the rate plot does indicate that they have already or have begun flattening the curve. US has also begun the flattening process albeit at a much larger rate commensurate with it being a bigger country with many parallel outbreaks.

Now, let us take a closer look at Italy for which was the first to be hit in Europe and for which I was able to find detailed data made available on GitHub by Umberto Rosini et al:

italy_mort.Sun.Apr.12.2020

Figure 3. The daily progression of various disease statistics is presented starting from 24th Feb 2020. On Mon Mar 09 2020 after a steady increase in death rate and mortality percentage Italy went into lock-down. The effects were first seen approximately 2 weeks later and it took nearly 21 days for sustained flattening of the infection rate, decline in hospitalizations, and gradual decline in death rates to be achieved. So Italy is test case of a country with a severe epidemic where the lock-down did do its job. A similar trend is being observed in the other hard hit countries. So in the least this shows that the lock-down can consistently control the epidemic and perhaps there is light at the end of the tunnel.

Noting this we are seeing the “economists” raise their head again saying all this shutdown is an over-reaction after all and it is not as threatening as it was made out to be by the “epidemicists”. Their opinion is buoyed by preliminary reports of the serological survey coming from the Gangelt town in Germany that many of the asymptomatics (~15%) have developed antibodies against the virus. While we would all be happy to be done with this thing, we should add that one can hardly get over-confident or think that this thing is over from the above evidence of the turning of the curve. The curve has turned because of the logical and easily demonstrable consequence of limiting social interaction and movement in the population and not by itself. The one country that is for most part letting the virus run through it is Islamic Iran and we do not have reliable information of what really has happened to them. While entirely anecdotal, statements by their head Ayatollah that they do not want to lock-down even if a “2 million” Islamic Iranians die suggests that they are facing much higher tolls than acknowledged. Indeed the figure stated by the head Ayatollah might be close to the order of the deaths in an unmitigated epidemic.

In this regard let us consider the following:
(1) At this stage letting up on the lock-down because the curve is beginning to turn would simply waste all the effort that has gone in. The case of Italy (Figure 3) shows that the climb down is a gradual one and even after 5 weeks of lock-down things are hardly close to life as usual.
(2) Let us assume there are lot more asymptomatics and untested individuals than the actual numbers. For Italy, as of writing we have: 156363 cases. It has tested 1.674% of its population at the time of writing. Let us assume that the true infected numbers are 50 times the detected cases: in our opinion a reasonably relaxed assumption for the asymptomatics and those not tested (taking that Gangelt study at face value: in reality could lower). That means \approx 13\% of the population of Italy has been infected. Let us next assume they are all immune to the disease in the near future. That is a lot but still not enough to give you the so-called herd immunity. Hence, we have our doubts that even assuming massive under-detection Italy has reached state of herd immunity against this virus, which would need 50-70% of the population to be immune.
(3) The presence of antibodies does not necessarily mean complete immunity: we still do not have a clear idea of: (i) how protective are these antibodies; (ii) how dependent is such immunity on infection dose; (iii) how long this immunity lasts.
(4) Finally, when the infections finally fall to a manageable level how well can a country take the next step of more specific tracing and containment of the holdouts? If that cannot be worked out, given the above that we are likely not near herd-immunity, there is considerable risk to at least some countries (more on this below).

All these are considerations at the gross country level but what about more circumscribed systems like New York City? 103208 people in this giant city of \approx 8.242624 \times 10^6 were reported as being infected at the time of writing. That makes it \approx 1.25\% of the city. Now, if they were under-testing by 50 times then the city might be closer to herd immunity. However, given that they comprise roughly 15% of the tests done in the US, they are likely not under-testing by 50 times but perhaps more like 15 times. So, even in NYC probably only about 18..20% of the population has been infected, which is still short of herd immunity. Serological tests will tell us how far these estimates are from the reality. In light of all this uncertainty, as the Lebanese thinker N. N. Taleb has often pointed out, there is much reason to be conservative and cautious, i.e., keep over-reacting rather than the opposite in the coming days despite the fatigue we are facing.

Now we come to some geopolitical considerations. There are some natural systems where a parasite is mild in one host and severe in another. These might be used by the first against the second host if they are in a biological conflict with each other. One classic example is the Dhole-Sarcocystis-herbivore system. Here, part of the population of the dhole, a social canid found in Indian jungles, is infected by the apicomplexan parasite Sarcocystis which it sheds via feces. The effect of Sarcocystis in the gut of the dhole seems to be relatively mild. However, when herbivores like hare or deer consume the fecally contaminated plants they get infected by the parasite, which invades their heart muscles and weakens their stamina and probably speed. This makes them easier prey for the dholes and allows completion of the parasite cycle. Bats harbor lots of coronaviruses and filoviruses among others to which some of them might be immune. Bats are also social and engaging in communal behaviors such as guarding of pups and harems. One wonders if their partly domesticated viruses help in the warfare against rival assemblages of bats competing for similar nesting resources or harems.

In the current crisis it is possible that some nations have used their relative success against the virus in a similar capacity as above. Some hold that the Urheimat of the virus, Cīna-deśa, at first sight, might seem to have benefited in the net from unleashing this on the world by way of their lies and manipulation of information. However, this is not certain for now because of their Galtonian linkage with the Occident. The Occident being in lock-down is no longer serving as a good market for the Cīna-s. Further, there is at least some impetus from the pandemic in the Occident, Japan and India to decouple from the Cīna-s. If this goes through then the Cīna gains from this might be limited and it could even prove detrimental for them. While the Cīna-s are trying to use the opportunity to acquire ailing assets in Europe or Australia, there is a concerted effort to try to limit this. Hence, their gains remain unknown. An election result in the US presidential elections that unseats the current ruler could, however, change this and place the Cīna-s in an advantageous position. This can be amplified by the Cīna assets among the left-liberals and academics in the West who are ceaselessly batting for them as we speaking. Indeed, the removal of dogs from the list of consumable livestock by the Cīna-s rather than being a public health measure is a signal to these assets in the West who are typically cynophilic in their orientation. In Europe, the relative success of Germany might mean its even greater ascendancy in the continent perhaps along with the German-speaking Austria — this was not certain until recently given that they could have been badly hit by the large number of vallūra-dasyu-s streaming in from the marusthala. Their traditional rivals the English are also doing poorly suggesting that they ability to limit the śūla-puruṣa-s in the near future might be limited. But we must confess that the long-term prognostics of this event are not entirely clear for we are hardly done with it and the aftermath could have its own fluctuations. In Asia, so far Vietnam, Taiwan, S. Korea and Japan have done relatively well at differing levels and this might help them to differing degrees. The low-fertility Koreans might not be able to gain much beyond surviving this, but the Vietnamese and Taiwanese could benefit from any manufacturing shifts from China. The Japanese might have shown a middling success against the Wuhan disease but, as ever having the islands for themselves, they are probably going to suffer economically to a lesser degree than some of their rivals.

Finally, we come to India, which has done well in two ways: (i) It started mitigation early with the Dillīśvara realizing the threat from this virus early. (ii) For reasons which are not entirely clear India has kept relatively low rate of infection (Figures 1 and 2). This has puzzled a lot of people in the Occident, even as the traditional constituents of the mleccha-marūmattābhisaṃdhi are hoping that Bhārāta goes down to the virus. We routinely see that public Western liberal/academic types egged by their iriṇavātūla friends are barely concealing their wish for the worst to happen to India on the internet — just ask yourself if these types would hope the same for Cīna-s, śāntidūta-s or mūlavātūla-s. Initially, people thought it was the weather and more recently the theory regarding the role of the BCG vaccine has gained some traction. We are not certain of either of these because a counter-argument can been made for both proposal based on the countries which are doing badly.

However, we are not entirely unsympathetic towards the BCG explanation. It reminded us of an inquiry we did with Aurvaśeyī as undergrads. She was at that point still in the MBBS course and asked me what I thought about something she had dug up: Albert Calmette one of the inventors of the vaccine had noted that the mortality of children who had received the vaccine was significantly lower first few years of their life than that of children who had not. She pointed out that this had since been reproduced in India, where both of us had received the vaccine. I did not have an immediate explanation but became curious about this. We soon dug up some information that in some places people receiving BCG even had reduced mortality from malaria and other febrile illnesses. Other studies have shown that BCG vaccination reduced hospitalizations from respiratory diseases other than TB in children under 14 years. Thus, it became clear that the BCG vaccine was activating a more general immune response that had a broadly protective value. Hence, this can in principle be a factor in the resistance to the Wuhan disease in India. Our review of the BCG immunology revealed that a lot of good stuff seems to happen upon vaccination: First, the adaptive T-cell based immune response involving both CD4+ and CD8+ T cells with Interferon-\gamma production and enhancement of anti-mycobacterial activity of macrophages occurs. This IFN-\gamma also activates B-cell based immunity via antibody generation against the TB actinobacterium. Second, 1-2 months after BCG vaccination there is induction long-lived memory B cells that can provide long-term TB protection. More non-specifically, it might also induce IgG and IgA production against other pathogens. Third, it seems to have a complex role in regulating various immune responses, such as one hand inducing TH1/TH17 responses to unrelated pathogens. On the other it might induce a Treg based immunomodulation or one where activated CD4+ T cells undergo IFN-\gamma-induced apoptosis. This could control the hypercytokinenemia seen in SARS-CoV-2 pathology — indeed BCG has shown promise in certain autoimmune conditions. There could be other actions that we poorly understand like the immunity-enhancing roles of Mycobacterium indicus pranii or Mycobacterium vaccae. Indeed, we have wondered if M. vaccae might have general health benefits that can be widely exploited further along with a BCG vaccination.

Whatever the case, so far the basic growth of the Wuhan disease in India is at a lower rate. However, in India it has become a weapon for bioterrorism in the hands of the ārtanarākṣasa-s who have been the primary cause for countering the otherwise rather effective measures of the Lāṭanareśa. Hence, we fear this could cause the disease to fester in the deśa because contact-tracing and enforcement of isolation is difficult in the deśa and particularly so in the mini-Pakistans that dot it. The śāntipravartaka-s having seen an opening could continue to use it to their advantage even as their rākṣasotsava breaks out. However, there are things which cannot be stated in public and if the opportunity is seized in conjunction with mollifying the mleccha-s with hydroxychloroquine and the like the Hindus could still come out relatively less harmed by the pandemic. At this time it is we remain cautious about the prognosis of the deśa and the conflict between the opposing forces hangs in balance.

Posted in Life, Politics, Scientific ramblings | Tagged , , , , , , ,

The culmination of Galtonism or pandemic days-2

Ensconced in the apparent safety of the 4 walls the mind looks out into the completely silent streets with hardly a soul or even a passing ratha — a mere 120 days have made the world look and sound different. With that realization, we place here the 3rd in this series of notes recording these times. This note will cover some very basic stuff on visualizing disease progression in a population which we used to explain things to laypeople (bālabodhana).

We hear laypeople express surprise over things like: (1) “Just a few weeks ago there were just 10s of cases of the Wuhan disease now it has just exploded.” (2) “What is the point in staying at home? How can it reduce the disease?” (3) “There are all these predictions of millions of people dying and now it has come down to tens of thousand. There must be some over-reaction due to X or Y.” As long as the person is willing to have a patient discussion we found that we could explain the basics relating to these using some illustrations which don’t use any sophisticated mathematics and just a simple program which we wrote for the purpose. One can easily reproduce similar illustrations with ones own assumptions in whichever is the language of ones choice. When we try to clarify point (3) we tell people that what we are showing them is not like any of the complex models that they hear of in the news. We explain to them, extrapolating from our simple program, that the forecasting models have many, many ways in which they can be parameterized. We do not know the correct values of several of these parameters and as result of that there can be tremendous diversity of outcomes due to the combinatorial interaction of the alternative values of parameters. Hence, we simply cannot be certain which forecast of the model is going to actually play out. However, the simple program does explain some of the essential outcomes that a layperson needs to understand.

Our model for how disease can progress in a population is a very simple one: (1) It assumes a population starting a certain fixed size n, which undergoes decrement from deaths due to infection but beyond that there are no other factors changing its size. This simple assumption is good enough for a fast-moving epidemic like the one we are saddled with. In what we illustrate below n=40000. (2) We initiate the disease with a single infected individual introduced into the population and let the disease run for a certain maximum number of time units. One can take them to be days if one wants; m = 25 in our example. (3) If a person in the population catches the disease he remains infectious for 7 days, i.e., can transmit it to someone who is in the vicinity. Within that period the infected individuals can die with a certain probability; d=.02 in our below illustration. (4) After that period if the person has not died he is considered as recovered and is immune to reinfection. This again is a reasonable assumption for a fast-moving disease. Thus, our process features 4 states for individuals in the population: Uninfected; Infected; Recovered; Dead. These are respectively shown in different colors: white; brown; green; black. (5) Not every person who has gotten the infection spreads it uniformly to others in the vicinity. Here, we assume a power-law distribution of the capacity of an infected individual to infect others in the vicinity in an encounter. One example goes thus; an infected person in an encounter infects:
0 persons 0.747475 of the times
1 persons 0.204900 of the times
2 persons 0.025200 of the times
3 persons 0.008725 of the times
4 persons 0.004550 of the times and so on till,
124 persons 0.000025 of the times. They all add up to 1.
This assumption captures the fact that most encounters do not actually result in an infection but some rare people are “super-spreaders” so they infect a large number of people in an encounter.

In its default state, people in our population randomly move at each time unit with little restriction on movement. Infection is transmitted with the power-law-defined frequency to k neighbors when they encounter an infected individual. This goes on till the set maximum of time units m is attained. But we can set a restriction parameter, which decides how freely the people in the population can randomly move to encounter others. The greater this restriction parameter the less they can move about to bump into others.

For each run, the upper panel shows the actual numbers of the 4 states in the population at each time unit of the simulation. The lower panels illustrates the state of the population at each of the 25 time units as 1 dot per person, colored based on the individual’s state.

disease_prog1

Figure 1 shows a run where there is essentially no restriction on movement. It begins innocuously with most of the population in the uninfected state and within 25 time units the entire population goes through infection and ends up as recovered or dead. This helps bring home to a layperson how the disease progresses: the initial stages always look innocuous and how an explosion happens by exponentiation and can overwhelm any treatment system. An analogy can also be made to a fire: as long as there is fuel it continues to burn but as the fuel decreases there is a down turn in the burn rate and finally the fire ends.

disease_prog2

Figure 2 shows a run where right from early on free-movement of the population is somewhat restricted. Here, we see some mitigation of the deaths and infections. Still majority of the population gets infected.

disease_prog3

Figure 3 shows a situation where much stronger restriction is imposed on the free movement of people. Here, a considerable fraction of the population is uninfected, deaths are considerably reduced and the infection can be brought to zero in the end.

The latter two scenarios illustrate to the layperson the value of early mitigation measures in the form of restriction of social interactions or free movement in controlling such epidemics. Of course, this is a very simple illustration and in no way should be taken as a forecasting model. However, as noted above forecasting models suffer from great parametric uncertainty and a combinatorial increase in search space when the multiplicity of alternatives are taken into account. Thus, a simple model like this, which provides some basic qualitative understanding of what seems puzzling to laypeople is good enough for them to get some intuitive feel for the situation.

Posted in Life, Scientific ramblings | Tagged , , , ,

The culmination of Galtonism or pandemic days-1

While many thoughts crowd our mind we have hardly felt the drive to put them down on a more visible medium. Such is the nature of the times when things happen reminding you of the inexorable turning of the yuga-cakra. Though you know what is going to play out in advance, when you watch the event unfold, and have it impinge on your life and finally open up a new horizon of uncertainty you are left with a certain paralysis where you are a spectator who is unable to narrate the unfolding spectacle. There are nights when the rush of thoughts break into your hypnogogia and there are days when you get up thinking of the steps you need to take while heading out for the purchase of essential groceries. Then the passing of days and weeks start to feel like a blur and all the omissions and commissions you might have done in the preparation knowingly or unknowingly unfold under the 20-20 vision of hindsight. Though proximal time might feel indistinct and yuga-cakra is ever-turning it is not often that we feel its turn like this: “vartate vartate iti kālavādino vadanti |”

There is no doubt that the utterly evil ways of the Chinese Communist Party, the contra-urban food and medical customs of the Cīnaka-s, and their questionable scientific practices are at the root of this. If someone were to object that the second point is “racist”, all one needs is to note the tacit admission of the CCP itself in the form of a cosmetic ban on the bush-meat and “wet” markets. Instead of providing proper information of the epidemic that had started in their midst they doctored the numbers and used the WHO to spread propaganda even as they rushed to publish crappy papers in the much sought-after scientific tabloids to the adulation of their gullible peers in the Occident. But history does not care and on the other side is particularly harsh to those who despite having prior intelligence fail to take steps of self-preservation.

As the event scrambled our existence, in the backdrop were the progressivists who used to be convinced of the “success” of humans. In our youth we saw them in the form of the Soviets and the socialists inspired by them. They used to talk breathlessly about Man’s triumph over nature and the end of the superstitions like the many gods of the Hindus. “An unimaginative crowd” we used to think. Later, we saw their counterparts in the academe of Krauñcadvīpa. They talked of the end of human disease being at hand, if only a little after driverless cars. Some spoke of the great revolution of artificial intelligence in classifying chest X-rays or micrographs of sputum smears as the biggest thing in medicine; others chattered about “big data” as the next savior almost on par with the messiah whose psychosis had memetically infected their ancestors; yet others were almost ready to celebrate the dawning of an “Aquarian age” with personalized precision medicine ensuing from a USB drive sequencer plugged to their smartphones. Their voices found resonance far and wide, well beyond their political sphere. The layfolk in my family, mouthing biological and biochemical jargon they hardly understand, would ask me when [insert some technical jargon here] would be bringing the great panacea for all humanity. Seeing my unexcited responses or words like “Time will tell” they would cock a barely concealed snoot as to why my work didn’t concern itself with things which might place my name in the newspapers. The general triumphalism apart, there as a specific type which largely saw infectious disease as “something that mainly troubles Africans” who needed their benign favor and was something which was mostly conquered by the “triumph of evidence-based medicine”.

In contrast to these were those who either had some real biology education (experience has shown us that this is not a very widespread commodity) or those with an uncanny intuition, which again is not necessarily common. The former know by virtue of their saṃskāra that the battle between the viral and cellular genomes is a quintessence of life itself and is essentially an eternal process even as our Iranic cousins imagined the struggle between Ahura Mazdha and Angra Mainyu. Once one gains the upper hand and then other and it goes on. Having been a student of the work of Leigh Van Valen in our youth, followed by investigations of our own into these matters for several years we had come to realize that biological conflict is an inseparable feature of life. The apparent state of conquest of infectious disease that many of the “progressivists” saw was merely a passing phase — it was an unstable state which could easily breakdown some day and shift to another more dismal equilibrium. Of course we did not know when it might happen and certainly in December of 2019 CE we had no inkling that it was at hand.

What we got was the ultimately the fruit of Galtonism, the consequence of entanglement of China and the Occident. While seen as rivals at the superficial level the two have had a certain symbiosis, whose early expression was seen in Galton’s admiration for the Cīna-s and willingness to concede to them a sphere of their own which the rapacious Anglosphere had no intention to concede to any other people (his letter to The Times in 1873). In turn the Cīna-s have come to intensely value the standards set by the Occident as their own. Even as the Kauśika is said to have sought above all the validation of the Kauṇdinya the Cīna-s value that of their Occidental rivals. As we noted before when this is combined with a hunger for large lives in the Occident it led to an intimate entanglement of the incompatibles: One sheds no tear while gulping down a cow or a turkey, all while going apoplectic at the other devouring a roasted dog or a cat. That coupling exploded in the form of a virus whose invasion of Homo sapiens stems directly from this unstated intimacy. The Galtonian spirit which has spread beyond the Anglosphere to much of the Leukosphere was hale and bold when the depredations of the virus were still in the distant Orient. Several in the Western academe had in the recent years come to be recipients of the Chinese largess — jobs, grants, pleasure trips to the Great Wall of the father of the nation Chin Shi Huang, sometimes even sexual favors, and above all a fulfilling validation of their hope of a rightly enacted socialist paradise. Thus, there was an Ivy League virologist who was shrilly decrying any sort of barrier to travel relating to the Cīna-s, while others were enacting ritual embracing of the Cīna-s, even as a deśi physician, signaling his imbibition of Occidental virtues, was admonishing his listeners about their racism. But the world changed even before people processed any of that.

Even as the virions were permeating the atmosphere in the Orient, we saw two camps emerge in the lands distant from it. There were those who kept telling us ad nauseum that the risk from the virus to the average person was minimal or that it would fade away like SARS — after all the virus was (almost) a boring brother of the old SARS-CoV with an even lesser mortality rate. Some bought into the Cīna numbers and said that it was not particularly fatal — you know 1000-3000 old people might die, at worst a few times that of SARS. Even people whom we know personally were telling us that it was just a “cold” or in the worst case a “flu” or that it does not seriously affect “healthy” people. If this was one camp, some of us belong to the opposite camp, which saw the obvious and tried to tell them that it was not a joke and that we were actually on the brink of the precipice. It did not matter whether we convinced them or not for soon they had the pratyakṣa drop on their heads dragging all of us together into house-arrest.

Even as one of these camps was rallying around Michael Levitt or whoever, nations throughout the world were going into lock-down. In some places like Islamic Iran, which had embraced the Cīna-s for reasons distinct from the Occident, the true price of dependence on the Cīna-s was brutally revealed. Even as the viral fires were ablaze in Islamic Iran they were more bothered about taking shots at the Hindus, despite the fact that India is one of the few nations that shows them some sympathy (perhaps due to some vāsana of our pre-Islamic shared ancestry). The virus was to soon knock them down, leaving them fighting for survival. Likewise, in faraway Ecuador which was almost being purchased by the Cīna-s corpses have been literally piling on the streets under the unstoppable viral assault. In Europe, the Italians, who had in the past decade gotten closer to the Cīna-s than everyone else and signed up for latter’s imperialist Belt and Road Initiative, were the first to be struck. As of the time of writing 1 in every 485 Italians is known to be infected by the virus (the real number is definitely higher) and 1 in every 3937 Italians is dead from it. Then the Spanish (while writing 1 in every ~358 Spaniard is infected) and the French (1 in every 725 Frenchman is infected) went down to the invasive RNA. The English and Dutch still thought life could continue as usual but they were soon kicked into submission by the virus with notable losses. A similar case played out in Sweden which responded slower than its Scandinavian neighbors, showing that no nation which ignored this virus could hold out against the punch it delivered via the exponential function. Mysteriously, while the Germans, Austrians and Portuguese registered a lot of infections like their neighbors, they have had significantly fewer fatalities than the Italians or the Spanish. The European Leukosphere was mostly defeated by the virus and it is not over yet.

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The sole superpower of the world was also dealt its worst defeat in recent memory by the virus, exposing it as the frontier nation rather than a complete superpower. As a frontier nation it enjoys the great defensive moats of the Atlantic and the Pacific that separate it from the rest of the world but the Galtonian entanglement meant that these moats were no defense against the viral invasion. The results are devastating — at the time of writing this one person is dying of this Wuhan disease nearly every minute in the US and that is only going up for now. The causes for this dismal failure of the US are many, and a grim reminder that military superiority does not guarantee safety from a pathogen:
1) Even as the virions were swirling through the Eastern air the US had ample time to prepare but smug in it narrative of exceptionalism failed to take critical steps.
2) Instead, its leftward political elite were lost in fighting the emperor and trying to impeach him. For them the virus was for most part a means of getting at their political opponents rather than a real cause of fear. In fact they were more eager to bat for the Cīna-s than take care of their own home.
3) While a military superpower, as is typical of a frontier nation the US is imminently ill-equipped to handle serious medical and natural crises. We see this on a small scale when calamities like hurricanes or tornadoes strike. But the generally low disease load of the North American continent has meant that there is no pressure for a good medical system. People pay a crushing amount for substandard care, often by incompetent physicians. The physicians are given enormous autonomy and power in medical decisions but they are not very responsive to the patient’s conditions or actual needs beyond mechanical prescription of medications and procedures. Further, many are part of a triumvirate with the insurance companies and the drug makers as the other two nodes who ensure that they get rich while the patient suffers. Such a system can easily come apart when hit by something as devastating as this virus.
4) The decentralized nature of the States with each state having its own style of doing things can prove to be advantageous in certain circumstances but in a situation like this it is a recipe for disaster. While one arm of the American elite was saying how a totalitarian system like China cannot handle this epidemic they did little self-introspection of whether they were capable of handling it themselves. On the other side the liberal elite, which is an antithesis of dharma, was watching with bated breath if India might go down to the virus while caring little to look into whether their own house was secure.
5) Major failure of the political leadership: at every level there were failures in the political leadership across party lines. Politicians did not prepare the people for the onslaught. As a frontier nation they had little in place to ensure that all people got basic protective supplies such as hand sanitizers, disinfectants, gloves and masks. In such an epidemic population-wide protection is critical not just that of a few who bought such in advance. These critical supplies were scraped by price-gouging hoarders and others who might have even been Cīna agents (we know for certain they did in Australia). Even today there are people who don’t have these critical supplies. Further, several politicians showed little will to enforce lock-downs and put an appropriate testing plan in place.
6) Given that they had outsourced manufacturing to the Cīna-s, key medical equipment like ventilators or N95 mask are not available in sufficient quantities. If the Cīna-s block raw material supply several critical drugs could be in short supply.
7) Failure in public medical advise and action: People were seriously misinformed by health agencies at various levels. People were told that the risk to them was low and they were asked to go about their lives as usual, including attending large gatherings. Even after the risk of cruise ships became apparent people were not advised against taking them for while. To this date we have sick people from cruises trying to get back home. They were told to wear masks only recently after the infections had crossed 300000. Screening at the airports was a disaster causing more people to congregate in close proximity in several places. Testing was not gotten off the ground quickly with a lot of problems ranging from faulty kits to shortfalls of RNA extraction reagents. Importantly, proper planning to screen, sort and treat the infected patients from those seeking other treatments or tests was not properly managed. Here, they could have learned from the Cīna-s to set up a chain of action where first one screens for fever, followed by a flu test, followed by a CAT scan and finally tops it with a SARS-CoV-2 test with those testing negative being taken out of the pipeline in each step.
8) Social irresponsibility: far too many people took this to be a party. Not just them but everyone around them is now paying the price.
9) The politicians and administrators have no plan for what to do after the lock-down is released.
10) The above are all rather apparent to any observer and are mistakes in part committed elsewhere in the Occident. However, there is something more subtle which we cannot articulate fully in public. It concerns a certain ideology which has taken deep grip of the Occidental institutions and forcibly drives people farther from reality towards a certain solipsism which can only be compared to the state of the New Zealand paradise of birds before the coming of their Austronesian butcher. We have been noticing this for while and felt that it will result in the ultimate demise of the Anglospheric ascendancy. The inability to properly respond to the virus in large part stems from this — when you have a system that rewards and lavishly funds insubstantial falsehoods as education rather than basic scientific education that helps people understand such diseases, survivorship in face of them, the result can be similar to what we are seeing now. Sadly, the purveyors of this ideology are mostly well-heeled and perhaps more insulated from the blows falling on the population. This virus may not by itself bring the end of the system but it will definitely have reverberations that are not going to die out anytime soon.

Finally, we come to the peculiar situation of India. While there is lot of trade between India and China the degree of entanglement is not exactly Galtonian. This shielded India in the initial phase of the pandemic. However, with fires lit all over the world it was just a matter of time before new sparks fell on India and sure they did. Now they are being spread by the arsonist śantidūta-s supported by the Occidental liberals even as they spread the viral infection alongside their memetic infection. What turn it would take the coming days would tell us.

In all this, as a good part of the world is confined within the four walls (so much so that the earth is apparently quaking less from our confinement), we again see the emergence of two camps. We can call one the “economists” and the other the “epidemicists”. The “economists” oppose the lockdown as it rather obviously demolishes the economy. The “epidemicists” hold that there is really no way out of it for now. While it is undeniable that most of people are losing money (us included) or worse have no employment, there is really no alternative the “economists” can offer. The pratyakṣa of the Cīna-s is there for all to see. The CCP has nurtured a nation of arthasādhaka-s who value kārṣāpaṇa-s over all else. Now if they shutdown their country then this thing must be really bad. As we noted above, those who failed to do so elsewhere were soon confronted with mounting deaths and overwhelming of the medical system such that the economy could fall apart like in a catastrophic military defeat.

Does that mean “epidemicists” like us are suggesting a perpetual lock-down. No we all understand that there has to be a plan to restore continuity of economic activity and the lock-down is only a means to buy time. But you cannot really buy time unless you have a clear plan for you are going to do next. This leaves us with tremendous uncertainty about the future. It is not all clear if most nations have a clear plan forward. This is especially true of countries with: 1) low social trust due to juxtaposition of fundamentally incompatible populations; 2) those with governmental structures which cannot quickly divide their population into “dirty” and “clean” zones and restrict internal movement between them. In the absence of such drastic or alternatively smart measures we are more or less destined the let the virus ravage the population at some rate before a significant population acquires some kind of immunity. While this might sound alright on paper, even with low mortality estimates it would mean a large absolute count of deaths and long-term morbidity that would seriously limit the functioning of modern urban systems. Indeed, a crisis like this reveals true alliances. One can see precisely why the Hungarians have decided to invest their leader with sweeping powers and break with their indignant fellow Europeans neighbors. In the immediate future there are urgent production issues, dealing with which is easier said than done: everyone needs to have good quality masks, sufficient supply of disinfectants and the health-care workers who are like soldiers being sent into the battlefield without proper gear need the appropriate supplies to do their difficult jobs. From a biological perspective the essential steps can be plainly stated:
First, on the prevention side, yes, the virus is highly infectious but there needs to be better understanding of its transmission so that it can translate to engineering solutions: 1) What is the relationship of the probability of contracting infection to the duration and distance of contact with an infected individual? 2) What is the fraction of asymptomatics and how infectious are they? 3) What are the weak-points of the virus? What are the best ways to inactivate virions on surfaces and in the air? The final point would require a quick translation to engineering solutions: right irradiation treatments, redoing fittings using copper etc. This is where our modern technological achievements can actually help in a substantial way; however, it should be kept in mind that the engineering solutions would span an entire range from simple things like making masks to tasks needing complex programmatic automation.

Second, in terms of actual biology, we need to understand the immunity against this virus. How long does protective immunity last after exposure to the antigen? Being a coronavirus it is likely to be mostly slow-evolving; hence, the vaccine strategy might work but on the other hand its main cause of mortality and morbidity is from an immunological sideshow. Hence, this could in theory place an impediment to easy development of a safe vaccine. So, the next question is there a genetic background that predisposes individuals towards a bad immunological response? There is prior knowledge that natural genetic variants are associated with predisposition to different forms of hypercytokinemia. Whether any of these or others have a role in the hypercytokinemia associated with the morbidity from SARS-CoV-2 needs to be investigated on a war-footing. Finally, the an immediate step for restarting economic activity is combining serological detection of viral exposure with the existing RT-PCR based methods. This should be done in setups emulating the Korean model that has been one of the few which has had a measure of success in this war. If the protective immunity is reasonably long-lasting then seropositive individuals can be at the forefront of re-entering the workforce without fear and restart the frozen economy to a degree.

There are lots of other thoughts that crowd our mind but we shall stop here. We have not covered the geopolitical consequences or the socio-biological factors. The latter have some important implications and need to be seen from the viewpoint of the “public health” responses of social arthropods such as hymenopterans and blattodeans which have had the longest lived social systems on earth. The conclusions, while deeply troubling to the Occidentally conditioned minds, have considerable consequences for the emergence of disease responses in social groups. If the gods favor us we might cover that in the future.

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We end with a plot of number deaths versus number of infections for countries with over 10 deaths (names abbreviated by first 4 letters). The lines of different colors represent 1..5% mortality rates. The results show that not all countries have fared the same. While some of this is from trivial stuff like amount of testing, overwhelming of the system or plain lying, not all the differences are purely from these factors. Hence, we need understand the underlying causes better as we go forward.

Posted in History, Life, Politics | Tagged , , ,

Snatches

Vidrum had called on his friends Somakhya and Lootika on a quiet afternoon to accompany him for a climb on the trails of Vidrumavistāra that lay beyond Viṣṭhaparvata. The fierce fighting arising from a surprise ghazvat of the makkha-viṣaya-dānava-s had been repulsed and the region was going through a quiet interlude. Vidrum and Lootika’s sister Vrishchika had been kept busy by the peculiar case of the European ambassador’s sudden klazomania. He was keen to take break from that high drama. As they reached the plateau on the top of their trail they paused at a megalithic circle of rocks to take in the air and chat. Vidrum: “Somakhya, hope Lootika is treating you well.”
Somakhya chuckled: “Ask the mahiṣī yourself if she is following the śruti-vākya of making me the ekādaśa.”
Lootika: “By the Ṛbhu-s, you already want to be the ekādaśa! That comes only after something like a daśa. But Vidrum, you know I’m more rational than any other girl I know perhaps barring my youngest sister. What more would a man as Somakhya want from his vāvātā?”
V: “It is interesting you say rational. That you are but you were always rather full of prickly edges like a cactus in bloom; at least till you left for a higher degree. That’s why I was checking with my friend about how living with you was going, though I must say you seem more relaxed nowadays.”
L: “Well, you know that it is an innate tendency of women to put on a show, often without knowing it themselves. I’d hangout with the guys mainly to be with Somakhya and the fact the discussions in his maṇḍala were always more interesting than those back with the girls. But there was too much cloying attention in the male circles from others. Hence, I had to put on that persona. Now that I’m united with my own vīra there hardly any further need for that old veṣa.”
V: “Well Somakhya, I have heard you say that rati-kallolādi are not the most important objectives of such associations. You two have had a lot of things to do together since childhood; so, I believe that keeps you occupied.”
S: “Vidrum, there are broadly two types of people. Those for whom the attainment of the woman is the end it itself and those for whom attaining their woman serves as the pathway attain other ends which were previously inaccessible or difficult for one or both. We belong to the second type — the Gautamī and me have a substratum of a shared experience on which new quests can be launched.”
V: “It is a truism that her kids and then her parents matter more to a woman than her mate. After all, Lootika, you had told me that is but natural given the foundational principles of biology. So, is the second path a superior one for continuing to get along in face of that?”
S: “Different strokes for different men. But it is quite conceivable that the first path results in certain: `what next?’ situation after the pleasures of maithuna become common-place or flounder against the need to attend to the harsh realities of life or upon fulfillment of their ultimate purpose. But then not all people may have the cerebral proclivities to synergistically explore the nooks and crannies of knowledge as Spidery and I might be inclined to do. The second path is not an option for the people lacking such tendencies. For them the best is to go with whatever favors gene-furtherance.”
L: “The second path also works well when the people involved need to have some time in their own space: like Somakhya needs his Saturday afternoon solitude devoted to recreational mathematics, whereas I need mine to make myself clothes and perfumes and some utilitarian things like wickerwork. Then we can have an accentuated experience when we do our communal activity such as astronomy or microscopy or even discussing the results of that mathematical exploration in the evenings.”

V: “That brings me to a point that both of you have alluded in the past that a teacher needs an appropriate student to produce knowledge. I wonder if that aspect needs the two of you together to keep up the motivation in your arcane pursuits?”
L: “Yes, that secret is laid out in the upaniṣat of the Taittirīya-śruti which describes the production of knowledge thus:
athādhividyam | ācāryaḥ pūrvarūpam | antevāsy uttararūpam | vidyā sandhiḥ | pravacanaṃ sandhānam | ity adhividyam ||
Now regarding knowledge: The preceding word is the teacher; the following word is the student; the union is knowledge; their link is instruction. So it is regarding knowledge.

So it is for most producers of knowledge — just as the old śaiva-s of the Bhairava-srotas would state, Śiva cannot exist by himself. He needs the Śakti who is like a reflective surface to reflect himself in and that reflection is the universe. Likewise, a paramparā of knowledge needs the synergy of individuals for it is a reflective process.”
V: “You seem to hint something more philosophical than just plain symbiosis.”
S: “Yes. It goes beyond. You can have a purely transactional cooperation as we have with our collaborators. There is no serious dimension to it beyond barter that characterizes the process — for example, that happens when we work we various other peoples like mleccha-s or cīna-s to get a piece of scientific research done. But there is no deeper reflection beyond that and nothing more to that interaction. In another battle we might be ranged against the very same former coauthors. But the production of knowledge needs a certain inspiration — this can come only from a more entwined synergistic interaction across many domains of thought — like performing kratu-s in unison. Or, our ancestors would have likened it to the fire-drill and the fire-board. Indeed, many producers of knowledge lose steam and eventually splutter and choke because they lack the partners for prolonged synergistic interactions.”
V: “But both of you all had remarked the need to spend a phase of your life where each acted as a lone-wolf with absolutely no contact with the other. Indeed, you did so. I almost feared you two may never see each other again. How does that fit in?”
S: “A man must prove himself that he is fit for the quest he has placed before himself. This he should do entirely on his own and even in somewhat hostile settings. Only then he will know if he is really fit for the battle of life. He must show that he can perform all survival tasks while on the hunt by himself like a lone leopard or a tiger. In our national epic we see Arjuna go on his own to seek the weapons of the gods or Bhīmasena proceed by himself to slay Baka or Kirmīra. Thus, they had proven themselves for the great war. Only under this test his true capacity will be revealed. It is much like how the the great rājan of the marāṭha-s had to kill the monstrous Afzal Khan by himself before he could become the aindrābhiṣikta commanding a great army with many talented generals.”
L: “Moreover, such a solitary phase is very important for those who practice mantra-s — it is important to shut oneself off from sexual desires and pleasures and practice with a focused mind to achieve any sort of success. But returning to the ways of routine life I think women generally are in the position of choice-makers, choosing men who prove themselves. So they would wait to see that men prove themselves. However, if they are born in high clans there is some onus on them to prove their svādhīnatā in certain capabilities that are not common among the lay. Finally, I would add that a knowledge-producer needs to be like a practitioner of yoga or tapas for a while — as the solitary wanderer who wanders among the high peaks like Umā Haimavatī. Some remain in that state: they need no further interaction for expanding their knowledge: they are lone pursuers never transitioning from a leopard to a dog or a lion, hunting by themselves for all their lives.”
S: “That latter state is particularly common among mathematicians.”
L: “I would even say that for most knowledge-producers, as they age, there is a greater need for reflective interactions with those who can play a bit of foil, unless they are very conscientious about exercising their mental skills much like one keeps up ones body with physical training. Thus, in the V_1 tradition an old reciter who is declining can still collectively keep up the oral tradition by co-reciting with a younger one. Together they can reconstitute the totality of the tradition.”

V: “To summarize, you see the need to prove oneself in solitude but then to further your greater objectives you need each other acting synergistically, mirroring what happens with a teacher and student. But what about transmitting it to real students? You all have gathered a mass of knowledge across different domains of inquiry, the sciences, the performance of rituals, darśana, the way to experience the gods and national vision. How would you transmit all of that? Surely, your secular students don’t get the whole package from you.”
L: “Oh certainly not. Most of them get a very limited transmission of our knowledge. Only those whose are closely linked to us like my sisters or their vīra-s like Indrasena or his brother Pinakasena have complete transmissions to the point they are mostly our samāna-s.”
S: “Why Vidrum? You and Sharvamanyu might also consider yourselves receivers of a limited transmission. But would you really want the full thing? I’ve not seen you evince much interest in the entire package.”
V: “Yes, dear friends I don’t think I’m capable of receiving the whole thing even if I were not uninterested but I guess you would admit that I’ve at least been a bit of a facilitator in certain domains by playing student. But would you not want it to be transmitted in toto to others like yourself.”
L: “The first part we would admit…”
S: “As for the second, it is not an easy thing. Those like Lootika’s sisters or Indra may never be seen again in our midst — it is a rare coincidence. Even my own jāmi-s, like Mandara or Saumanasa don’t receive anything for they are mostly unfit vessels. Plus, resonance does not happen with all those who possess the spark, for more often than not they emerge as bhrātṛvya-s on whom we need to invoke Indra’s vajra.”
L: “May be it will happen if our kids don’t regress to the mean. Even if they do not regress in the future generations dilution and dispersal is almost a given unless one is able to retain an institution like that of brāhmaṇatvam that sustains over the ages. Then after some generations of submergence the fire can shoot up again as it has done among the Bhṛgu-s or the Gotama-s. But today the vipra-s have all but cashed their check and others are on the path to being Rāvaṇa-s and Kumbhakarṇa-s. Hence, it could pretty much end with us becoming samidh-s for the noose-wielding Vaivasvata in the final iṣṭi.”

V: “That is a rather pessimistic outlook on your part! Why do you take such a position?”
L: “Well it is simply realism based on what has happened to others before us. The contrast is visible when we look at other nations and the transitions they made. If you look at the European situation, you notice that once intellectual activity was reinitiated after the Dark Ages brought on by the cult of Christ, there was a large body of resonant intellectuals who could provide the reflective surface for their ace knowledge producers. Thereafter, every now and then from among those reflectors a new hub of production would emerge. You can see Leibniz had others to extensively converse with on diverse matters, like Huygens, Leeuwenhoek, Spinoza and von Tchirnhaus. From that matrix eventually an Euler was to arise. If you look at Japan in the late 1800s of CE, you see the great Kitasato who of his own endeavor seeking microbiological knowledge went to Germany to study with Robert Koch. He then returned to his homeland to found a vigorous school that continues to carry out studies marked with great originality and good caliber. But in our midst the schools of such towering intellectuals have invariable folded up with them or at best after a generation or two. We hold that this is because there is not much of a samāja beyond the founder which can understand the pioneering and original studies to take it forward. It may be transmitted to a select few who are inspired by the great producer but beyond that it stagnates.”

V: “I recall Somakhya tell us in a conversation we had before you all left about the complexity of building and sustaining institutions. Is what just Lootika described a consequence of a purely institutional issue or something more than that? There is a tendency among our people to keep bitching about the lack of good institutions. That is not at all untrue but institutions ultimately depend on people so one cannot have institutions emerging in vacuum without people of good character. So, what is this brahminical institution you all have talked about?”
S: “It is indeed true that institutions arise from people and if there are structural problems in the populace institutions may not arise easily. But if there are good institutions then you can keep going to a degree even with a relatively lower quality of the participants. Further, the participants are not a constant. The population they come from changes over the ages with selection pressures from biological disease, memetic disease, the effects of war/genocide and gene-flow. So, a once good stock can decline and vice versa but these things take a long time and we are where we are after a long history of such events. In my opinion, the issue we are talking about has relatively deep roots going back to the time when the Ārya-s had completed their conquest of northern India and were well-settled and well-mixed. It became apparent to their intellectuals that situation and times no longer supported the Ṛgveda-style spontaneous creativity but required a consolidatory approach of the Ādhvaryava tradition. This gave rise the roots of the tradition we term ‘brāhmaṇatvam’. In general terms, such consolidatory traditions were probably not the unique feature of the V_1s. Even in the times we are talking about, it was perhaps routinely practiced among the vaṇik and the service jāti-s like the ambaṣṭha. Across the jāti-s, it perhaps had some contribution from the mysterious Indus people who have participated in our ancestry but about whom we know very little beyond their extensive material productions. The basic premise of the consolidatory approach was that great knowledge-producers are few and far between in the then newly constituted Hindu society in India. However, it recognized the fact that the knowledge they produced was of great importance and needed to be preserved accurately until another great producer could come along. Hence, the institution which was developed was one that could accurately preserve the knowledge created by the old preservers and transmit it to the next generation. Thus, the focus was on creating such preservers — they might not have been original thinkers themselves but they were smart individuals who could apprehend the existing knowledge and transmit it faithfully to the next generation. Every now and then a great original man would then come along and produce the next leap because he had a good foundation of well-preserved knowledge. However, this meant things proceeded at a slow pace for it had to wait for what were mostly rare events. Thus, what could have happened within a century or two of Āryabhaṭa had to wait several centuries till the great nampūtiri-s came along in Southern India. But even that would not have happened if the knowledge of Āryabhaṭa had not been faithfully preserved in face of counter-currents like Brahmagupta. However, this system fragmented due with irruptions of the monstrous marūnmatta-s and was completely destroyed with the mleccha conquest of Bhārata.”
L: “One could add that complete destruction by the mleccha-s went beyond the knowledge systems of the V_1s and affected the other systems of transmission such as manufacture of fabrics, high-end wickerwork and high-grade steel.”

V: “How does the minimalistic lifestyle you all follow relate to the brahminical system you mention?”
L: “We are not true minimalists in the sense that we keep everything at bare minimum. As you can see from our house, we keep what we deem important in a fairly extensive form. However, what others may think to be important are often not important for us and are kept in a minimal state. The aspect of V_1 spirit that we adhere to in this regard is that: 1) there is no need to acquire and display things that are meant to display status, impress beholders and seek encomium from others; 2) there is no need to accumulate anything that you will not use extensively. I believe this is an retention from the ways of our mobile IE steppe ancestors.”
V: “Ah! I can see that being a real V_1 is status in itself — so you guys don’t need other things. One could say that made the V_1 system quite robust…”
S: “Systems have different kinds of properties: 1) Robustness, where it is relatively immune destruction of its bulk elements. 2) Anti-fragility, where it develops greater strength against future challenges only when challenged in the first place (e.g. adaptive immune systems of vertebrates) \footnote{a term described the Lebanese thinker N.N. Taleb}. 3) Reconstructability, where the system can reconstitute itself repeatedly in different places and environments at different times while retaining an undiluted essence. One could also say that that the old V1 system had a certain capacity for reconstructability due this minimalism — they could move from place to place in Aryanizing waves by carrying relatively little with them.”

V: “I can indeed picture the most extreme version of this — a V_1 of old moving with just his animals, weapons, shovel, a basin, a few clothes and vessels. He even carries no books for he keeps his knowledge in his head. But returning to our own times we can clearly see that knowledge-production is a cosmopolitan one, much like cricket where some of the most exciting games are T20 games played by cosmopolitan multiethnic teams with no heed to their national origins. You have yourselves participated in such multiethnic ventures in science. In contrast, the brāhmaṇatvam you talk about is very much an intra-ethnic affair. So do such models have any relevance at all other than a limited transmission, as you all talk about, within your own family network.”

L: “Having worked in multi-ethnic armies of science we would still say that ethnicity remains a major factor. While I don’t watch cricket, unlike you guys, I know that the multiethnic teams come together due to the quest for money — in essence they are mercenaries. Similarly, in the past Hindu rulers, like the Marāṭha-s, deployed mleccha mercenaries who had they own motivations and played by their own coethnic networks. Similarly, our participation in multi-ethnic scientific armies as mercenaries was with our own interests and motives in mind. In fact, within those seemingly multi-ethnic scientific armies ethnicity plays a subtle and much greater role than would be apparent to the casual observer. Sadly, the Hindu happens to be among the most blind in this regard due to rampant deracination. But it is strong ethnicity-based networks that has helped certain groups prosper within the international scientific ecosystem, such as groups of prācya-s and the mūlavātūla-s. Further, the true belonging in this cosmopolis, which is essentially ruled by the mleccha-s and mūlavātūla-s, is achieved by admitting and surrendering to the precepts of the pracchanna-ekarākṣasa-mata run by their ‘liberals’. If you do not align with it you cannot be anything more than a mere transactional participant in that cosmopolis. Thus, despite the impression of the cosmopolis it is merely a collection of people who look different but think pretty much the same.”
S: “I’d add that the cosmopolis is not a new thing. We had our own Sanskrit cosmopolis that stretched over much of Asia before its disruption and collapse due to the dānava-s of makkha-viṣaya. It had participants of multiple ethnicities but unlike the mleccha-dominated cosmopolis it did not have single dominant ideology. There were various dominant foci such as the śaiva-s of the Bhairava- and Saiddhāntika-srotas. In those times too, just as mleccha scientists plagiarize our original work, we had the tāthāgata plagiarists in the system, who despite their plagiarism, showed some prolific productivity. There were comprehensive synthetic schools like that represented by the Yogavāsiṣṭha among others and the emergence of encyclopedic polymaths like king Bhojadeva. Ideas were widely exchanged, people moved from one school to another, and there were major intellectual battles like today; however, unlike in the mleccha-system there was no subservience to the mleccha and mūlavātūla cores. People took ownership of the productivity in their locales. For example, in the Indonesian, Tibetan, Chinese and Indo-Chinese centers their own boldly distinctive local productions emerged while never losing the connection to their Jambudvīpīya foundations. The absence of such ownership and the enslavement or dominance by the mleccha often leads to malpractice and plagiarism, especially among our people.”

V: “That is rather interesting. But should we call this knowledge-production or religious piety? After all most of the creativity you mention seems to fit in the domain of religion or metaphysics rather than science.”
S: “As V_1s, we see religious and metaphysical knowledge as parts of the whole. Of course unlike many of our more recent traditional counterparts we privilege science greatly because any philosophy that separates itself from science is destined to be less-effective and incomplete and there branches of metaphysics, which result in hairsplitting among the uttaramīmāṁsaka-s, that do not apply to our darśana. Indeed, some of the religious savants did contribute to our science-like philosophical system, like Vyomaśiva-deśika the Saiddhāntika-śaiva’s work on work on Vaiśeṣika. Moreover, I should point out that at the acme of this cosmopolis, which came just before its abrupt end, also saw unprecedented acceleration in the otherwise slow pace of scientific/mathematical discovery — we see this in Udayadivākara’s Cakravāla, Mañjula’s discovery of basics of differential calculus during his study of the moon’s orbit, King Bhojadeva’s works and culmination of gola in Bhāskara-II. At one level the integration of science with philosophy was also visible as can be seen in the synthetic Yogavāsiṣṭha.”

V: “But the transmissions you two are talking about seem to be about specialized knowledge – something that might be transmitted between advanced or self-driven students. That is different from educating the masses with basic stuff. People frequently blame our education system, Jawaharlal Nehru and the like but is there a deeper problem of which these are merely symptoms? Can a great institution of knowledge production help if the student is bad in the first place?”
L: “Of course that would not help a bad student. But as the puruṣa said good institutions can to an extant mitigate the defects of the populace. I believe at their zenith our knowledge production systems were capable to doing so — this was caricatured by the grammatical fidelity of King Bhojadeva’s woodcutter or the ladies from the 4 varṇa-s. However, the systems that exist now are no longer capable doing so. You yourself used to complain about the physician who graduated from a college where his only qualification was the purse of his parents. Likewise, we have engineers less than the worth of a palāṇḍu-guccha. Now why have we come to such a pass – may be we can blame pūjanīya śrī Jawaharlal Nehru for some of that. But we would say that indeed the deeper problem is the lack of a proper knowledge-production system. If such existed, then even though at its cutting-edge it would cater to the advanced seekers, it will also produce as a by-product a standing crop of lesser but competently trained individuals who can serve as teachers for the masses. Thus, the knowledge will trickle to the masses. However, this can happen only if as in the now gone Hindu system the teacher was a respected and socially supported profession. In the system our people have transplanted from the Occident it is quite the reverse — they are low-paid and hardly respected. Thus, they will seek other avenues. Anyhow we have been chatting for long; let us get moving to reach home in time for dinner.”

Posted in Heathen thought, History, Life | Tagged , , , , ,

The Plague: historical, biographical and current: a brief roundup

Globalization is not a new thing. The Indo-European empire of the steppes was perhaps the first one. In addition to having a serious component of our genetic ancestry and most of our memetic inheritance in it, we can still see its impact worldwide. It perhaps laid the foundations for the human global system (the idea that a world system could exist) but there is not much that remains in terms of the precise memory of it. However, the empire founded by Chingiz Khan and his capable successors is still very much a part of our historical memory. At its peak under the grandsons of Chingiz Khan it resulted in a globalization not entirely different from modernity. The Qubilai Khan even imagined the Phags-pa script based on the Brāhmī family as a means to write all languages of the world. The support for paper currency, a postal system, efficient passport control and the ability to conduct trade through the known world under Pax Mongolica created a certain globalization that contributed to the prosperity of the Mongol empire and its neighbors. But concomitant with its collapse a great plague came out of East Asia and spread westwards to destroy anywhere between 1/3 to 1/2 of Europe’s population with the trading cities in Italy being particularly hard hit. The effects of this Black Death plague caused by Yersinia pestis on Europe are widely documented but it is not entirely clearly if it seriously affected India. There is little evidence for such from inscriptional records even though we have documents like the Vilāsa copper plates which record the contemporaneous destruction on the scale of the Black Death in India caused by the śānti-dūta-s bearing the third West Asian disease of the mind. The plague did strike India later around the time we were locked in a life-and-death struggle for independence from the English in 1857 CE and later after their conquest was complete. Nevertheless, this combination of the destruction caused by the Black Death, that spread by the throat-slicing śānti-dūta-s and the collapse of the Mongol empire brought an end to this globalization of the 12-1300s.

In the 11th year of our life we became interested in deciphering the grand picture of evolutionary history of viruses. We have off and on returned to that project to this date whenever we have had resources with a degree of success. From that time on a more informal thought used to run through in our head: Could a repeat of the scenario like the Black death bubonic plague happen again? We wondered about the mleccha use of Bacillus anthracis in Africa and if that might go out of hand or if Yersinia pestis might gain antibiotic resistance. Such a fear still remains as the plague outbreaks in Madagascar cause death on the island and some AB resistant strains have been found there. However, given antibiotics we wondered if a more likely modern scenario for such an outbreak was a viral one. There was the excitement over HIV in those days but it was clear that it was mostly avoidable with behavioral corrections. The stories of the outbreaks of Lassa fever and the Crimean–Congo hemorrhagic fever caused by Bunyaviruses, Marburg and Ebola caused by filoviruses raised the possibility of such a viral outbreak being reality. This fear came back when we were in the deśa about 9 years ago and the Crimean–Congo hemorrhagic fever sent the patient along with his nurse and doctor to the abode of Vivasvān’s son. In another direction it also led us to thinking about whether such highly infectious and fatal viruses might eventually evolve to self-limit even as the Myxoma poxvirus introduced by the mleccha-s to kill the rabbits that they had brought along to Australia. This in turn got us interested the mathematics of the logistic map.

In our early explorations of viruses we only briefly studied the nidoviruses in 1995 CE being largely focused on other positive-strand RNA viruses in which we discovered multiple families of thiol peptidases. It was at the dawn of the new millennium that we revisited them in course of discovering a new RNA-processing enzyme in a branch of nidoviruses known as coronaviruses. Shortly, thereafter there was a major outbreak of a coronavirus in the form of the SARS epidemic. This again brought to mind our old musing on a viral replay of the Black Death plague but it mostly died out without being anything of such proportions. However, it added the coronavirus as another candidate to the list which was otherwise impressively populated by negative strand RNA viruses. This was only reinforced by the Arabian disease cause by another related clade of coronaviruses MERS. But this apparently was just the beginning and these viruses were destined to hit the world in a bigger way.

In what follows we give some background for a lay reader: coronaviruses belong to a clade of RNA viruses (i.e. have RNA as their genome) known as nidoviruses. The nidoviruses are among the RNA viruses with the largest genomes that are currently known to us (e.g. the 31 kb genome of the Mouse Hepatitis Virus). Their genomes appear to have grown in size because of acquiring a proof-reading exoRNase enzyme, similar to what cellular organisms use to proofread their DNA during replication, and some RNA end-processing enzymes like the one we discovered. The nidoviruses in turn belong to a vast and ancient radiation of positive-strand RNA viruses (i.e. their genome can directly serve as a template for translation into proteins) that are marked by a particular type of RNA polymerase (enzyme that replicates the virus) which in turn belongs to a vast radiation of replicative polymerase enzymes that include those that replicate the retroviruses like HIV and also DNA in our own cells. Thus, these viruses have their ultimate roots in an ancient world of replicators from which all life has arise. Within the nidovirus clade, there are viruses like: 1) Plasmopara viticola nidovirus which infects an eponymous oomycete, which in turn infects the grapevine to cause the downy mildew; 2) the Aplysia californica nido-like virus which infects the eponymous slugs; 3) Yellow Head Virus which infects arthropods like shrimps; 4) Dak Nong, Cavally and Nam Dinh viruses which infect arthropods like mosquitoes; 5) the White Bream Virus (WBV) which infects fishes; 6) Coronoviruses which infect mammals and dinosaurs. 7) Toroviruses related to WBV Ateriviruses which infect mammals. In mammals and dinosaurs these mostly cause gastrointestinal and respiratory tract diseases.

There is some information in the public domain that gives some misleading claims that the known coronaviruses have all diverged relatively recently from their common ancestor: like in the past 10000 years and for the more terminal divergences in the past few centuries. As some researchers have already pointed, out this is false. Given this distribution of nidoviruses they are very unlikely to have the recent divergence times, which are likely the result of the application of wrong molecular clock assumptions to infer branching times on their tree. Instead we think they probably have been around from early in animal evolution with the coronavirus branch itself being over 20-30 \times 10^6Y in age. The scenario we favor is thus: the nidoviruses first entered vertebrates from them feeding on invertebrates like molluscs or crustaceans. Alternatively (less likely), blood-feeding invertebrates like mosquitoes transmitted them to vertebrates. Once in vertebrates they have mainly transmitted through the oral and respiratory route (that’s why latter alternative is less likely). The prevalence of coronaviruses in bats and birds raises the possibility that these flying vertebrates contributed to their wide dissemination with a likely early exchange between birds and bats perhaps from shared nesting sites.

Fast forward to 2020 CE. The civil year began with the news report of a SARS-like pneumonia disease in Wuhan, China. The first cases were noticed in December of the earlier year. By Jan 10th it was clear that we were seeing something big which the Cīna-s were probably covering up. In the month that followed the Cīna cover-up and the epidemic only kept increasing. Now, it is poised to be a pandemic. The hit to the globalization of our age is already being seen. As of today people are talking of breaks in the supply lines and the stocks are in a state of volatility. A researcher mentioned in the Cell journal that it is shaping much like the pathogen that we had imagined in our scenarios of the replay of the Black Death plague (termed pathogen X in the report). This pathogen is a coronavirus related to SARS dubbed SARS-CoV-2 and the disease has been named COVID-19. The attention is shifting away from the Cīna epicenter, perhaps much as the Cīna-s had hoped, because as of the time of writing, the virus is now rapidly spreading in Japan, Korea, Iran and Italy and sporadic cases are turning up throughout western Europe bringing home the global threat it poses [between the time we started and finished this note a situation has unfolded at a hotel in Tenerife and had established a bridgehead in Spain]. Countries like the US which despite their wealth look rather ill-prepared to handle it hitting the country. Supposedly, the 3 Indian cases have recovered and there are no new ones. Is this because of the increasing temperatures a known susceptibility of the lipid-coated coronaviruses? Africa is another strange case which despite intimate contact with the Cīna-s in several nations has not reported much dissemination. We suspect they might not be closely monitoring or reporting the cases and we could hear of them in the near future (But could sub-Saharan Africans have a degree of natural immunity?).

A serious blame for this disaster lies with Cīna-s for their cover-up. But the Cīna cover-up comes directly from the fact that they are deeply entangled with the Occident. Since Francis Galton the Occident has had a certain kind of Cīna-fetish: Galton had hoped that the Cīna-s, whom he held to be superior to the Hindoos, could be placed in Africa where they could take over the land from the Black Africans evidently under the benign stewardship of the English. The American component of the Occident first used the Black Africans as slaves to get their dirty jobs done as they lived in relative prosperity. As slavery ended and the post-World War-2 world unfolded they need a new mechanism to lead large lives. This mechanism was Galtonian in its conception as it shipped skilled labor to the Cīna-s who provided the Occident with modern goods at a low price. This in turn helped raise the Cīna-s out of poverty but also created a serious dependency on the western consumption of finished commodities. This dependency is entwined with their national pride and calls for not halting production at any cost. This facilitated the suppression of the news of the seriousness of the outbreak with outright denial in the beginning followed by fudged numbers. With long supply lines of the Occident and even of traditional trading partners like India leading into the depths of China the spillover of this agent was a given. This raises one of the biggest questions about the sustainability of the long supply lines and argues for the value of self-contained localism. The other side of this surfacing of the fear of the alien which happens under such crises. During the Black Death plague various trading and mobile communities came under attack and were massacred in different European cities. Even with the current episode we are hearing reports of attacks on Cīna-s in the West and even associated bystander Hindus.

Like SARS, COVID-19 seems primarily to kills via respiratory failure. However, going by the reports outside China, which are likely more reliable, its mortality rate seems lower than SARS but certainly much higher than influenza. Ironically just a few weeks back people were saying the influenza is a bigger danger than COVID-19! In any case it has several nasty features: 1) It seems to have a fairly long incubation time which could go over 2 weeks. 2) It is highly infectious as the infections on the cruise-ship docked in Japan showed. We would say it has a higher mean R_0 than SARS, i.e. the mean number of individuals an infected individual transmits the virus to. 3) Several patients are asymptomatic but appear potentially capable of shedding and transmitting the virus. 4) It is still not very clear if means of transmission other than through droplets exist. This combination of features make it a rather dangerous thing: it has all the stealth features for wide dissemination on one hand and enough of a mortality rate to be fear-inducing on the other.

Finally, we comment on one point that ignorant or lay people often ask us about. Is this an engineered biological warfare agent? People could ask what about the experiments which were done several years ago on the high mortality bird flu H5N1 to partly engineer it and partly select it to turn it into one which transmits between mammals (ferrets). Could such a thing have been done with SARS to get this virus. Given that we have now spent sometime closely examining its genome and have even found some interesting features we can say this is not the case. It is very much a natural virus in all ways particularly close to a SARS-clade virus that infects bats. Now a milder variant of the question is whether it has escaped from a lab in China. The proponents of such a hypothesis would point out that the epicenter of the epidemic, Wuhan, is a famous center for virological research in China. Indeed, most the previously published SARS-like coronaviruses were sequenced and published by the lab in Wuhan. Further, in China there are incidents of the unscrupulous consumption of lab animals after the experiments are done on them. However, the key argument against this is that the Wuhan group published several viral sequences in the past few years and none of them is identical to SARS-CoV-2, even though, as mentioned above, one of them is close. However, even this close relative could not have evolved into SARS-CoV-2 in the lab from the time of its relatively recent isolation given the number of substitutions in their RNA. However, one possibility which cannot be entirely ruled out is that they had only recently isolated this particular virus and it escaped the lab even before they could publish its genome. However, given the timeline of their research on SARS-like viruses this is not evident in anyway.

One could point out that the Russians have a long history of developing biological weapons. During WW2 they deployed Francisella tularensis against the Germans. Few years later while assisting the Indian smallpox vaccination program they look a particularly virulent Indian strain to develop smallpox weapons in Russia which apparently caused some deaths during accidental deployment. Another case was their development of the Marburg Virus weapon which accidentally killed one of their virologists involved in the program. Could the Cīna-s have been doing something like that and could this be an escape like the Russian cases? If so it would imply considerable sophistication on part of the Cīna-s for which we do not any evidence currently.

Whatever the cases this promises to be a serious global threat in the coming days. We had an old fictitious story about the deployment of an engineered virus. Something from the last part of that tale applies to these SARS-like coronaviruses. Hopefully, with time it can see the light of the day.

Posted in History, Life, Scientific ramblings | Tagged , , , , , ,

Two squares that sum to a cube

Introduction
This note records an exploration that began in our youth with the simple arithmetic question: Sum of the squares of which pair integers yields a perfect cube? Some obvious cases immediately come to mind: 2^2+2^2=2^3; 5^2+10^2=5^3. In both these cases we can see that the addition of a square the to the square of the first number yields its cube. Hence, we can ask another related question: Which are the integers whose squares when added to another perfect square yield their cubes? In general terms the answers to these questions are the non-trivial (i.e. where x, y, z \ne 0) solutions of the indeterminate equation:

x^2+y^2=z^3 \;\;\; (\S 1)

Figure 1 shows the first few solutions lying on the surface defined by \S 1. Due to the 8-fold symmetry we restrict ourselves to positive solutions such that x \le y for a given z.

sqrs_cube_xyz_Fig1Figure 1

Since z is explicitly defined by x, y we can study the solutions of \S 1 by plotting x, y on the x-y plane (Figure 2). At first sight, a mathematically naive person might perceive only a limited order in this plot of the solutions. However, as we shall see below, a closer examination and some algebra reveals a deep structure.

sqrs_cube_xy_Fig2Figure 2. Some of the more obvious structure is indicated in different colors and is discussed below in detail.

Solution circles, 2 square numbers and Brahmagupta’s identity
Some simple algebra can provide an understanding of the structure of the solutions of \S 1. From \S 1 it is obvious that all solutions would be lattice points (x,y) on circles of radius z^{3/2}. Based on this, we can determine a set of x, y, z that are to solutions of \S 1 thus: let z be an integer that is the sum of two perfect squares, z=m^2+n^2. Then,
\left(m^2+n^2\right)^3= m^6 + 3m^4n^2 + 3m^2n^4 + n^6\\[6pt] =(m^6+2m^4n^2+m^2n^4)+(n^6+2m^2n^4+m^4n^2)\\[6pt] =m^2(m^4+2m^2n^2+n^4)+n^2(n^4+2m^2n^2+m^4)\\[6pt] =(m(m^2+n^2))^2+(n(m^2+n^2))^2 \; \; \; (\S 2)

Thus, if z=m^2+n^2 (i.e. it is a 2 square number) \S 2 gives us an expression for z^3 as the sum of 2 squares. Hence, the valid solutions to \S 1 will be all
x=m^3+n^2m\\[6pt] y=n^3+m^2n\\[6pt] z=m^2+n^2

For example, if z=5 = 1^2+2^2 we have m=1; \; n=2. Thus, x=1^3+2^2\times 1=5 and y=2^3+1^2\times 2= 10. Hence, z which belong to the 2 square sequence, i.e. numbers that can be non-trivially (i.e. one of the terms is not 0) expressed as the sum of 2 perfect squares, are valid solutions: 2, 5, 8, 10, 13, 17, 18, 20, 25, 26… The corresponding x, y can be derived as above. One also notices from the above expressions that m^2+n^2=z is a common factor for x, y, z in all such solutions to \S 1.

While this method yields x=5, \; y=10, \; z=5, we notice that the same z=5 also yields a second pair of x=2, y=11. Now how do we account for these additional pairs and for what values of z they arise?

The most obvious set of these correspond to z being equal to special numbers such as 5, 13, 17, 29… One notices that these are both 2 square numbers and primes of the form 4n+1. Fermat first noticed that such primes are always 2 square numbers. This attracted the attention of some of the greatest mathematicians like Euler, Lagrange and Gauss, who proved it to be so by using different methods. What is important for us is that such primes are also the karṇā-s (hypotenuses) of primitive bhujā-koṭi-karṇā triples. Thus, we have z=m^2+n^2 from the fact that they are 2 square numbers and z^2=a^2+b^2 from the fact that they are bh-k-k hypotenuses. Hence, we can write,
z^3=z \cdot z^2 =(m^2+n^2)(a^2+b^2)\\[6pt] =a^2m^2 + a^2n^2 + b^2m^2 + b^2n^2\\[6pt] =(a^2m^2 + b^2n^2 +2am \cdot bn) + (a^2n^2 + b^2m^2 - 2am \cdot bn))\\[6pt] =(an-bm)^2+(am+bn)^2 \; \; \; (\S 3)

Alternatively,
(a^2m^2 + b^2n^2 -2am \cdot bn) + (a^2n^2 + b^2m^2 + 2am \cdot bn)\\[6pt] (am-bn)^2 + (an+bm)^2 \; \; \; (\S 4)

\S 3 and \S 4 give us the famous identities of Brahmagupta from which we get two expressions for z^3 as the sum of a pair of perfect squares when z is a 4n+1 prime. From \S 3 we get:
x =|bm-an| \\[6pt] y = am+bn\\[6pt] z = m^2+n^2=\sqrt{a^2+b^2}

From \S 4 we get:
x= |am-bn|\\[6pt] y= an+bm\\[6pt] z=m^2+n^2=\sqrt{a^2+b^2}

For example, if we take z=5, we have m=1, \; n= 2, \; a=3, \; b=4. From first set of expressions we get x=|4-6|=2, \; y=3+8=11. From the second we get x=|3-8|=5; \; y= 6+4= 10. The second pair x, y is equivalent to what we can get via \S 2. Thus, when z is a 2 square prime its cube can be expressed as the sum of distinct two pairs of squares.

Now, one can get a further set of z corresponding to solutions with two pairs of x, y when say z=2p where p is a 4n+1 prime. Here z^3=2p \cdot (2p)^2. From above we can write,

p=m^2+n^2\\[6pt] \therefore 2p= (m+n)^2+(m-n)^2

Further,
p^2=a^2+b^2\\[6pt] \therefore (2p)^2= (2a)^2+(2b)^2\\[6pt] \therefore z^3= ((m+n)^2+(m-n)^2)((2a)^2+(2b)^2)

Thus, from Brahmagupta’s identity we have:
x = 2a(m+n)-|2b(m-n)|\\[6pt] y = 2b(m+n)+|2a(m-n)|

Alternatively,
x=2a(m+n)+|2b(m-n)|\\[6pt] y=2b(m+n)-|2a(m-n)|

For example, if z=26= 2 \times 13 then m=2, \; n=3, \; a=5, \; b=12. Thus, x=74, \; y=110 or x=26, \; y=130. Similarly, one can derive formulae for z=4p, 8p...

When z=p^2 =a^2+b^2 where p=m^2+n^2 is a 4n+1 (2 square) prime we can show via repeated application of Brahmagupta’s identity that there 3 possible x,y pairs whose squares can compose z^3. We can derive the below formulae for them:
x=a^3+b^2a\\[6pt] y=b^3+a^2b

Or,
x=p(a^2-b^2)\\[6pt] y=2pab

Or,
x= |m (b^2 n + 2 a b m - a^2 n ) - n | a^2m + 2 ab n - b^2 mb| | \\[6pt] y= (a m - a n + b m + b n) (a m + a n - b m + b n)

As an example, by applying the above formulae we can see that for z=25 = 5^2 its cube, 25^3, can be split up into 3 distinct pairs of squares: 35^2+120^2 = 44^2 + 117^2 = 75^2 +100^2.

The nested combinatorial application of Brahmagupta’s formula can thus result in increasingly complex formations with multiple alternative partitions of a perfect cube into perfect squares for different multiples of the 2 square primes. Thus if z=50=2 \times p^2 we get 4 distinct pairs of x, y; for z=125= 5 \times 5^2 we get 5 distinct pairs of x, y.

When a number is product of two distinct 4n+1 primes then its cube can be partitioned into 8 distinct pairs perfect squares. For instance,
65^3= 7^2 + 524^2 = 65^2+ 520^2 = 140^2 + 505^2 = 191^2 + 488^2= 208^2 + 481^2 = 260^2+ 455^2= 320^2+ 415^2 = 364^2+ 377^2
The same applies to the multiples of these numbers by 2, 4… When a number is a product of a 4n+1 prime and the square of a distinct 4n+1 prime then the cube of that number can be partitioned into 14 different pairs of perfect squares. Thus, 325=5^2 \times 13 is the first number whose cube can be thus partitioned. The next is 425 = 5^2 \times 17. Figure 3 shows some circles with multiple pairs of x, y whose squares sum to the same z^3.

sqrs_cube_xy_Fig3Figure 3

Ramanujan and Landau had independently discovered that the number of positive integers, N(x) \le x, that can be written as a sum of 2 squares including 0 defines a constant:

\displaystyle K_{RL} = \lim_{x \to \infty} N(x)\dfrac{\sqrt{\log(x)}}{x} = 0.7642236535...

Based on this, we can also look at how many unique z \le x, i.e. N(z) by defining:

K=N(z)\dfrac{\sqrt{\log(x)}}{x}

We empirically observe that this K \approx 0.8071 (Figure 4). The K_{RL} has a closed form which has deep connections to the Riemann \zeta(x) and the Dirichlet \beta(x). However, we are not aware of a closed form for the constant K in our case or even its exact value as x \to \infty. This K seems to reach a fairly stable value around that reported above but K_{RL} converges very slowly.

sqrs_cube_RL_Fig4Figure 4

Solution families on curves
In addition to the arrangement of solutions as lattice points on circle of radius z^{3/2}, there are also other patterns that become apparent from a closer look at the solutions in the x-y plane (Figure 2). The most obvious is the set of points lying on the y=x line at the right diagonal boundary of the plot. These are defined by a family of the form:

x= y =2n^3; \; z= 2n^2

This family defines the sequence of integers twice whose square equals a perfect cube: 2, 16, 54, 128, 250, 432, 686, 1024 … (blue line in Figure 3).

Then we see pairs of regularly positioned points that eventually lie closer to the right diagonal boundary (emphasized in red and blue in Figure 3). These are families that lie on one of two parallel curves. The first is defined by the parametric equations:

x = 2t^3 + 6t^2 + 3t - 2 \\[6pt] y = 2t^3 + 12t^2 + 21t + 11

This curve has a lobe (Figure 5; gray) and generally resembles the shape of the curve of von Tschirnhaus (see below). The second curve takes the form:

x=2t^3+6t^2+9t\\[6pt] y=2t^3+12t^2+27t+27

This curve has no lobe and lies inside the divergent arms of the first one (Figure 5; dark orange).

sqrs_cube_xyz_Fig5Figure 5

A further family lies on the curve defined by the below parametric equations (emphasized in red in Figure 3):

x=3t^2-1\\[6pt] y=t^3-3t

This curve has a single lobe and two divergent arms (Figure 5, light orange) and is a version of the eponymous curve discovered by Ehrenfried Walther von Tschirnhaus, the famous German polymath, who among other things reinvented porcelain in the Occident. It crosses over the above two curves and proceeds closer to the vertical left boundary of the x-y plot.

Finally, we a family of families lying on the family of curves (shown in blue in Figures 3, 5) defined by the parametric equations:

x=kt^2 + k^3\\[6pt] y= t^3 + k^2t,
where k=1, 2, 3 ...

The first member of each of the new families on this family of curves starts from where the curve intersect the y=x line. Thus, each starts with the points x=y=2k^3 (Figure 3). The first of this gives yields the answer to the second question posed in the introduction. When the square of x=n^2+1 is added to the square of y=n^3+n we get the cube of z=n^2+1, i.e. x=z. It remains unknown to us if there are any further families beyond these.

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Difference of consecutive cubes, conics and a Japanese temple tablet

Introduction
In our part of the world, someone with even a nominal knowledge of mathematics might be aware of the taxicab number made famous by the conversation of Ramanujan and Hardy: the smallest number that can be expressed as the sum of two distinct pairs positive cubes: 1729=1^3 + 12^3 = 9^3 + 10^3. This number is just one of a family of such taxicab numbers with deep connections to other objects in the mathematical world that were discovered by Ramanujan long before anyone knew of their significance. There are several other interesting questions, which, in a similar vein, relate to the sum and difference of cubes. From Fermat’s last theorem we know that the indeterminate equation x^3+y^3=z^3 cannot have any non-trivial integer solutions (non-trivial being where none of the solutions are 0). However, this still leaves open other possibilities with sums of 3 cubes and differences of the cubes and the like. For example, one popular and widely investigated one asks which integers can be expressed as a sum of any three cubes. When we learned that Fermat’s last theorem precludes integer solutions for x^3+y^3=z^3 in our early youth, we wondered if there are non-trivial positive integer solutions indeterminate equation: x^3+y^3+z^3=w^3. Soon we became interested and investigated one specific version of it of the form:

x^3+y^3=w^3-z^3, where w=z+1; thus, we have x^3+y^3=3z^2+3z+1 \;\;\; (\S 1)

This led us to discovering and proving for ourselves a simple arithmetic theorem concerning such cubes. Later, we read to our amazement that a related problem had been considered on a Japanese temple tablet in a beautiful “mystical”-sounding verse and solved in a commentary on it. This inspired us to look at the problem again and we discovered further interesting links between conics and these cubes. We detail these explorations and the Japanese temple tablet below.

The difference of consecutive cubes and an arithmetic theorem
We first asked when does the difference of consecutive cubes result in a sum of two cubes. This idea came to us as a parallel to the bhujā-koṭi-karṇa triples. While we have 4 cubes in (\S 1), given that two are consecutive, we only have a triple of distinct positive integers x, y, z bearing the relationship:

z = \dfrac{\sqrt{12x^3 + 12y^3 - 3} -3}{6} \; \; \; (\S 2)

Thus, we have to search for the all the cases where (\S 2) evaluates to an integer given dyads of positive integers x, y. If we order the resultant triples such that x < y < z we get a unique set of triples, e.g. 1, 6, 8: 1^3+6^3=9^3-8^3. Figure 1 shows a plot of x, y, z for all x, y<20000, a total of 1173 triples.

cube.diff3D_fig1Figure 1

The plot initially reminds one of a first time observer of the sky with a great mass of stellar points with some vague patterns among them. Given that z is dependent on x, y and the points lie on a single curved surface in 3D, we can reduce dimensions for simplicity and plot just x, y (Figure 2).

cube.diffsolns_fig2Figure 2.

As we have ordered our triples x, y all points obviously lie above the y=x line. We then notice the first clear pattern. There are set of regularly arranged points (in red in Figure 2) that mark the left margin of the plot. These points lead us to the following theorem:

The cube of every positive integer is equal to the difference of the cubes of two consecutive positive integers minus the cube a third positive integer.

To prove this let us consider those regularly arranged points. The first few of them are tabulated below:

x y
1 6
2 17
3 34
4 57
5 86
6 121

We observe that the x values of these points include each positive integer in order. The corresponding y grows rapidly and follows a peculiar pattern. This pattern is represented by the numbers lying on the 5th spoke hexagonal spiral (Figure 3) where 6 spokes separated by the rotation angle of \tfrac{\pi}{3} radians pass through the origin.

cube.diffspiral_fig3

Figure 3.

The numbers lying on the 5th spoke of this spiral can be represented by the formula: 3n^2+2n+1. Thus, we have x=1, 2, 3...n and the corresponding y=3n^2+2n+1. Hence,

x^3+y^3=n^3+(3n^2+2n+1)^3= 3(3n^3 + 3n^2 + 2n)^2 + 3 (3n^3 + 3n^2 + 2n) + 1 \;\;\; (\S 3)

If we write z=3n^3 + 3n^2 + 2n, the right hand side of (\S 3) becomes (z+1)^3-z^3

\therefore x^3=(z+1)^3-z^3-y^3, where x=n;\; y=3n^2+2n+1;\; z=3n^3 + 3n^2 + 2n \;\;\; _{...\blacksquare}

Thus, this theorem illustrates one deep connection between the cubes of numbers the hexagonal number spiral.

The Japanese temple tablet
The Samurai intellectual Shiraishi Chochu recorded a problem inscribed in tablet hung at a temple in the 1800s by the poorly known mathematician Gokai Ampon:

“There are three integral numbers, heaven, earth, and man, which being cubed and added together give a result of which the cube root has no decimal part. Required to find the numbers.” – translation from Smith and Mikami

In essence, Gokai Ampon wants us to find integer solutions to indeterminate equation x^3+y^3+z^3=w^3, which is the same question that had originally prompted our quest. His solutions recorded by Shiraishi Chochu are a particular class of solutions to x^3+y^3=3z^2+3z+1, i.e. the sum of the cubes of two positive integer being the difference of the cubes of consecutive positive integers.

If we look at Figure 2, we find that among the mass of points with apparently no discernible order there are a group of regularly arranged points coming in pairs with the same x value and lying on a slim parabola (colored purple). It was this group of points that caught Gokai Ampon’s attention. The first few of the pairs are listed below:

x y1 y2
3 4 10
12 19 31
27 46 64
48 85 109
75 136 166
108 199 235

We observe that for these points x takes the form 3n^2; \; n=1, 2, 3.... For y_1 we get a fit with 6n^2 -3n +1 and for y_2 with 6n^2 +3n +1. Thus, with y_1 we have:

x^3+y_1^3=(3n^2)^3+(6n^2 -3n +1)^3 = 3 \cdot 3^2(2 n^2-3 n^3 - n)^2 + 3 \cdot 3 (2 n^2-3 n^3 - n) + 1 \;\;\; (\S 4)

By writing z_1=3(2 n^2-3 n^3 - n) and plugging it in (\S 4) we get x^3+y_1^3+z_1^3= (z_1+1)^3, which is a family of valid solutions to the Japanese problem. Given that we are only considering positive integers, the final parameterization will be z_1=3(3 n^3- 2 n^2+ n)-1. Similarly, with y_2 we get:

x^3+y_2^3=(3n^2)^3+(6n^2 +3n +1)^3 = 3 \cdot 3^2 (3 n^3 + 2 n^2 + n)^2 + 3 \cdot 3 (3 n^3 + 2 n^2 + n) + 1 \;\;\; (\S 5)

By plugging z_2= 3 (3 n^3 + 2 n^2 + n) in (\S 5) we get x^3+y_1^3+z_2^3= (z_2+1)^3, the second valid family of valid solutions to the Japanese problem.

One can see that these two solutions are lattice points on a parabola whose parametric equation is (x=3t^2, y=6t^2 \pm 3t +1). Thus, both our solution which covers the cubes of every positive integer and the Japanese solutions are 2 distinct parameterized families corresponding to parabolas in the (x,y) plane.

The elliptical families
When we learned of the Japanese solutions, we wondered if there might be any other families of solutions hidden within the apparent disorder of the total set of all solutions. Returning to the x-y plot, we noted that several points lie on arcs of increasing size (shown as orange and pink points in Figure 2). Examining these, we discovered that they are integer points lying on pairs of related ellipses of eccentricity \sqrt{\tfrac{2}{3}} that have equations of the form:

\begin{cases} x^2-xy+y^2-ax-by+c=0 \\[6pt] x^2-xy+y^2-bx-ay+c=0 \end{cases} \; \; \; (\S 6)

Where the 3 parameters a, b, c are defined by:

c=27u^4+9u^2+1\\[6pt] a=c+(9u^2+1) = 27u^4 + 18u^2 + 2\\[6pt] b=c-(9u^2+2)= 27u^4 - 1

The lattice points of (\S 6) are solutions to (\S 1) and emerge at special values of u. First few values of u and the number of solutions they yield on the 2 corresponding ellipses are shown in Figure 4 and Table 1.

cube.diffelliptical_fig4Figure 4

Table 1

cubediff_table1

The ellipses first corresponding to the few values of u that yield solutions numbering \ge 2 are show in Figure 5. Not all values of u are equally rich in terms of solutions for (\S 1). The integers (1, 2, 3…), thirds i.e. n \pm \tfrac{1}{3} consistently yield solutions but the integers tend to be clearly richer than the thirds (Figure 4). Other than integers, certain irrational values of the form u=\sqrt{\tfrac{6n+1}{6}} are particularly rich in solutions. Other than those, u=\sqrt{12} and certain values of the form u=\sqrt{\tfrac{6n+1}{18}}, u= \sqrt{\tfrac{6n+2}{18}} are also rich in solutions. Thus, 12a, 12b, 12c are integers for all u, yielding solutions lying on ellipses. However, it remains unclear if there is a general rule to determine which of the quadratic surds that take the above forms will be u that yield solutions for (\S 1). In any case the given that integer u yield solutions for (\S 1) that lie on defined ellipses the family of such elliptical solutions is infinite.

cube_Ellipse_complete_workoutFigure 5

Finally, we may note one special feature of the family of elliptical solutions that are specifically associated with the integer values of u. Given that the solutions x, y, z define a simple curved surface (Figure 1), i.e. the surface does not show any folding, there can utmost be 2 pairs of x, y that yield the same z. There are 29 z which can be derived from 2 distinct pairs of x, y< 20000. For example: 9^3+ 58^3= 22^3 + 57^3=256^3-255^3. Thus, (9, 58) and (22, 57) form a pair that yield the same z=255. The majority of such pairs of solutions with the same z lie on the ellipses arising from integer u. Further, each such ellipse contains a pair whose whose x values are respectively defined by x_1 =9u^3; x_2= 9u^3+9u^2+3u+1. The corresponding y values can be obtained by plugging x_1 into the second ellipse and x_2 into the first ellipse in (\S 6). Thus we get y_1= 27n^4 + 18n^3 + 9n^2 + 3n + 1; y_2=27n^4 + 18n^3 + 9n^2 + 3n; thus, for these cases y_1=y_2+1. These pairs are shown below for u=1, 2, 3, 4.

Thus, these paired values define an infinite family by themselves lying on in the curves defined by the below parametric equations (Figure 5, blue curve):

\begin{cases} x= 9t^3\\[6pt] y= 27t^4 + 18t^3 + 9t^2 + 3t + 1 \end{cases}\\[10pt] \begin{cases} x=9t^3+9t^2+3t+1\\[6pt] y=27t^4 + 18t^3 + 9t^2 + 3t \end{cases}

There are a minority of pairs which lie outside of the ellipses. We do not know as yet if they define any other families of solutions. More generally, it is also not clear if there are any other families of solutions beyond the above parabolic and elliptical families.

Sum of cubes of 2 positive integers that equal the difference of cubes successive same-parity positive integers
In this final section we shall briefly consider a related indeterminate equation:

x^3+y^3=(z+2)^3-z^3=6z^2+12z+8 \; \; \; (\S 7)

One can see right away that some of the regular families of solutions of (\S 7) are related to those of (\S 1). The first relates to the above theorem regarding the cubes of every positive integer. In this case the equivalent is:

The cube of every positive even number is equal to the difference of the cubes of two consecutive even numbers minus the cube of another even number.

These correspond to the solutions to (\S 7) of the form (x=2n, y= 6n^2 + 4n + 2 (Figure 6, violet curve). Further, the equivalents of the solutions to Gokai Ampon’s points in this case lie on the parabola defined by the parametric equation: (x=6t^2, 12t^2 - 6t + 2) (Figure 6, purple curve). However, the solutions to (\S 7) feature a unique parabolic family of solutions with no equivalent among the solutions of (\S 1). These lie on the parabola defined by the parametric equation: (3t^2 - t + 1, 3t^2 + t + 1) (Figure 6, orange curve). These correspond to x, y such as:
(3, 5); (11, 15) (25, 31) (45, 53)… Thus, x=3n^2 - n + 1 and y=3n^2 + n + 1 provide another link to the hexagonal number spiral as they correspond to numbers that respectively lie on its 2nd and 4th spoke (Figure 3). With this in hand, we can show that for these x, y give rise to z=3 n^3 + 2 n - 1, which defines the sequence: 4, 27, 86, 199, 384, 659, 1042…

As with the solutions to (\S 1), here too we have the equivalent elliptical families corresponding to the integer lattice points on ellipses of eccentricity \sqrt{\tfrac{2}{3}} that have equations of the form:

x^2-xy+y^2-ax-by+c=0 \\[6pt] x^2-xy+y^2-bx-ay+c=0

Here the 3 parameters a, b, c are defined by:

a=54u^4 + 36u^2 + 4\\[6pt] b=54u^4 - 2\\[6pt] c=108u^4 + 36u^2 + 4

However, the u which yield elliptical solutions for (\S 7) are the same as those that yield solutions for (\S 1) and there is an equivalence in the corresponding solutions. Figure 6 shows a few elliptical solutions \left(u=2, \sqrt{\tfrac{37}{3}}, \tfrac{\sqrt{73}}{3}, 3\right).

In conclusion, this exploration reveals connections between a certain class of cubic indeterminate equations and families of solutions defined by particular parabolas and ellipses. It is not known to us if any one previously studied these elliptical families or reported any other families beyond those considered here.

cube2_ellipse_workout

Figure 6

 

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The Mātrā-meru and convergence to a triangle

What is presented below will be elementary for someone with even just the mastery of secondary school mathematics. Nevertheless, even simple stuff might present points of interest to people who see beauty in such things. Consider the following question:

Given the first 2 terms 0 \le x_1, x_2 \le 1, what will be the behavior of the sequence defined by the recursive relationship:

x_{n+1}=x_n\sqrt{1-x_{n-1}^2}+x_{n-1}\sqrt{1-x_n^2}

Answer: It will converge to a cycle of length 3, where x_n, x_{n+1}, x_{n+2} will be the sines of the 3 angles of a right or an acute triangle. Further, let M be the well-known Mātrā-meru sequence: 1, 1, 2, 3, 5, 8, 13, 21…, then:

x_n=\sin\left(M[n]\arcsin\left(x_1\right)+M[n+1] \arcsin\left(x_2\right) \right);
x_{n+1}=\sin\left(M[n+1]\arcsin\left(x_1\right)+M[n+2] \arcsin\left(x_2\right) \right);
x_{n+2}=\sin\left(\pi-M[n+2]\arcsin\left(x_1\right)+M[n+3] \arcsin\left(x_2\right) \right)

Where M[n+1]\arcsin\left(x_1\right)+M[n+2] \arcsin\left(x_2\right) is the largest such angle that is \le \tfrac{\pi}{2}

This can be easily proved thus:

1) Since 0 \le x_1, x_2 \le 1, we can write x_1=\sin(A), x_2=\sin(B).

2) Thus, given the recursive relationship the next term becomes,
x_3=\sin(A)\cos(B)+\sin(B)\cos(A)=\sin(A+B)

3) Continuing this way, we can write,
x_4=\sin(A+2B); \; x_5=\sin(2A+3B); \; x_6= \sin(3A+5B).

We notice the multiplicands of A, B are the successive terms of the Mātrā-meru sequence. Thus, x_n=\sin\left(M[k]A+M[k+1] B \right).

4) This will continue till M[k]A+M[k+1]B comes closest to \tfrac{\pi}{2}. Then the next term M[k+1]A+M[k+2]B \ge \tfrac{\pi}{2}. But due the symmetry of the sine function,

\sin(M[k+1]A+M[k+2]B)=\sin(\pi-M[k+1]A+M[k+2]B).

Since, M[k+1]A+M[k+2]B=M[k-1]A+M[k]B+M[k]A+M[k+1]B,

we get M[k-1]A+M[k]B+M[k]A+M[k+1]B+\pi-M[k+1]A+M[k+2]B = \pi.

Thus, at this stage the three successive terms x_n, x_{n+1}, x_{n+2} are sines of the 3 angles of an acute or right triangle and they will settle into a cycle of those 3 values _{...\blacksquare}

Hence, the above recursive relationship results in any pair of x_1, x_2 converging to the 3 sines of an acute or right triangle. As a corollary if you start with x_1, x_2 which are already sines of an acute or right triangle then you stay on that triangle. Let us consider some special cases below (Figure 1).

Contri_Fig1Figure 1. The angles are given in degrees for ease of representation

When x_1=x_2=\tfrac{\sqrt{3}}{2} then the values are on an equilateral triangle and the iterates remain fixed on that triangle (Figure 1, panel 1).

When x_1=\tfrac{3}{4}; \; x_2=\tfrac{4}{5} then the values are on the 3-4-5 right triangle and the iterates remain fixed on that triangle (Figure 1, panel 2).

When x_1=\tfrac{1}{\sqrt{2}}; \; x_2=\tfrac{1}{\sqrt{2}} then the values are on the half-square right triangle and the iterates remain fixed on that triangle (Figure 1, panel 3).

When x_1=x_2=\tfrac{1}{2} then they are not an acute or right triangle. However, within one iteration the iterates converge to the sines of a right triangle, namely the 30^\circ-60^\circ-90^\circ \; \triangle (Figure 1, panel 4).

It is easy to see that if \tfrac{1}{\sqrt{2}} \ge x_1=x_2 \le 1 then they are on an isosceles triangle and remain on that. However, if \arcsin(x_1)+\arcsin(x_2) < \tfrac{\pi}{2} then can we converge to an isosceles acute triangle? This happens in special cases which can be determined by solving an equation. In order to do so we shall take x_2=x; \; x_1=k-x; \; k< \sqrt{2}. From the above proof the successive angles corresponding to the iterates of x_n are:

\arcsin(k-x); \; \arcsin(x); \arcsin(k-x) + \arcsin(x); \; \arcsin(k-x) + 2\arcsin(x); \; 2\arcsin(k-x) + 3\arcsin(x); \; 3\arcsin(k-x) + 5\arcsin(x)...

Thus, we have to look for real solutions of the equations such as:

\arcsin(k-x)=\pi - (\arcsin(k-x)+\arcsin(x))
\arcsin(x) = \pi -(\arcsin(k-x) + \arcsin(x)
\arcsin(x) = \pi -(\arcsin(k-x) + 2\arcsin(x)) ...

Let consider the example of k= 1: with either of the first two equations we get degenerate triangles (e.g. x_1=0, x_2=1, x_3=1). However, if we instead take x_1=\tfrac{1}{m}, x_2=\tfrac{m-1}{m} for some large m we get near-isosceles triangles (Figure 1, panel 5).

The one equation with a real solution for k=1, which gives a unique isosceles triangle, is seen when:

\arcsin(1-x) +3 \arcsin(x) = \pi, \; x \approx 0.83756543528332

This x is the greatest root (r_1) of the cubic equation 4x^3-4x+1=0. Thus, x_1=1-r_1, x_2= r_1 yields an isosceles triangle with its equal angles \arcsin(r_1) \approx 56.88^\circ (Figure 1, panel 6).

Next we shall consider the evolution of certain special sequences of triangles. The first is where x_1, x_2 are constituted by successive terms of the Mātrā-meru sequence (Figure 2).

Contri_Fig2_meruFigure 2.

Here, the triangles start with the 30^\circ-60^\circ-90^\circ \; \triangle and converge to a unique scalene triangle with angles \arcsin\left(\tfrac{1}{\phi}\right) - \arcsin\left(\tfrac{3\phi-1}{2\phi+1}\right) - \arcsin\left(\tfrac{2\sqrt{\phi}}{\phi+1}\right) \approx 38.17^\circ - 65.48^\circ - 76.35^\circ, where \phi= \tfrac{1+\sqrt{5}}{2} is the Golden ratio.

The last panel here shows an interesting numeric coincidence. If you start with x_1=r_1, x_2=1-r_1 (see above for r_1) you converge to a triangle close to that emerging from the Mātrā-meru sequence. Is there more to this than the coincidence of values?

Finally, let us consider 2 other special triangles that emerge as convergents for 2 related types of operations based on the Mātrā-meru sequence (Figure 3).

Contri_Fig3_meru_densFigure 3.

The first 2 rows (in light green) show triangles emerging from x_1=\tfrac{1}{M[k]}, x_2=\tfrac{1}{M[k+1]}, where k=2, 3, 4.... Here again, we start with a 30^\circ-60^\circ-90^\circ \; \triangle and converge to a triangle of the form \approx 70.82^\circ - 65.41^\circ - 43.77^\circ.

The second 2 rows (in yellow) show triangles emerging from x_1=\tfrac{1}{M[k]}, x_2=\tfrac{1}{M[k]}, where k=3, 4, 5.... These converge to a triangle of the form \approx 87.29^\circ - 57.30^\circ - 35.41^\circ.

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The Aśvin-s and Rudra

The twin Aśvin-s and Rudra are both Indo-Aryan reflexes of two deity-classes which can be reconstructed as likely being present in the Proto-Indo-European religion. Both are likely to have even deeper roots going back to even earlier religious traditions across a wide swath of humanity. Indeed, the divine twins feature even outside the IE religions. At the face of it it is not obvious if these two deity classes show any special links. For example, with Rudra as a focus we can use the Ṛgveda to sample his association with other deities. One proxy for association can be how often Rudra is mentioned with another deity in the same pāda or hemistich of a ṛk. Below is a table showing his co-occurrences by this metric.

Table 1. The association of various Devatā-s with Rudra in the RV

Devatā #
Marut-s 21
Vasu-s 18
Āditya-s 13
Agni 10
Ṛbhu-s 7
Indra 6
Aśvin-s 6
Soma 6
Varuṇa 5
Mitra 5
Bhaga 5
Puṣaṇ 4
Viṣṇu 3
Bṛhaspati 2
Sarasvatī 2
Tvaṣṭṛ 2
Aryaman 1
Savitṛ 1
Aditi 1
Vāyu 1

Not surprisingly, he is most closely associated with his sons the Marut-s who are often referred to as the plurality of Rudra-s. The other close associations come from the fact that the Rudra-s are mentioned as a group of gods along side to two other big groups of gods: the Vasu-s and the Āditya-s. He further shows notable associations with one of the archetypal deities of the Vasu-group, Agni with who he shares a duality. Beyond that his mentions along with the Aśvin-s is no more frequent than with Soma, Indra or the Ṛbhu-s (the divine craftsmen whose masters are said to be the Rudra-s: RV 8.7.12). That said a closer look reveals a deeper link between Rudra and the Aśvin-s.

uta tyā me raudrāv arcimantā
nāsatyāv indra gūrtaye yajadhyai |
manuṣvad vṛktabarhiṣe rarāṇā
mandū hitaprayasā vikṣu yajyū || RV 10.61.15

Also these two sons of Rudra, the Nāsatya-s [worshiped] with ṛk-s, are to be welcomed and made offerings by me, O Indra. The two who are liberal to him, who as Manu [had done, invites them] to the woven grass (the twisted barhiṣ in the vedi), the delightful [twins] for whom pleasing offerings are made, the twins who seek the ritual among the people.

Here the ṛṣi Kakṣīvān calls the Aśvin-s the sons of Rudra in a manner similar to the Marut-s. Further, this sūkta mentions our ancient ancestor Cyavāna as the one who had measured out the vedi for the for the Aśvin-s. Cyavāna is mentioned in tradition as the one who instituted the rite where soma is offered to the Aśvin-s. Indeed, in the above ṛk it appears that Kakṣīvān seeks Indra permission for the same as there are indications from later narratives that Indra was not entirely on board with that. In any case, what is important for our current discussion is the relatively unambiguous link of the Aśvin-s and Rudra in this ṛk.

This is not the only instance the Aśvin-s are called Rudra-s or the son-s or Rudra. Indeed, we encounter another such example elsewhere in the RV:

tāv id doṣā tā uṣasi śubhas
patī tā yāman rudravartanī |
mā no martāya ripave vājinī-
vasū paro rudrāv ati khyatam || RV 8.22.14

Just these two in the evening, these two auspicious lords at dawn, the two who follow the tracks of Rudra in the course. Don’t look over and beyond us to a roguish mortal, O Rudra-s with booty-bearing mares.

In this ṛk of Sobhari Kāṇva, they are not just called Rudra-s but also specific described as Rudravartanī. This word is of considerable interest in regard to this connection between these deities. It belongs to class of compounds of the form “x+vartani” that are found throughout the RV and used for different deities. The word vartani means wheel (typically of the chariot) or track of the chariot wheel. Thus, x+vartani compounds are usually interpreted as bahuvrīhi-s. Below we list all the instances of such compounds in the RV along with the gods they denote and the number of occurrences of each case:

  • Rudravartani
    • Aśvin-s, 4
  • Hiraṇyavartani
    • Aśvin-s, 6
    • Sarasvatī, 1
    • river goddess, 1
  • Raghuvartani
    • Aśvin-s, 1
    • Soma, 1
  • Ghṛtavartani
    • Aśvin-s, 1
  • Vṛjinvartani
    • Agni, 1
  • Kṛṣṇavartani
    • Agni, 1
  • Gāyatravartani
    • Indrāgnī, 1
  • Dvivartani
    • Agni, 1

It is immediately apparent that this class of compounds are special descriptors of the Aśvin-s for 12 of the 19 occurrences of them are used for the Aśvin-s. This is likely a special allusion related to the oft-mentioned speeding tricycle (tri-cakra) chariot of these gods. However, it should be noted that such compounds, while most frequently used for the Aśvin-s are not limited to them; e.g. Hiraṇya-vartani is use for at least 2-3 distinct deities (river goddess Sindhu could be a cryptic allusion to Sarasvatī). Some of the usages, such as Hiraṇya-vartani or Raghu-vartani can be simply interpreted as the ones with the golden wheels or one with swift wheels and appear to apply to other deities than the Aśvin-s. Indeed, a related term “Hiraṇya-cakra” is used for the actual Rudra-s, i.e. Marut-s (e.g. RV 1.88.5) or for that matter Hiraṇya-ratha used for Indra (e.g. RV 1.30.16). Further, some might be even typical of other deities: e.g. Kṛṣṇa-vartani (with black tracks: alluding to the smoke) and Vṛjina-vartani (with curving tracks, alluding to the flames) are apt for Agni for whom they are used. The form dvi-vartana used Agni is simply indicative of two tracks left behind by the two chariot wheels.

However, of all these Rudra-vartani is specifically used only for the Aśvin-s on multiple occasions and by multiple composers. This suggests that this term has a special connection with the Aśvin-s. Drawing the cue from the more frequent Hiraṇya-vartani, also used for the Aśvin-s, Rudra-vartani has been tradition interpreted as either Rudra= ruddy or Rudra = fierce. Thus, the compound is understood as the Aśvin-s with ruddy tracks, or those with fierce tracks — perhaps as an allusion to their speeding chariot that is frequently seen in the RV, or as those who go along terrifying paths. Entirely, independently of our investigations, we learned that this line of reasoning was first explored in detail by the great patriot Aurobindo Ghose. But the key is the observation that this term is specific to the Aśvin-s. Rudra is not used in the sense of “ruddy” elsewhere in the corpus. “Fierce or terrible tracks” would have implied the form raudra-vartani, which we do not ever encounter in this corpus. Further, ruddy horses or chariots are described by terms like aruṇayugbhir aśvaiḥ (RV 6.65.2) and fierce chariots by terms like tveṣa-ratha for the Marut-s (RV 5.61.13; also perhaps a personal name of a Mitanni ruler among the Indo-Aryans of West Asia). Indeed, the ferocity of the Marut-s’ chariots with ruddy horses are alluded to elsewhere too (e.g. RV 1.88.2) but the term Rudravartani is never applied to them even if it might be natural in this sense.

Thus, taken together with the instances where the Aśvin-s are called Rudra-s or the sons of Rudra (see above and also below) we conclude that Rudra-vartanī specifically indicates the association of the twins with Rudra and means “the two who follow the track of Rudra”. After we reached this conclusion, a search revealed that such a translation had been independently conceived by the German indologist Hermann Oldenberg. Notably, this link to Rudra is further strengthened by another instance where they are called both Rudra-s and Hiraṇyavartanī:

ā no ratnāni bibhratāv
aśvinā gacchataṃ yuvam |
rudrā hiraṇyavartanī
juṣāṇā vājinīvasū
mādhvī mama śrutaṃ havam || RV 5.75.3
Bearing treasures to us, Aśvins, come here, you two, O Rudra-s with golden wheels, with booty-bearing mares, being pleased, the holders of the honey-lore, hear my invocation.

We believe there are many dimensions to this connection:
1) The ancient name of the Aśvin-s is Divo Napatā. The twin sons of Dyaus. This is an equivalent of the name of their Greek cognates the Dioskouroi (the national deities of the Spartans), meaning the youths of Zeus (the cognate of Dyaus; Skt Divaḥ Kumārau) or their Lithuanian cognates Dievo Suneliai (Sons of Dieva = Dyaus). Now Dyaus on occasion is identified with Asura Rudra in the RV:

tvam agne rudro asuro maho divaḥ (RV 2.1.6)
You, O Agni, are Rudra the Asura of heaven (Dyaus)

yathā rudrasya sūnavo divo
vaśanty asurasya vedhasaḥ | (RV 8.20.17)
It shall be [just] as they wish, the sons of Rudra, the Asura of heaven (Dyaus) are the wise ones.

And like in this case too:
indrota tubhyaṃ tad dive tad rudrāya svayaśase | (RV 1.129.3)
Indra, this is for you and that heaven (Dyaus), for that self-glorious Rudra.

The above indicate that there was an early Vedic tradition that identified Rudra with Dyaus, in which sense he was also seen as the father of world by the Bharadvāja-s (RV 6.49.10). This, together with the appellation Divo Napātā for the Aśvin-s, hints a parallel Vedic tradition which saw them as deities in the Rudra-class associated with the leader of that class Rudra, the great Asura of Dyaus. Their “Raudra nature” is clearly brought out in the ṛk RV 10.93.7: uta no rudrā cin mṛḻatām aśvinā : Also, though being Rudra-s, may the Aśvin-s be merciful. This plea for mercy to them is comparable to that typically made to Rudra or the Marut-s. Consistent with this, they share their medical prowess with Rudra (as physicians of the gods) and even more tellingly also their knowledge of poisons with Rudra (RV 1.117.16: where they either kill the brood of Viṣvāc with poison or destroy the poison associated with the brood of Viṣvāc). Thus, across the RV, composers from different clans occasionally saw the Aśvin-s as Rudra-s or Rudra’s sons and allude to their Rudrian properties.

2) One of the notable aspects differentiating the early Atharvaṇ tradition from the RV tradition with regard to Rudra is the use of the twin appellation Bhavā-Śarvā for the deity in the former. These names of Rudra also persist in the ādhvaryava tradition preserved in the Yajurveda-s but the twinning is less prominent relative the AV tradition. Notably, in the celebrated Mṛgāreṣṭi ritual the AV tradition features a sūkta to the twin Bhavā-Śarvā whereas in its place the KYV tradition has ṛk-s to the Aśvin-s. Further, in the incantations for the Śūlagava ox sacrifice laid out in the Śāṅkhāyana-śrautasūtra (4.20.1-2), Bhavā-Śarvā are called the sons of Mahādeva.

tasya te dhanur hṛdayam mana iṣavaś cakṣur visargas tam tvā tathā veda namas te astu somas tvā avatu mā mā hiṃsīḥ | yāv araṇye patayantau vṛkau jambavantau iva | mahādevasya putrābhyām bhava-śarvābhyām namaḥ ||

The heart is your bow, the mind is your arrow, the eye is your shooting. Thus we know you. Obeisance to you. May Soma protect him and may you never ever harm me. The two who roam around in the forest like wolves with jaws wide open; obeisance to the two sons of Mahādeva, Bhava and Śarva.

This points to two parallel streams within the early Vedic tradition which featured Rudra in singular form (apart from the plurality of the Rudra-class) as seen in the RV or in twin form of Bhavā-Śarvā as seen in the AV and the Śāṅkhāyana-śrautasūtra (as the twin sons of Rudra). This suggests that the Rudra-class had an ancient intrinsic twin nature shared with the Aśvin-s which lingers in the Bhavā-Śarvā dyad. Notably, even in Greek tradition the deity of the Rudra-class, Apollo is born with twin (albeit female), Artemis. While the RV Aśvin-s are identical twins in character, sometimes, in the IE world one sees some differentiation of the the twins with one of them associated with healing and animal-husbandry and the other with warfare. For example, among the Yavana-s, one of the pair, Castor is a horse-trainer while the other one Polydeuces is a boxer. A similar differentiation is perhaps reflected in the twin Rudra-s, with the name Bhava indicating welfare and health, and the and Śarva meaning and archer and indicating the warrior nature of the second twin. Thus, it is likely that Bhava and Śarva were part of the parallel Vedic tradition where they played the role of the Aśvin-s as twin Rudra-s.

3) A later reflex of this twinning in the Rudra-class appears to have emerged via developments in the form of the Kaumāra tradition emerging in the Indo-Iranian borderlands. There we see the dual form of the god Kumāra as Skanda-Viśākha. This dyad is earliest attested in the Atharvavedīya-Skanda-yāga from the AV-pariśiṣṭha-s and has a prolonged presence in the Hindu tradition. We find an allusion to this in a simile used in the Mahābhārata when Kṛṣṇa describes the twins, Nakula and Sahadeva, to his brother Halāyudha:

yau tau kumārāv iva kārttikeyau
dvāv aśvineyāv iti me pratarkaḥ |
muktā hi tasmāj jatuveśmadāhān
mayā śrutāḥ pāṇḍusutāḥ pṛthā ca || (Mbh “critical” 1.180.21)
The two who are youths, like twin Kārttikeya-s,
are the sons of the two Aśvin-s; so I infer.
It has been heard by me that the sons of Pāṇḍu and Pṛthā
have indeed escaped from the burning of the wax-house.

Here the sons of the Aśvin-s (their earthly manifestations) are explicitly connected to the twin Kārttikeya-s — an allusion to the twin deities, Skanda and Viśakha. The grammarian Patañjali mentions metal images of Rudra along with the Skanda-Viśākha dyad in the Mauryan age (~322-185 BCE). The persistence of this tradition is illustrated by a much later attestation of this dyad, evidently drawn from a now lost early tradition, seen in the Kālikā-purāṇa with a predominantly East Indian locus:

dahano ‘pi tathā kāle prāpte gaṅgodare svayam |
retaḥ saṅkrāmayāmāsa śāṃbhavaṃ svarṇa-sannibham ||
Agni himself, in due course of time, transmitted the semen of Rudra, which shone like gold, to the womb of Gaṅgā.

sā tena retasā devī sarva-lakṣaṇa-saṃyutaṃ |
pūrṇa-kāle ‘tha suṣuve putra-yugmaṃ manoharam ||
Then by that semen the goddess, upon completion of pregnancy, gave birth to charming twin sons endowed with all [good] features.

ekaḥ skando viśākhākhyo dvitīyaś cāru-rūpa-dhṛk |
śakti-dvaya-dharau dvau tau tejaḥ kānti-vivardhitau|| KP 46.82-84
The first was Skanda and the second bearing a beautiful form was known as Viśākha. The two held a spear each and two shone with their radiance.

Thus, we see Rudra siring the twin Kaumāra deities Skanda and Viśākha who are described as bearing spears much like the depiction of the spear-bearing Dioscuri in the yavana tradition. Interestingly, their beauty is specifically described much like that of the Aśvin-s in Vedic tradition. Further, the Kālikā-purāṇa recommends the worship of this dual Kaumāra form for the ṣaṣṭhī night:

rātrau skanda-viśākhasya kṛtvā piṣṭa-putrikām |
pujayec chatrunāśāya durgāyāḥ prīyate tathā || KP 60.50
In the night having made images of Skanda and Viśākha from flour one should worship them for the destruction of enemies and for pleasing Durgā.

In archaeological terms we find depictions of the Skanda-Viśākha dyad on Kuṣāṇa coins and also Kuṣāṇa age images from a lost Kaumāra shrine from the holy city Mathura (now housed in the collections of the Mathura museum). What is notable about their numismatic appearances is their resemblance to the twin gods on Greek Dioscuri coins. Thus, like Bhava-Śarva in a parallel Vedic tradition, Skanda-Viśākha are likely developments of the ancient Raudra twins in a Para-Vedic tradition that then entered the Indo-Aryan mainstream.

4) Finally, it is plausible that the term Rudravartanī and the association of the Aśvin-s with Rudra have an astronomical significance. In the classic nakṣatra system that developed by the time of the AV and the YV the Aśvin-s are associated with the constellation of Aśvayujau which corresponds to part of Aries. However, the obvious constellation that resembles the divine twins is Gemini, which was recognized as the Dioscuri in the Greek tradition. While it was termed the constellation of Aditi, the ārya-s too recognized the dual nature of the asterism Punarvasu made up two stars Castor and Pollux — it is occasionally used in dual like: punarvasū nakṣatram aditir devatā ||. Early on we see the recognition of the twin nature in the statement that Aditi is two-headed in the Yajurveda (Taittirīya Saṃhitā in 1.2.4; Śatapatha Brāhmaṇa 3.2.4.16). Ironically, Pāṇini reinforces the dual nature with the sūtra: chandasi punarvasvor ekavacanam || PAA 1.2.61. In the Veda Punarvasu might be [optionally take a] singular declension. The grammarians clarify that this is limited to the Veda while in common speech it is always dual indicating its twin nature. The only direct allusion to the asterism in RV (along with Revati) appears to be in RV 10.19.1 (by our ancestor Cyavāna), which associates it with Agni and Soma in a cryptic hymn whose actual meaning has been hard to discern. However, a potential connection is seen in RV 10.39.11 where the Aśvin-s are called Rudravartanī and explicitly linked Aditi — pairing that is otherwise rather unusual:

na taṃ rājānāv adite kutaś cana