## Johannes Germanus Regiomontanus and his rod

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

Figure 1. The rod of Regiomontanus

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

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

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

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

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

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

Figure 3. Construction to solve the Regiomontanus problem.

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

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

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

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

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

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

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

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

tripañcāśaḥ krīḻati vrāta eṣāṃ
ugrasya cin manyave nā namante
rājā cid ebhyo nama it kṛṇoti ॥

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

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

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

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

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

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

Integer Draws Number
1 L 1
2 D, LL 2
3 T, DL, LLL 3
4 K, TL, DD, DLL, LLLL 5
5 KL, TD, TLL, DDL, DLLL, LLLLL 6
6 KD, KLL, TT, TDL, TLLL, DDD, DDLL, DLLLL, LLLLLL 9
7 KT, KDL, KLLL, TTL, TDD, TDLL, TLLLL, DDDL, DDLLL, DLLLLL, LLLLLLL 11

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

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

Figure 2. First 18 integer triangles

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

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

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

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

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

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

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

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

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

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

Figure 4. Triangles with an angle approximating a radian.

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

## Modulo rugs of 3D functions

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

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

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

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

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

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

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

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

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

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

## A guilloche-like trigonometric tangle

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

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

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

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

Figure 1.

Here is another run of the same.

Figure 2.

## Huntington and the clash: 21 years later

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## The phantoms of the bone-pipe

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

Back home from the test drive, Vidrum rang in his butler and informed him that he had obtained dinner from outside and offered the butler and his wife a packet of the same too. He then asked the butler to prepare cold turmeric and almond milk for the night and dismissed him. As he was enjoying his mouth-tingling dinner he lapsed into a train of thought: “I wish Somakhya was around to cast a spell of protection on my car. Time and again my mind goes back to my lamented first car. I remember that day clearly.” As his mind drifted there, his joy from the tasty dinner flattened quite a bit. Having concluded his meal, he went over to the little shrine Somakhya had installed for him to worship the 16-handed Vīryakālī, whose original was enshrined at the edge of the cemetery in his ancestral village. When Somakhya and Indrasena had learned of that shrine they were excited. They told him of the great significance of that goddess as per some ancient text whose name he had forgotten. He had promised to take them along with Lootika and Vrishchika to his village someday. He felt his meditation gave him some focus to put a few words to paper reporting a fatal case involving the infection of a sanitation worker by Burkholderia mallei. As he was doing so, he remembered that Somakhya had given him a monologue on the bacterium when they were in college but Vidrum was not yet a suitable vessel to have imbibed any of it. He kicked himself for the same because he knew it was something important for his current investigation but he just did not recall anything. But Somakhya and his gang were far away and in limited contact; hence, he had to contend with whatever he had.

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

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

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

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

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

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

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

What followed was an exciting phase of my life. I soon got a small, on-campus residence — a quaint little tiled roof house. I also managed to secure my friend Jang Bahādur a job as a security man for the campus museums. He had supplied me a manuscript from Nepal on Kiranti temples. I studied and translated it and published an article in a white indological journal. I followed it up with another paper in such a journal reflecting the linguistic knowledge that I had acquired from Somaśiva Śarman on the substrates in Indo-Aryan. The I went with Jayasvāmin, the curator of our museum, and his wife Śilpikā to study a Gupta temple of Bhairava and the 8 mothers in Mālava. In that expedition, we found a cache of Gupta gold coins that we brought back to the college museum to study. Jayasvāmin and Somaśiva with their epigraphic expertise worked out some worn inscriptions in the temple and based on that we wrote a paper that described the early phases of the Bhairava-srotas and how the worship of Kārttikeya followed by the 8 mothers was an important facet of that tradition. As part of that study, I reconstructed a verse to Skanda in the Vasantatilaka meter that was inscribed in that temple during the reign of emperor Vikramāditya, which described a fierce form of Skanda. This tradition was to be incorporated as Baṭukanātha in the later Bhairava tradition. All this work earned me some reputation in the college and mleccha visitors from aboard came to pick my knowledge. It was then that my patrons at the college suggested that I submit a dissertation for the doctoral degree at the university. Thus, I could upgrade myself from a mere staff paṇḍit to a professor. As I was wondering what I should submit as a dissertation, Śilpikā had invited a young brāhmaṇa lady to talk about a paper she had recently published on a comparative analysis of the substrates in the different Yajurveda saṃhitā-s and the implications it had for the āryan conquest of the northern India.”

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

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

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

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

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

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

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

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

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

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

Lootika told her mother-in-law about the story and asked if she had any recollection of this deceased man. Somakhya’s mother: “I never took the claims of the phantasmagorical encounters of you kids seriously. But I must say this one comes about as close to being believable as any. Yes, Bāẓ Nayan was a learned man who was borne away by the inexorable force of fate — after all, even the Yādava-s, be it the mighty Sātyaki or the brave Pradyuṁna, had to clobber themselves out of existence when their time came.” She went to the study and brought out a huge volume and gave it to Lootika: “However, here is a copy of his dissertation. While nobody knows this, it was not lost for good after all. I had made a copy of it for my own study. I wanted to give it you all for it still contains insights that will help you in your own study and practice of the śruti but I kept forgetting. While no photo or belonging of Bāẓ Nayan Śarman seems to have survived the one who chomps through the Vanaspati-s, after all, my dear, the śruti has said: na tasya pratimā asti yasya nāma mahad yaśaḥ|‘”

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

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

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

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

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

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

“apāṃ puṣpaṃ mūrtir ākāśaṃ pavitram uttamam” iti ācamyābhyukṣy ātmānam anumantrayata | [sūrya jīva devā jīvā jīvyāsam aham |
sarvam āyur jīvyāsam ||]
“The flower is the form of the waters, the empty space [and] that which the most pure”. Thus, he sips the water and having sprinkled water (practically mārjanam) he recites the incantation indra jīva etc: Enliven, o Indra; Enliven o Sūrya. Enliven, o gods. May I live. May I complete my term of life.

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

iti brāhmaṇam ||
Thus is the brāhmaṇa.

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

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

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

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

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

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

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

Thus is the brāhmaṇa.

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

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

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

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

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

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

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

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

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

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

## Some notes on the Henon-Heiles Hamiltonian system

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

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

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

Figure 1.

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

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

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

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

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

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

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

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

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

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

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

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

Figure 3.

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

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

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

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

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

Figure 4.

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

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

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

Figure 5.

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

Figure 6.

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

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

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

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

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

Figure 8

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

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

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

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

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

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

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

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

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

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

## RV 10.5

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

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

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

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

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

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

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

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

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

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

ṛtasya hi vartanayaḥ sujātam
ghṛtair annair vāvṛdhāte madhūnām || 4

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

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

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

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

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

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

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

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

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

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

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

## Bṛhaspati-śanaiścarayor yuddham-2020 ityādi

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Figure 1

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

Figure 2

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

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

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

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

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

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

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

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

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

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

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

Figure 4. Successive oppositions of Jupiter

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

Figure 5. Successive conjunctions of Venus

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

Figure 5. Successive oppositions of Ceres

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

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

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

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

Figure 1

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

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

Figure 2

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

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

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

Figure 3

Figure 4

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

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

Figure 5

Figure 6

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

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

Figure 7

Figure 8

## Prakīrṇaviṣayāḥ: Life, brains, warfare and society

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

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

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

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

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

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

We have wondered if this phenomenon extends beyond nucleic acid replicators. The bloated Microsoft and Adobe Software come to mind. Another possible example is the growth of the LaTeX system. Thus, one would expect that large complexes of memes are likely to be invaded by smaller memetic parasites that they cannot get rid of. Similarly, generationally transmitted social structures and strategies (overlapping with memes) could also become home to freeloading smaller parasitic systems. We first got a hint of its wider relevance from governmental bureaucracies, where you find various, “inserts” whose absence would make little difference to the functioning of the system. We then observed the same within the otherwise robust structure of religions. This would appear to be particularly so to a pure mīmāṃsaka for whom every word in religion goes towards a ritual injunction. Thus, the presence of extraneous stuff, what he may dismiss as “arthavāda”, would be seen mostly as such textual freeloaders transmitting themselves much like an intron or an intein in a genome. This comparison led us to realize that even as the larger genomes might not be able to rid themselves of such freeloading genetic material, these other systems like texts or bureaucracies might also not be able to do so. Why? one might ask, should selection not purge them? First, as we saw with the inteins in the DNA polymerase, the cost of purging them might be higher than letting them remain if some kind of “compromise” is reached. Sometimes a cost against purging is imposed by an “addiction module” (e.g. in toxin-antitoxin systems). Here, the intein splices itself and thus make sure the DNA polymerase is functional after all. Second, the robustness of the invaded systems might smear the effect of selection into a band rather than a discrete line — therefore within that given band, the variability (e.g. presence or absence of a freeloading genetic parasite) results in no selectable difference in fitness. This implies that the freeloaders can stay on as long as the cost imposed by their persistence mechanisms does not exceed the allowed bandwidth. While we began our example by comparing large viral genomes to small ones, this is taken to an extreme in cellular genomes. The self-splicing machinery of the introns was built into a more complex spliceosomal machinery that confers a certain resistance to introns for the cellular genomes. On the intron side, it allowed them to spread relatively benignly — freeloaders but not imposing a cost enough to derail the system. In ciliates, we see another such “compromise arrangement” taken to an extreme. The cellular RNAi machinery together with the transposases from some of the freeloading elements ensure that a functional “cellular genome” from which they are excised is put together in the macronucleus. However, they are passed on for posterity via the micronucleus where they remain intact but inert.

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

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

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

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

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

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

## The old teacher

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

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

S: “So, what have you all been expounding at the temple?”
LM: “Your mother does the Bhagvadgītā. But as you know, over the years I’ve come over to your side. Little Lootika and I had a tiff after she was lambasted by a learned Uttaramīmāṃsā anchorite for aggressively upholding the doctrine of the owl, which evidently you had introduced to her. You may recall that while talking about this incident when we were visiting your place, your father had brought to our attention that the last common ancestor he and I shared had written a text in 1657 CE titled the Ulūkāvabodhanam, expounding the great doctrine. Since then I started studying the doctrine more closely and came over to position closer to you kids. Hence, I introduced it to the audience by way of some variety.”
S: “I’m always inspired by its opening: nama ulūkāya rudrāya aṇuvide |
LM: “I can see that, my dear. It is indeed that opening story that I narrated to them — where Rudra appeared in the form of an owl to teach the doctrine to the great Kāśyapa. In any case, other than that, at some point when all of you kids were out of home, we also combined our efforts to start a second course, a purāṇa exposition. While there are many expositors of Vaiṣṇava-bhakti themes, we decided to go off the beaten track and have readings from the old Skandapurāṇa — to our surprise, it has gathered much greater interest than the philosophical one. It was in that regard that I needed to consult with you regarding the tale of how Skanda helped his mother in creating Vināyaka.”

Somakhya and Lootika’s mother continued on the topic of her intended exposition until she was done with her cooking and she asked Somakhya to get ready for lunch in a short while. Just then, Lootika and her companions arrived home and Lootika reintroduced them to Somakhya, reminding him of the long past days when they had first met. Then they performed a namaskāra to Lootika’s mother, who asked them to stay on for lunch; Kalakausha and Kallolini said they had to be in the hospital shortly but agreed to take some packed food along from her. Lootika’s mother admonished them to eat properly and on time. Just as they were at the gate to leave, Kalollini remarked: “Our history teacher from school periodically asks about you all. She remarked that she would like to talk to you, Lootika or your husband, or one of your sisters, whenever you’ll are in the city. I think it is something specific that she might want to talk about. Please meet her if you get the chance and are so inclined. It might really help her.” Saying so, the two ladies sped away. After they had left, Lootika’s mother remarked: “These kids of humble provenance have come a long way. Lootika, they often recognize the role you and your sisters played in the process of raising them from an unimaginative ground state to aspiring to be part of the elite. Kallolini was quite close to Varoli just like her sister to you in the old school. So, at some point, when their financial status improved enough to afford the school bus, they transferred her to your school as Varoli’s classmate.” L: “Dear mom, I suspect you ascribe more than a required role to nurture in their case. I think there was some spark that came together in them by the churn of genetics despite their plebeian roots; without that our inspiration would have amounted to nothing. Moreover, there is not much friendship between asamāna-s; the fact that they could get along with us without a display of vulgarity indicates some deeper saṃskāra in them.” LM: “You would know better. Though, I must remark that Kallolini did even better than Kalakausha in the entrance exams. I guess that was because Varoli and Jhilli could directly help her with the intricacies of science and math. She was my student in college, and she was not bad at all. Maybe some vāsana from a past janman.”

L: “What is the deal with our old history teacher? As Somakhya would say, most of the teachers and the classmates from school with whom we have lost contact have passed out of our ken like vāsāṃsi jīrṇāni.”
LM: “Somakhya, do you recall anything of your interactions with her? I remember that she was generally, good to my other three daughters, but was among the multiple teachers who would repeatedly complain to me that your wife is rather arrogant due to her varied knowledge that was so atypical of the girls.”
Somakhya looked at Lootika and chuckled: “Actually, she was one of those teachers who was not bad to me at all. She had a Nehruvian bent of mind and liked to downplay the sultanate or the Mogol tyranny and boost the English superiority over our culture. That said, she was rather generous to me and I’d say to Spidery too when we pressed the counterpoints to these issues in class. But why would she want to see us? So much has gone by and we should have passed out of her memory too.”
LM: “I don’t know for sure, but I have a bit of gossip in her regard. Her husband, a lawyer, disappeared one day without warning and nobody ever found him dead or alive. Since then she has been rather beaten down and comes to the Śivālaya to attend your mom and my expositions. She has aged rapidly and looks bent. Hence, we did not recognize her right away. One day, Kallolini brought her up to us after the exposition and asked if we knew who she was. While I was struggling to place her face, your mother correctly called her out. She told us the tale of her husband’s sudden vanishing. It was then that we remembered that we had encountered that story in the news but had not connected it to her. It looked as though she wanted to say something more in that regard but then hesitated and went quiet. Then she asked about you kids and when she could see you all. She periodically keeps pressing your mother and me about that very point. I never made much of it and simply took it to be some form of a polite inquiry.”
Lootika waded in: “I sensed that Kallolini seemed to know something more; I was about to ask but they ran away. I’ll call her and ask.”
LM: “Lootika, you can be so callous! They would be busy with patients.”

An orthogonal thought running through Somakhya’s mind made him change the topic inadvertently as he thanked his mother-in-law for the bhiṇḍītaka dish she had made: “This brings pleasant old memories. You may not know this but now I can say this freely. This was the first dish Lootika ever shared with me. I remember the day clearly — there was a torrential downpour; hence, we had the misfortune of needing to eat in the school halls. I always dreaded that, as one would invariably end up eating in the vicinity of someone with the wafting odors of abhojya food. We were also supposed to be preparing for the talent show that was in the coming week and we had a couple of periods off after lunch. Several of our schoolmates had already begun doing so during the lunch break itself in the front of the hall. Therefore, I retired to the last bench of the hall and still not feeling like eating due to the odors in the surroundings and continued working on the unusual small serine-peptidase chaperone domain of tailed bacteriophages that I had recently uncovered. It was then that Lootika sneaked up to sit beside me and shared this dish with me. We spoke of what I was doing and that sparked a remarkable idea in her head for creating a biotechnological reagent. Taking advantage of the free time, we worked out that idea to completion. She eventually did make it a reality in the lab and used it effectively.” Blushing a bit, Lootika remarked as though to distract away from the story: “It remains a useful weapon in our armory for the purification of functional proteins.” LM: “Lootika, no wonder you asked me to make this today. Now I know its link to your romance…”

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

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

Soon the two of them got ready and took the relatively long walk back to their old school reminiscing about the happier and adventurous side of the old days. As they walked along, they paused at the spots where they had discussed memorable old findings like the primases of viruses, the RNA-modifying enzymes they had put on the map and the like. The place had changed quite a bit since they had left — in parts swankier and in parts dirtier than before. Finally, they reached the school and rung in their history teacher who with great excitement let them in. As Lootika’s mother had remarked, she looked more haggard than her age and health would suggest. She did not linger for much of an inquiry, as would be typical of a such a meeting after so long a time, confirming their suspicion that she had some specific reason for meeting them other than catching up with past students. She broke into her story: “I have something to ask of you all. Kallolini had told me that, if anyone, it would be one of you all who could help me deal with this. Kallolini’s elder sister, who apparently knows you Lootika, affirmed the same. My husband, you may or may not know, was a respected lawyer who practiced not only at the high court in our city but also in the higher courts. I really took no interest in his cases or his clients and nor did he mention any of those at home. Our conversations were on entirely different matters of common interest. One thing I did know but again did not take much interest in was his fancy for unsolved cases. He occasionally played detective and helped people with cases where they had no recourse through the official channels. He had his connections from his long practice and could get some of these cases reopened or receive official attention due to his efforts. He was no busybody but found great satisfaction in helping people reach closure in incidents where the system had failed them. I mention this because it might have some relevance to what happened to him but I don’t know the exact connection.

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

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

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

H.T: “I do not keep track of the lunar calendar, but I have a strong feeling that it was not the case. The periodicity set in only after the second visitation.”
L: “Pray, continue with your story.”
H.T: “That day the girl led me to the temple of Hanūmat. I had never been religious and only rarely visited that temple, but I do know my husband was quite a devotee. In fact, when I went to the temple with the girl that night I had a vague recollection of seeing her there in life on the rare occasions I did go there and giving her the prasāda for I used to always feel uncomfortable about eating it. In any case, her phantomhood was confirmed as she vanished at a door just outside the circumambulatory path around the main deity. Around that time, a strange change overcame me. I felt drawn to religion and started attending the expositions of your mothers in the old language at the Śiva temple. The visitations of the phantom-girl continued, and she kept calling me. I feared I might be going nuts and wanted to consult a psychiatrist. I wondered if I should get in touch with your mother, Lootika, to get a good referral but felt rather embarrassed to do so. By some chance, one day I saw my old student Kallolini, your younger sister’s classmate, who had just finished medical school, at the temple. I told her this story. She was accompanied by her elder sister who mentioned knowing you well, Lootika. Both of them asked me many medically relevant questions and I got a referral via her sister. The visit to the shrink was of no consequence as I was certified as merely having some anxiety and prescribed some drugs. They made no difference whatsoever and I discontinued them out of safety concerns. Your friends gave me a symptom-sheet and asked me to talk to them if I had any of the reported symptoms. I did not have any other than the persistent melancholy from the odd turn my life had taken and a sense of dread whenever the phantom girl came knocking. I mentioned that to them again. They at once looked at each other and said that I should try to talk to one of you four sisters or you Somakhya. I asked if I could mention it to your mothers. They specifically said not to tell that part of my tale to your mothers but simply ask to meet with one of you when in town.”
S: “Is there anything more you would like to add to your story?”
H.T: “Yes. I did follow the phantom girl on two other occasions. The second time she led me again towards the Hanūmat shrine but instead of going inside vanished at the cart of the śṛṇgāṭaka vendor who stands at the street just where the bridge over the river ends. The third occasion was the most frightening. She led me by the railway station and my legs were wearying from the long walk. She then turned into a narrow alley with Pakistani flags fluttering on either side. A cricket match seemed to be going on and the dwellers were all chanting in unison for the victory of Pakistan. She then stopped at the break in the wall that would take you right to the train tracks and beckoned me to enter. I just could not get myself to proceed any further and turned around to scoot out of that alley known to be a dangerous place at night and call a cab. The phantom girl cried out there!’ pointing beyond the broken wall and vanished.”

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

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

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

V: “A young female civil servant in the police service was rather friendly with the suave strongman choṭā Dawood who reigned like a little shaikh of the large slumland of Kabirwadi near the railway station. But during a marūnmatta festival, she arrested some chaps for the slaughter of a camel on the street not knowing that they were Dawood’s buddies. Dawood issued her a friendly warning for her impudence; unfortunately, having taken her power to her head she arrogantly retorted that she would have the strongman spend the rest of his life in jail at a flick of her finger. Soon thereafter, she disappeared much like our teacher’s lamented husband. The force which had no leads despite vigorously pursuing the death of their commissioned officer suddenly got an unexpected tip-off and found her corpse in a deep old railway maintenance manhole beside the tracks. They also recovered the remains of two other people long deceased. DNA tests revealed them to be a well-known city lawyer who had mysteriously vanished, who was none other than our teacher’s husband, and a girl. It turned out that the girl was kidnapped while procuring śṛṇgāṭaka-s when returning from school for the heinous pleasure of choṭā Dawood. She died at the hands of him and his henchmen and was thrown down there. She was the daughter of the poor arcaka of the temple of Hanūmat. Just to remind you, Dawood’s man had also tried to abduct our friend Sharvamanyu’s now-wife Abhirosha in the days of our youth — but we had fought him off — I’m thankful I had my grandfather’s billhook with me that day. Our former teacher’s husband had apparently tried to help the arcaka by trying to get the case to the police. That evening he had arranged a meeting with the arcaka to help him make a complaint with the material he had uncovered. However, that was not to be for Dawood and his men intercepted our teacher’s husband and dumped him in the same place. The reports mentioned that he feared their ghosts and had the well-known fuckeer Ata mulq Alla al-Din cirāg-vālā bābājī to help him ward them off. For some reason the bābā himself claimed to be possessed by the girl’s ghost and spilt the beans — his tip-off led the māmu-s to his client’s victims.”

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

## Bhāskara’s dual square indeterminate equations

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

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

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

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

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

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

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

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

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

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

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

The second parametrization he offers is rather simple:

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

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

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

Bhāskara’s first fractional parametrization takes the form:

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

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

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

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

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

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

Figure 3. First few integer solutions.

Table 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Similarly, we get:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Figure 4. The relationship between families.

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

## Pandemic days-6: Genetic risk factors

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Counting pyramids, squares and magic squares

Figure 1. Pyramidal numbers

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

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

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

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

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

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

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

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

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

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

For bālabodhana:

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

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

3) The shell thus constructed has:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplifying we get:

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

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

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

Thus, we get,

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

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

Figure 4. Squares defined by packed circles.

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

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

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

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

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

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

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

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

Figure 5. Magic choice.

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

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

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

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

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

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

## The tale of the dolmen-dweller

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

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

At the library he sighted Vrishchika and went up to her: “Vrishchika, thanks for all those cheat-sheets you generated for pharmacology and biochemistry — it really clinched the day for me”. Vri: “Well, is good to hear that. Unlike you, I still have a couple of stupid exams to finish before I’m a free bird.” Vid: “Why would you need to study?”. Vri: “Well I could certainly pass them without studying but if you are in this business of beating others it takes more effort, just as the guys say you need to keep up the practice if you have to be in fighting shape.” Vid: How is your sister Lootika? is she done with her exams?” Vri: “Yes, if you wait for half an hour, you will see her swing by — today we are riding back with our father.” Vid: “What happened to her aśva?”. Vri: “You wouldn’t believe it. We were attacked two days ago by those loutish gangster boys Samikaran et al. They broke the spokes of her aśva; so, it is under repair now.” Vid: “That is awful of them. What happened?” Vri: “When we were returning home they whistled at us and displayed some lewd gestures. Lootika asked me to ignore them but I lost my cool and lunged at them with my bike and hit one. 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.

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.

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.”

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.

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.”

## 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.

Figure 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.

## 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.

A little later 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 of 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 Sufi 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.”

## 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).

Figure 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 $f_1$:

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

Figure 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.

Figure 3.

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

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

Figure 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.

## 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:

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:

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:

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

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 ||

## 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).

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):

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.

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:

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:

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 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.

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

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.

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.

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:

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

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

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

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

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.

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.

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 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.

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

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.

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.

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:

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

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

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

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

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.

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.

## Generalizations of the prime sieve and Pi

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)?

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 $\textrm{jab}(n)$.

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 $\sqrt{\tfrac{2n}{\pi}}$

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

## 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.

## 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.

$\dfrac{dx}{dt}=ay$

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

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.

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.

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.

## 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 ।
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 ।
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.

## 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.
Figure 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.

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