Big fruits and dead giants

The fruits of the elephants
nāgo bilvam ivākramya pothayiṣyāmy ahaṃ śiraḥ |
alabhyām icchatas tasya kīcakasya durātmanaḥ ||
Like an elephant seizing a bilva [fruit] I will crush the head of that evil Kīcaka who desires that which ought not to be attained.

This striking simile occurs in our national epic the Mahābhārata in the words of the 2nd Pāṇḍava Bhīmasena to his wife. Herein, he promises to strike her tormentor Kīcaka even as an elephant strikes a bilva fruit. The bilva (Aegle marmelos) and similar-looking fruits greatly fascinated us in our childhood (Figure 1). Among the other fruits that appear to have convergently evolved a comparable structure are Limonia acidissima which produces a fruit (commonly called the wood-apple) that resembles the bilva in terms of its woody shell. Then in our youth we ran into the elephant apple, Dillenia indica, in a great botanical garden, which also produces a monstrous, hard fruit with parallels to the above two. Such fruits with hard rinds which might need to be broken by a hammer to access the pulp puzzled us — which animals other humans (who are only recent feature of the Indian landmass) could deal with such a formidable fruit. We then read as a youth of an account of humans and elephants in conflict for this fruit in Northeastern India, which has since then only intensified apparently resulting the killing of some elephants.

elephant_fruits_smallFigure 1

It was shortly thereafter that we came across this remarkable simile from the Mahābhārata. The kavi as a naturalist was likely recording a real ecological observation via this simile. Suddenly everything became clear in our minds: These cannonball-like fruits did not evolve for human consumption at all in the first place but for a symbioses with large herbivores, the foremost among which were the proboscideans. Proboscideans enjoyed a widespread distribution in recent Cenozoic past across India and probably played a major role in the dispersal of the seeds of these plants. Their hard shells were an adaptation for specifically being consumed by these animals thereby allowing their seeds to be deposited far from the parent plant in their fertilizing dung. So strong was the selection for such a mode of dispersal that we have at least three independent cases of convergent evolution of comparable fruit morphology among these Indian trees. One may also add that the mango might have also benefited in the past from a similar proboscidean dispersal.

The above woody fruits also made us curious about the mysterious cannonball tree (Couroupita guianensis), which we encountered with much excitement in our youth in a temple of Rudra in the Karṇāṭa country. Indeed, its flowers are offered to Rudra, with its reproductive whorls resembling a liṅga sheltered by a hooded serpent. The woody fruit itself is quite reminiscent of the morphology of bilva or the wood-apple. So it appeared to us that it was yet another case of convergent evolution of such a fruit type. The only issue is that we soon learned that it is not native to India but to Central and South America. This got us wondering as to how and when it reached India. Given that is called the nāgaliṅga or the śivaliṅga tree across both Indo-Aryan and Dravidian languages, and its wide usage in folk Indian medicine, it appears that it might have even reached India in pre-Columbian times. Thus, it might join a select band of mysterious plants like the custard apple, the sunflower, datura and the maize that made their way to India from the Americas in pre-Columbian times. How exactly these made their way from the pre-Columbian Americas to India remains a matter of debate to date.

India still has the elephant and has a deep and reasonable fossil record of megafauna, some of which were truly gigantic even relative to the modern Indian elephant: the extinct elephant Palaeoloxodon namadicus was probably one of largest land mammals ever, which might have grown up to 5.2 meters in height, and mysteriously became extinct sometime after 24000 years before present when Homo sapiens was already in the subcontinent. Therefore, the presence of several such fruits primarily accessible to megafauna, like the elephants (hereinafter colloquially used for proboscideans), is no surprise given their long history in the region. However, we see no elephants in the Americas; hence, the presence of fruits like those of the cannonball tree puzzled us. This puzzle deepened when we first reached the shores of Krauñcadvīpa and observed other such fruits such as the osage orange, which would have been most suitable for giants like elephants. Moreover, even other fruits like the avocado, which are today consumed by Homo, seemed more apt for a megafaunal consumer. But this was a stark contradiction for, unlike Africa or India, there are no megafaunae in the Americas. This raised the question to as to what were these fruits doing in the Americas?

The American elephants
The answer to this is tied to remarkable history of the mammalian megafauna of the Americas. We shall talk briefly about two distinct lineages of those here. The elephants as afrotherians had their provenance in the African landmass with some of the earliest members of the clade, like Moeritherium being found in North Africa. There they eventually radiated into at least 5 great lineages: the Deinotheriids, Mammutids (mastodonts), Gomphotheriids, Stegodontids and Elephantids. Between 18-22 million years ago multiple members of each lineage left Africa and invaded Eurasia to colonize it in a big way. By around 16.5 Mya the Mammutid Zygolophodon invaded the North American continent via the Bering strait. It was followed shortly thereafter by Gomphotherium which was in North America by 15 Mya. They were followed by likely multiple invasions of Stegodonts, with Platybelodon reaching the New World around 13 Mya and Amebelodon by around 10 Mya. The Elephantids were the last lineage to invade the New World with the mammoth — an initial species Mammuthus columbi emerged in NA after 2.6 Mya and then within the last million years Mammuthus primigenius followed. In the late Pleistocene, between 120000-13000 ya, there were representatives of three major elephant lineages in NA including, multiple mammoths of the Elephantid lineage, Mammut of the Mammutid lineage and several gomphotheriids.

elephants_pngFigure 2: The phylogeny of elephants according to Jeheskel Shoshani

Around 3 Mya the isthmus of Panama was complete connecting the North American continent to the isolated South. This sparked the great American biotic interchange, the main phase of which was between 2.7-1.8 Mya. During this period in the region corresponding to today’s Florida to Honduras there were members of all three lineages, namely the Mammutid Mammut, the Gomphotheriids Cuvieronius and Stegomastodon, and the Elephantid Mammuthus. A little after 2 million years a Gomphotheriid derived from the North American representative Rhynchotherium entered South America. It subsequently spread throughout South America and spawned 3 species, namely Cuvieronius hyodon, Notiomastodon platensis, and Haplomastodon waringi. Thus, until not very long ago there was an extensive elephant fauna across both the Americas, which provided the opportunity for an extensive co-evolution of certain fruit morphologies with them.

The evidence from the Arabian footprints suggests that the common ancestor of the Gomphotheriid and Elephantid lineage already had behaviors similar to those seen in extant elephants — matriarchal herds and roving males with a more solitary behavior. It is conceivable that such behaviors had already emerged even earlier prior to the divergence of the Mammutids or the Deinotheriids. The elephants have large brains and particularly well-developed temporal lobes. These features are consistent with their complex behavior and learning capacity which go hand-in-hand with their sociality. This, combined with their large size, made them relatively safe from conventional predation pressures and likely fueled their great march out of Africa that in the end spanned 5 continents and the entire range of ecosystems from wet-land forests, to dry-lands, to the frozen Arctic. But this march that spanned over 20 million years suddenly came to a halt some time after about 1 Mya. A wave of extinction of the elephants was seen first in the Old World and sometime between 20000-7500 years before present in the New World. What was the cause of this? While some ecologists like to blame climate change, we hold that there is little doubt that things point to the hand of Homo.

elephants_vanderGeerFigure 3. The global spread of some extinct elephant skeletons from Van der Geer et al 2016; DOI: 10.1111/jbi.127431. 1.European Palaeoloxodon antiquus; 2. North American Mammuthus columbi; 3. Asian Stegodon zdanskyi; 4. Southern European Elephas mnaidriensis; 5. Mediterranean pygmy Palaeoloxodon falconeri; 6. Channel Islands Mammuthus exilis; 7. Japanese Stegodon aurorae

The roll-back of the elephants corresponds to the major expansion of Homo beyond Africa. Indeed, elephants are seen at human feeding and kill sites throughout this period. Notably, the last elephants of the lineages that became extinct are seen in refugia where humans did not initially reach: The remote Wrangel Island in the Arctic ocean had Mammuthus until around 4000 YBP and these finally vanished around the time Homo first settled on the island. Similarly, in SA the gomphotheres persisted in refugia until around 8000-7500 YBP before their complete extirpation by the paleo-South Americans. It is conceivable that there was a faint memory of recently extinct elephants in human tradition. For example, the Mahābhārata records a giant elephant-like animal in the legend of the brāhmaṇa in the pit. As a large species with a long maturation time and a slow rate of reproduction, what was once a key to their success became their Achilles heel when faced with a super-predator like Homo who wiped out the reproductive adults before they could be replaced. Thus, the empire of the giants came crashing to the floor.

One may ask but what about the elephants of Africa and India? How did they survive the assault of Homo. Some have offered the suggestion that the long presence of Homo in these regions allowed co-evolutionary responses that allowed them to cope with him. It is possible that some behavioral adaptation of such a type exists – e.g. aggressiveness towards Homo. However, as others have pointed out (e.g. Surovell et al) that the more likely reason was that only the elephants from forest environments, where sizable populations of Homo could not be supported, made it past the slaughter. Indeed, while Homo branched off from his closest cousins in a dense forest biome where they still grimly hang on today, the rise of the predecessors of Homo corresponded to his march into more open environments. Ironically, this opening of African environments into mixed wood- and grass-land habitats might have been triggered by the massive megafaunal expansion in the Pliocene, which also marked the emergence of the lineage leading to Homo. The work of O’Brien’s team using molecular data has shown that the savanna elephants of Africa, i.e. Loxodonta, experienced a greater population bottleneck than their forest counterparts, suggesting that being in more open habitats they experienced the heavy hand of Homo. Further, the extinction actually depended on the developments in the intelligence and technology of Homo. For instance, when archaic Homo first reached India he was probably not capable of completely wiping off all elephants. But probably sometime around 50 Kya the sapiens species of Homo was able develop hunting methods that allowed targeting of accessible elephant species. This was evidently the cause for the end of Palaeoloxodon namadicus by 24 Kya in India. Likewise, in the Americas the spread of the Clovis hunters and their successors probably was the key point in delivering the final blow to the elephants of the New World. This has gone on to our times as newer weaponry comes into being and newer races of H.sapiens reach places where they were previously absent. This is what marked the demise of the cheetah and the near end of the lion in Asia and the various faunae of Australia and New Zealand. Now that Homo’s gun-hand can penetrate the forest with ease, the remaining two species of elephants would also be wiped out in the relatively near future in the absence of a directed conservation effort.

The sloths and their demise
The elephants were a relatively late addition to the South American landscape, coming in only after the completion of the Panaman isthmus. South America had its own iconic megafaunae even when landmass was isolated from the north. The fate of one of those indigenous components of the fauna also paralleled that of the elephant-new-comers. These were the xenarthrans, remarkable components of the uniquely South American faunae, one of the great placental lineages, which included the sloths, the anteaters, armadillos and glyptodonts. These were enormously successful animals, which, during the great American biotic interchange, were one of few lineages of Southern mammals that successfully made it to and colonized the North. In fact, they were early movers, and reached Central and North America even before the isthmus of Panama was complete. On the other side they colonized the West Indies, with an extensive presence in the Greater Antilles and Cuba. Once in the North, they advanced further, with the Thomas Jefferson’s sloth (Megalonyx jeffersonii) reaching Alaska. Indeed, they might have even made it all the way to the Old World across Beringia had their world not come to an abrupt end as we shall see below.

Today we have only two genera of sloths, the two-fingered sloths (Choloepus) and the three-fingered sloths (Bradypus), small arboreal forest mammals living in regions not easily accessible to humans. They are a pale reminder of the once glorious radiation of the sloth clade of xenarthrans that spanned a great diversity of behaviors and sizes. We had the giant ground sloths like Megatherium that grew up to 6 meters in length. We had marine Megatheriid sloths like Thalassocnus, which progressively acquired aquatic adaptations and grew up to 3.3 meters in length. Then there were ground sloths which modified the very landscape of parts of Brazil and Argentina, where they dug up numerous burrows in hard substrates ranging in diameter from 1.5 to 4 meters. While the species which created them are not known, based on their size they could have been the work of sloths like Glossotherium, Scelidotherium, Ocnotherium or Lestodon. Further, they successfully survived the arrival of competing Northern herbivores, like the elephants, and new predators, like the cats and the bears, during the faunal interchange. Notably, some giant sloths like Eremotherium often co-occur with elephants after their arrival in South America suggesting that they had partitioned into different niches in the same habitat, thereby recreating a setting like the African bush.

sloths_smallFigure 4. Sloths by Jorge Blanco

The relationships of these sloths were enigmatic until recently, but as their demise was not so long ago it has been possible to use ancient DNA and paleo-proteomic approaches to address the problem. Recently published studies suggest an interesting evolutionary scenario for the sloths, wherein the Caribbean sloths Acratocnus ye and Parocnus serus form the basal-most lineage. The next to branch off was the clade uniting the Mylodontids and the 2-fingered sloths. The third clade includes all the Megatherioids and the 3-fingered sloths. Within it the South American Megatheriids are the basal-most lineage. The North American Megalonychids and Nothrotheriids form the immediate sister-group of the 3-fingered sloths. This suggests that either the arboreal adaptation evolved twice in the 2- and 3- fingered sloths or that there was an ancestral arboreal tendency from which the ground sloths repeatedly emerged. The latter is supported by the arboreal tendencies observed in the West Indian sloths (Figure 4). Further, this phylogeny also suggests that the sloths reached the West Indies via an early land bridge long before the connection of the North and South American continents.

The survivors of the Homo Blitzkrieg
Strikingly, the end of the sloths closely paralleled that of the elephants, which were only recent arrivals to the South. After an enormously successful run that spanned tens of millions of years, sometime around 10000 YBP, they all but completely vanished in the continental Americas. Subsequently, around 4400 YBP they vanished entirely in the West Indies. This pattern is telling as it parallels the fate of the last of the extinct Elephantids on isolated islands, where they vanished much after the mainland and typically only after humans arrived. Like elephants, sloths such as Megatherium americanum too have been found in butchery sites of Homo in the Americas from around 12600 YBP. There have been major climatic changes and concomitant shrinkage of ranges, loss of habitats and background extinctions going on all while before the origin of Homo. However, new species fill those that are lost through convergent evolution or the old lineages recover after a while. The pattern we observe here is nothing like that, especially given that many of the lineages that vanished so dramatically across continents had been successful in weathering the climatic changes and background extinctions over tens of millions of years. Thus, despite the fashionable statements among some ecologists and paleontologists pinning everything on climate, we have to conclude that Homo squarely stands accused.

All this happened with almost the suddenness of a mass extinction due to astronomical causes, but, unlike in those cases, many of the plants once adapted to the megafauna remain intact. Indeed, the ecologists Dan Janzen and Paul Martin, who first noted these anomalies in their brilliant paper “Neotropical Anachronisms: The Fruits the Gomphotheres Ate” noted several examples from the New World of what might be remnants of now lost adaptations of plants to megafauna. For example they propose that large clusters of thorns of the legume known as the honey locust tree might have been an adaptation against browsing by the now extinct elephants. Some of them, like the avocado, which was once on the menu of the great sloths, might have found a new partner in their extirpators, i.e. Homo. Others have proposed that the osage orange might have survived because some early Americans used the tree to make bows and subsequently due to the horses brought by Europeans taking up the role of their dispersers (see Barlow’s excellent popular account of the same). Hence, one would predict that new megafaunae might arise again once Homo himself goes the way of extinction.

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

Pearl necklaces for Maheśvara

Śrīpati’s pearl necklace for Maheśvara
The brāhmaṇa Śrīpati of the Kāśyapa clan was a soothsayer from Rohiṇīkhaṇḍa, which is in the modern Buldhana district of Maharashtra state. Somewhere between 1030 to 1050 CE he composed several works on mathematics, astronomy and divination, one of which is the Gaṇita-tilaka on basic arithmetic and algebra which has come down to us through incomplete manuscripts. In that he posses the following problem (The solution is provided by his Jaina commentator Siṃhatilaka Sūri in 1275 CE):

viśva-kha-sapta-bhujaṅga-navārkāḥ śaila-turaṅga-samāhata-dehāḥ |
syāt sphuṭa-tāraka-vartula-muktā-bhūṣaṇam atra maheśvara-kaṇṭhe ||
The bodies of all (13), the space (0), the seven (7), the snakes (8), the nine (9) and the suns (12) struck together (means multiplied in mathematical terminology) with the mountains (7) and the horses (7) may now be the clear, sparkling and globular pearls forming an ornament for the neck of Maheśvara.

The purpose of the problem is two-fold: (1) to make the student familiar with using the Hindu numerical code (the bhūta-saṃkhya) and (2) make the student do some elementary large number multiplication. Thus, the problem is actually a simple multiplication 12987013 \times 77 whose answer is 1000000001. Interestingly, Śrīpati offers a clue for the answer in the problem itself: he says the multiplication of the 2 numbers results in a necklace of clear, sparkling, spherical pearls for Maheśvara. This is clearly an allusion to the palindromic structure of the product, with the 0s forming the pearls, since they were written historically as circles and the two flanking 1s form the bindings of the necklace. Some numerical savants are known to exhibit a synesthesia with respect numbers — perhaps such was indeed the situation with Śrīpati for a palindromic number such as this could simultaneously produce in ones mind the vivid image of a pearl necklace.

Moreover, this is not just any pearl necklace but one for Rudra’s neck. It is in this regard we believe he encoded more into that number. As we can see from the above, one of the factors of 1000000001 is 11, which is the characteristic number of the Rudra-s. At the same time, the Rudra-s are also said to be 1000s upon 1000s (Yajurveda: sahasrāṇi sahasraśo ye rudrā adhi bhūmyām |). Thus, this product captures both those aspects. Further, the numbers that yield the product are described in bhūta-saṃkhya (itself eminently amenable synesthetic experience of numbers) as including viśvā (= all); kha (= space); 7, which symbolizes the heavenly realms or vyāhṛti-s, the 8 directional earth-bearing snakes, the 9, which symbolizes the planets, the 12 Āditya-s on one hand and the 7 continental mountain ranges and 7 solar horses on the other. Thus, the two numbers are described by the entities of entire universe pervaded by the 1000s of Rudra-s with their 11-fold essence and their product is seen yielding a necklace for Maheśvara. In this regard, the use of viśva for 13 is curious. In the bhūta-saṃkhya system, viśva represents the viśvedeva-s (all gods). In the gaṇapāṭha database from which Pāṇini constructed his grammar we find viśvadeva as entry 13 in the manojñādi-gaṇa (GP 177.13), thus lending viśva to encode 13.

Maheśvara’s necklace sequence and its factors
Taking the cue from Śrīpati, we can define a general integer sequence f[n] of Maheśvara’s necklaces thusly: f[n]=10^n+1, where n is an integer and f[n]^m, where m=1,2,3,4, is also palindromic. This implies that n=1, 2, 3... Thus, the first few terms of our sequence are:
11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001…

We can next ask what are the factors of each f[n]. The first few f[n] are factorized and tabulated below:

Table 1

From the above one may notice a few obvious things. The first 2 terms of the sequence are primes, 11 and 101, but all others are composites. It is also obvious that 3 can never be a factor of f[n] because the sum of the digits of f[n] will never be 3. We also observe that many of the f[n] have a tendency to have a mix of small factors with a very large one. Śrīpati’s original example f[9] is one such: f[9]=7\times 11\times 13\times 19\times 52579. We can then ask questions such as: 1) which primes will divide a given f[n]; 2) For which n will a given prime p be a factor of f[n]. 3) Knowing these, we can ask questions, such as, at what further n will we get a f[n] that will be again divisible by 52579, i.e. they will be further Maheśvara’s necklaces of the type specified by Śrīpati.

We notice right away that 11 divides every other term, i.e. whenever n is odd. We also notice that 101 is a factor of f[2] and every 4th term thereafter. Thus, we can formally write that 11 divides every f[n] when n=2k+1, where k=0,1,2,3.... Similarly, 101 divides every f[n] when n=4k+2. Thus, it also becomes obvious that 11 and 101 will never be co-factors of the same f[n]. With closer observation we can see that every prime p that divides a subset of f[n] does so at some n=m\cdot k+\tfrac{m}{2}, where m=2,4,6,8...: the reason for writing it this way will become clear below. The first few p that divide f[n] for some n are tabulated in the order of how often they do so along with the form n takes when f[n] is divisible by that p:

Table 2

At first site the order in which the primes which divide f[n] appear beyond 11 and 101 is puzzling — they wildly differ in magnitude and form. However, a closer examination reveals a striking pattern behind this: a prime p appears in the above list as per the multiplicative order of 10 modulo p: consider 10^j \mod p where j=1, 2, 3...; when for the first time 10^j \mod p = 1, that j is m the multiplicative order of 10 \mod p. Carl Gauss had famously shown in his Disquisitiones Arithmeticae that m is the length of the repeat pattern of the decimal expansion of \tfrac{1}{p}. Thus, it is also clear that p divides 10^j-1 for the first time when j=m. For example, for 7 we get m=6 because 7 divides 10^j-1 for the first time when j=6 to give \tfrac{999999}{7}=142857. Likewise, the repeat pattern in the decimal expansion of \tfrac{1}{7}=0.\overline{142857} which is of length 6. From the above we can easily see why a p will divide f[n] first time when n=\tfrac{m}{2}. Thus, the sequence of p that divide f[n] for the first time will be arranged as per the multiplicative order of 10 \mod p:

10^m-1= 10^{(m/2)^2}-1=(10^{m/2}-1)(10^{m/2}+1)

Now p divides 10^j-1 for the first time when j=m. Hence, it will not divide 10^{m/2}-1. However, because p divides 10^m-1, it therefore divides 10^{m/2}+1

Now what if p divides some factor of 10^{m/2}+1 which takes the form 10^j+1; j=1, 2, 3...? We can see from polynomial factorization that a polynomial of the form x^j+1 can frequently have two factors of the form x^j+1; j=1, 2, 3..., namely x+1 or x^2+1. For example, x^3+1=(x+1)(x^2-x+1) and x^6+1=(x^2+1)(x^4-x^2+1). Now, in our case x+1 \equiv 11 and x^2+1 \equiv 101. Those are primes and the first two terms of f[n]; hence, they will not be divided by any other p. Now, less frequently, other numbers of the form 10^j+1 are divisors of another such number for a larger j. For example, 10001 is a divisor of 1000000000001. So, let us assume for a moment that p divides some 10^l+1 which is a factor of 10^{m/2}+1, then l<\tfrac{m}{2}. If p does divides 10^l+1, then it also divides 10^{2l}-1. But 2l-1<m; hence, p cannot divide it because it will only divide a number of of the form 10^j-1 when j=m, i.e. the multiplicative order of 10 \mod p. Thus, p cannot divide any other 10^l+1 where l<\tfrac{m}{2}.

The Maheśvara’s necklaces f[n] are the sequence 10^j+1; therefore they would be divided by a given p for the first time when n=\tfrac{m}{2}, which is half the multiplicative order of 10 \mod p \;_{...\blacksquare}

By the procedure we followed above we can see that, after n=\tfrac{m}{2}, p would divide every f[n] where n=m\cdot k+\tfrac{m}{2}; k=1, 2, 3... A corollary to this is only p with even m can be factors of f[n] for only then \tfrac{m}{2} would be an integer. Hence, those primes with odd m such as 3, m=1; 37, m=3 etc will never be factors of any f[n].

Armed with the above, we can also tell which will be the next f[n] that will have 52579 as a factor as the original example of Śrīpati. For 52579, m=18 \; \therefore p | f[n] \iff n=18k+9; \; k=0,1,2... Hence, next term would be:
f[27]=1000000000000000000000000001

We can also see that some p will always come together as factors of f[n] because they have the same m. Thus, 7 and 13 with m=6 or 19 and 52579 with m=18 will always co-occur. Further, if a certain n satisfies the relationship n=m \cdot k +\tfrac{m}{2} for a certain p and p also divides that n then p will occur again as a factor of f[n]. For example, consider n=6 \times 3+3=21. Now, m=6 here; hence f[21] will be divisible by both 7 and 13 as they have m=6. However, 7 divides 21. Hence, 7 will occur again as the factor of f[21]. Thus, we have: f[21]= \underline{7} \times \underline{7} \times 11 \times 13 \times 127 \times 2689 \times 909091 \times 459691. Likewise, n=6 \times 6 +3=39 will correspond to a f[n] divisible by both 7 and 13. However, as 39 is divisible by 13, we will have 13 occur again as a factor of f[39]. Thus, f[39]=7 \times 11 \times \underline{13} \times \underline{13} \times 157 \times 859 \times 6397 \times 216451 \times 1058313049 \times 388847808493. All other f[n] would be square-free.

The families of Maheśvara’s necklaces

Figure 1

We can represent any given f[n] as a clique of its factors. For example, Figure 1 shows the dodecagonal clique formed by the factors of f[45], which the most composite f[n] for n=1..50. We then merge all cliques sharing common nodes for f[n], n=1..50. The edges and the nodes are then scaled as per their frequency of occurrence across all 50 cliques. The result is a factor graph for f[n] which is shown in Figure 2.

Figure 2 (click on figure to magnify)

We can see from Figure 2 that there are totally 6 families of f[n] in this range. These families can be described according to their founder member which is then the divisor of the remaining f[n] of that family. The founder member of each family can be described as the f[n] of the form 10^{2^l}+1, l=0,1,2..., where 2^l corresponds to a particular \tfrac{m}{2}:

● When l=0, \tfrac{m}{2}=1, we get the 11 family. 11, as we saw above, divides every f[n] corresponding to n=2k + 1. Thus, every f[n] corresponding to an odd \tfrac{m}{2} is drawn into this family, there making it the largest of them.
● When l=1, \tfrac{m}{2}=2, we get the 101 family. 101 draws all f[n] corresponding to n=4k+2. Thus it becomes the largest of the even \tfrac{m}{2} families.
● When l=2, \tfrac{m}{2}=4, we get the 1001 family. 1001 being composite is centered on its two factors 73 and 137 and corresponds to the terms where n=8k+4.
● When l=3, \tfrac{m}{2}=8, we get the 100000001 family. This number being composite is centered on its factors 17 and 5882353 and corresponds to the terms where n=16k+8.
● When l=4, \tfrac{m}{2}=16, we get the 10000000000000001 family centered on its factors 353, 449, 641, 1409 and 69857. This encompasses the terms corresponding to n=32k+16.
● When l=5, \tfrac{m}{2}=32, we get the 100000000000000000000000000000001 family centered on its factors 976193, 19841, 6187457 and 834427406578561. This includes the terms corresponding to n=64k+32.

Thus, we find that the even \tfrac{m}{2} terms are split up among the various families that appear as per the powers of 2.

The largest factor of Maheśvara’s necklace
Given the above information, we can cut down the time in which we factorize Maheśvara’s necklaces and gather the set of factors for the first 300 terms of f[n]. We can then ask which is the largest prime p_m[n] which divides the corresponding f[n]. Figure 3 shows the plot of \log_{10}p_m[n] against \log_{10}(f[n]).

Figure 3

We see that the general increase of \log_{10}(p_m) appears to be linear with \log_{10}(f[n]). It is bounded between lines y=a_ux, y=a_lx, where a_u=1 and a_l \approx 0.1831. The upper bounding slope a_u is easy to understand: as we observed above, some f[n] tend to have factors widely differing in magnitude; thus the large one is closer in magnitude to f[n]. Trivially, first two terms are primes. There after we get f[n] that are minimally composite. These tend to be of a particular form, e.g.:
f[19]=11 \times 909090909090909091
f[31]= 11 \times 909090909090909090909090909091
f[53]= 11 \times 9090909090909090909090909090909090909090909090909091
Thus, these have factors that approach the upper bounding line.

The median value of ratio of p_m[n] to f[n] is approximately 0.5006. This indicates an even distribution with half the number of p_m[n] being greater than the \sqrt{f[n]} and the other half being lesser than \sqrt{f[n]}.

We understand the lower bound is less clearly. Is there a way to derive it from theory alone? One can see that for n corresponding to multiples of 15 there is an increased propensity to be close to the lower bound. This is in part expected from the factorization of polynomial of the form x^n+1 where n is a multiple of 15. For example, we can see that:

x^{13}+1=(x + 1) (x^{12} - x^{11} + x^{10} - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) \\[5pt] x^{14}+1=(x^2 + 1) (x^{12} - x^{10} + x^8 - x^6 + x^4 - x^2 + 1) \\ [5pt] x^{15}+1=(x + 1) (x^2 - x + 1) (x^4 - x^3 + x^2 - x + 1) (x^8 + x^7 - x^5 - x^4 - x^3 + x + 1) \\ [5pt] x^{16}+1=x^{16} + 1 \\ [5pt] x^{17}+1=(x + 1) (x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

As one can see from the above example the polynomial where n=15 tends to have more factors than other polynomial adjacent to it. Thus, these tend to be among highly composite f[n]; hence, they are more likely to have smaller p_m[n] than their neighbors.

Tailpiece
Finally, let us look at some actual historical pearl ornaments of Maheśvara. The below were likely ātmaliṅga-s of a medieval South Indian ruler or a royal śaiva officiant.

Figure 4

In this example we see the first term of Maheśvara necklace sequence. 11 is of course the characteristic number of the Rudra-s. If we wish to add the bottom two pearls which belong to a different register we get 13 which is the next most frequent factor of f[n].

Figure 5

In this example we get two sets of pearl ornamentation one with 17 and another with 7. Both of these are factors of f[n], with 7 being the next most frequent and 17 the founder of a distinct family. Of course there are other numbers with other symbolisms in these ornaments.

For background also see:
Fermat’s little theorem and the periods of the reciprocals of primes
Visualizing the Hindu divisibility test

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

Nārāyaṇa’s sequence, Mādhava’s series and pi

The coin-toss problem and Nārāyaṇa’s sequence
If you toss a fair coin n times how many of the possible result-sequences of tosses will not have a successive run of 3 or more Heads? The same can be phrased as given n tosses of a fair coin, what is the probability of not getting 3 or more successive Heads in the result-sequence. For a single toss (n=1) we have two result-sequences H,T; so we have 2 result-sequences with no run of 3 or more continuous Heads. Let n be the number of tosses, f[n] the number of result-sequences satisfying the condition and p the probability of its occurrence. This can be tabulated for the first few coin tosses as below:

\begin{tabular}{|l|p{0.5\linewidth}|l|l|} \hline n & All result-sequences & f[n] & p\\ \hline 1 & H,T & 2 & 1\\ 2 & HH, HT, TT, TH & 4 & 1\\ 3 & HHH, HHT, HTH, HTT, TTT, TTH, THH, THT & 7 & 0.875\\ 4 & HHHH, HHHT, HHTH, HTHH, HHTT, HTHT, HTTH, HTTT, TTTT, TTTH, TTHT, THTT, TTHH, THHT, THTH, THHH & 13 & 0.8125\\ \hline \end{tabular}

Thus we get the sequence as f[n]=2,4,7,13... from which we can compute the probability as p=\tfrac{f[n]}{2^n}. The question then arises at to whether there is a general formula for f[n]. The answer to this comes from a class of sequences described by the great Hindu scientist Nārāyaṇa paṇḍita in 1356 CE. In his Gaṇita-kaumudī he provides the following (first brought to the wider public attention in modern times by Paramand Singh in his famous article on such sequences):

ekāṅkau vinyasya prathamaṃ
tat saṃyutiṃ puro vilikhet |
utkramato ‘ntima-tulya-
sthānāṅka-yutim puro vilikhet ||
First placing 1 twice f[1]=1; f[2]=1, write their sum ahead f[3]=f[1]+f[2]=2. Write ahead of that, write the sum of numbers in reverse order [and in] positions equal to the “terminal” (q).

utkramato ‘ntima-tulya-
sthāna-yutiṃ tat purastāc ca |
antima-tulya sthānābhave
tat saṃyutiṃ purastāc ca ||
evaṃ saika-samāsa-sthānā-sāmāsikīyaṃ syāt |
Write ahead of that, write the sum of numbers in reverse order [and in] positions equal to the “terminal” (continue process). (This means: If 3 \le n \le q then f[n]=f[n-1]+f[n-2]...f[2]+f[1]) In the absence of [numbers in] positions equal to the terminal write in the front the sum of those [in available places] (This means if q<n then f[n]=f[n-1]+f[n-2]...f[n-q] ). Thus, the numbers till the position one more than [the prior] may be known as the additive sequence (f[1], f[2]...f[n-q], f[n-q+1]...f[n]).

One will note that if one takes q=2 then we get the famous mātrā-meru sequence f[n]=f[n-1]+f[n-2] (known in the Occident as the Fibonacci sequence after Leonardo of Pisa). Nārāyaṇa then goes on to provide a numerical example for this class of additive sequences:
udāharaṇam –
samāse yatra sapta syur
antimas trimitaḥ sakhe |
kīdṛśī tatra kathaya
paṅktiḥ sāmāsikī drutam ||
Now an example: Friend, if we have 7 in the sequence and 3 as the “terminal” (q=3) then quickly say what will be the additive sequence under consideration.

The sequence in consideration in modern form starting from 0 will be f[n]=f[n-1]+f[n+2]+f[n-3], with the first three terms being 0,1,1 (in Nārāyaṇa’s reckoning they will be 1,1,2): 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705… Thus, Nārāyaṇa’s numerical example from f[3] onwards gives us the solution to the coin toss problem. In modern times this sequence has been given the name “Tribonacci”. With this, one can see that the probability of the getting a result-sequence without 3 or more continuous heads decays with 2^n asymptotically towards 0.

The convergent of Nārāyaṇa’s sequence and problem of the triple proportional partition of a segment
Consider the geometric problem: Partition a line segment \overline{AD} of length t_4 in 3 parts t_1, t_2, t_3 such that \tfrac{t_2}{t_1} = \tfrac{t_3}{t_2}=\tfrac{t_4}{t_3}=\tfrac{t_1+t_2+t_3}{t_3} (Figure 1).

Figure 1

We can see that the above geometric process is equivalent to the formulation of the above tryantima-sāmāsikā-paṅkti of Nārāyaṇa (Figure 1). This indicates that, at its limit, the convergent of the above sequence of Nārāyaṇa, \tau, will yield us the ratio in which the partition of the segment should be performed. Thus, we have:

\tau= \displaystyle \lim_{n \to \infty}\dfrac{f[n]}{f[n-1]} \approx 1.8392867552141611325518525646532...

This number, unlike the Golden ratio \phi, the convergent of the mātrā-meru-paṅkti, cannot be constructed using a straight edge and a compass. However, as we shall see below, it can be easily constructed using two simply specified conics which can in turn be constructed using the geometric mean theorem or other standard methods (Figure 2).

Figure 2

1) Draw the parabola specified by y=x^2 with its vertex C as the origin.
2) Draw a unit rectangular hyperbola with center C_h at (1,1). Therefore, its equation will be xy-x-y=1.
3) The two conics will intersect at P. Drop a perpendicular from P to the x-axis to cut it at B. t_1=\overline{CB}=\tau.
4) Extend \overline{CB} by a unit to get point A. \overline{BA}=1.
5) Draw a circle with C as center and \overline{CB} as radius to cut the x-axis at X.
6) Mark point Q along \overline{CB} at a unit distance from C. Join Q to X.
7) Draw a perpendicular to \overline{QX} at X. It will cut the x-Axis at point D. Join C to D. By the geometric mean theorem \overline{CD}=\tau^2.

This gives us the required partition of \overline{AD} into 3 segments each proportional to the other as \tau, and \overline{AD} proportional to the largest partition \overline{CD} again as \tau. Thus, we have:

\tau=\dfrac{\tau^2+\tau+1}{\tau^2}

Hence, \tau is the root of the cubic equation \tau^3-\tau^2-\tau-1=0. Thus, we see that the above construction is achieved by solving this cubic via the intersection of the said parabola and hyperbola. This cubic equation has only one real root which is \tau. We can take the help of computer algebra to obtain the exact form of this root as:

\tau=\dfrac{1}{3}\left(1+\left(19-3\sqrt{33}\right)^{1/3}+\left(19+3\sqrt{33}\right)^{1/3}\right)

The other two roots are complex conjugates:

\tau'=\dfrac{1}{3} - \dfrac{(19 - 3 \sqrt{33})^{1/3}- (19 + 3 \sqrt{33})^{1/3}}{6} + i\dfrac{ \sqrt{3}(19 - 3 \sqrt{33})^{1/3}-\sqrt{3}(19 + 3 \sqrt{33})^{1/3}}{6}\\[7pt] \overline{\tau'}=\dfrac{1}{3} - \dfrac{(19 - 3 \sqrt{33})^{1/3}- (19 + 3 \sqrt{33})^{1/3}}{6} - i\dfrac{ \sqrt{3}(19 - 3 \sqrt{33})^{1/3}-\sqrt{3}(19 + 3 \sqrt{33})^{1/3}}{6}

Comparable to the situation with \phi and its conjugate, these roots have a relationship of the form:

\tau=\dfrac{1}{\tau' \overline{\tau'}}

There are also some other curious identities satisfied by \tau like:
\dfrac{(1+\tau)^2}{1+\tau^2}=\tau

The convergent of Nārāyaṇa’s sequence and Mādhava’s \arctan(x) and \pi series
The triple partitioning of a segment leads us to a geometric construction that yields the relationship between \pi and \tau (Figure 3):

\pi=4\left(\arctan\left(\dfrac{1}{\tau}\right)+\arctan\left(\dfrac{1}{\tau^2}\right)\right)

Figure 3 provides a proof for this of a type the old Hindus termed the upapatti or what in today’s mathematics is a proof without words (to our knowledge never presented before). Nevertheless, for the geometrically less-inclined we add a few words below to clarify this.

Figure 3

In Figure 3, one can see how \angle{\alpha}=\arctan\left(\tfrac{1}{\tau^2}\right). It emerges once from the starting triply partitioned segment as \angle{\alpha}= \arctan\left (\tfrac{\tau}{\tau^3} \right). The construction creates segments t_1, t_2, t_3 in the proportion of 1:\tau:\tau^2. Thus, we get the second occurrence of \angle{\alpha}=\arctan\left (\tfrac{t_1}{t_3} \right). That in turn implies the occurrence of a vertical segment of size \tau^2. From the construction we also get \angle{\beta}=\arctan\left (\tfrac{t_3}{t_1+t_2+t_3}\right)=\arctan\left (\tfrac{1}{\tau}\right). Thus, \angle{\alpha}+\angle{\beta} add up to form the congruent base angles of an isosceles right triangle with congruent sides measuring \tau+\tau^2. This implies that:

\arctan\left(\dfrac{1}{\tau^2}\right)+\arctan\left (\dfrac{1}{\tau}\right)= \arctan(1)=\dfrac{\pi}{4}\\[7pt] \therefore \pi=4\left(\arctan\left(\dfrac{1}{\tau}\right)+\arctan\left(\dfrac{1}{\tau^2}\right)\right) \; \; \; _{...\blacksquare}

Likewise we can also see that:

\pi=\dfrac{4}{3}\left(\arctan(\tau)+\arctan\left(\tau^2\right)\right)

Approximately contemporaneously with Nārāyaṇa’s work, apparently unbeknownst to him, Mādhava, the great mathematician and astronomer from Cerapada, presented his celebrated infinite series for the \arctan(x) function:

\arctan(x)=\dfrac{x}{1}-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\dfrac{x^7}{7}...

We can use the first of the above relationships between \pi and \tau to obtain an infinite series for the former based on the latter:
\pi=4\left(\dfrac{1}{\tau}-\dfrac{1}{3\tau^3}+\dfrac{1}{5\tau^5}-\dfrac{1}{7\tau^7}+\dfrac{1}{9\tau^9}-\dfrac{1}{11\tau^{11}}+\dfrac{1}{13\tau^{13}}...+\dfrac{1}{\tau^2}-\dfrac{1}{3\tau^6}+\dfrac{1}{5\tau^{10}}-\dfrac{1}{7\tau^{14}}...\right)

Gathering terms in order of their exponents we get:
\pi=4\left(\dfrac{1}{\tau}+\dfrac{1}{\tau^2}-\dfrac{1}{3\tau^3}+\dfrac{1}{5\tau^5}-\dfrac{1}{3\tau^6}-\dfrac{1}{7\tau^7}+\dfrac{1}{9\tau^9}+\dfrac{1}{5\tau^{10}}-\dfrac{1}{11\tau^{11}}+\dfrac{1}{13\tau^{13}}-\dfrac{1}{7\tau^{14}}...\right)

One notices that all except the fourth powers are represented. One can compactly express this as:
\pi=\displaystyle 4\sum_{n=1}^{\infty} \dfrac{a[n]}{n\tau^n}
Here the cyclic sequence a[n] is defined thus:
a[n]=1, a[n+1]=2\cdot (-1)^{m-1}, a[n+2]=-1, a[n+3]=0; n=4(m-1)+1; m=1,2,3... a[n]=1, 2, -1, 0, 1, -2, -1, 0, 1, 2, -1, 0...

Using this series to calculate \pi results in reasonably fast convergence with a nearly linear increase in the correct digits after the decimal point with every 4 terms. Thus, with 200 terms we get \pi correct to 53 places after the decimal point (Figure 4). However, we should keep in mind that n actually includes a null term which removes every 4th powers; hence, the real number of terms is lower by \tfrac{n}{4}.

Figure 4

Using just the first 3 terms we get an approximation of \pi that works as well as \tfrac{22}{7} as:

\pi \approx 4 \left(\dfrac{3\tau^2+3\tau-1}{3\tau^2+3\tau+3}\right)

We can further compare this to the famous single angle \arctan infinite series for \pi provided by Mādhava using bhūta-saṃkhya (the Hindu numerical code):

vyāse vāridhi-nihate rūpa-hṛte vyāsa-sāgarābhihate |
tri-śarādi-viṣama-saṃkhyā-bhaktaṃ ṛṇaṃ svaṃ pṛthak kramāt kuryāt ||
In the diameter multiplied by the oceans (4) and divided by the form (1), subtraction and addition [of terms] should be repeatedly done of the diameter multiplied by the oceans (4) and divided respectively by 3, the arrows (5) and so on of odd numbers.

In modern notation that would be:

\pi=\displaystyle 4\sum_{n=1}^{\infty} \dfrac{1}{2n-1}

Figure 5

This series converges very slowly and in an oscillatory fashion (Figure 5) reaching just 2 correct digits after the decimal point after computing 200 terms. The oscillatory convergence features an alternation of better and worse approximations, with the latter showing a curious feature. For example, with 25 terms we encounter 3.1815 which is “correct” up to 4 places after the point (3.1415) except for the wrong 8 in the second place. With 50 terms we get 3.12159465, which is correct to 8 places (3.14159465) after the point except for the wrong 2 at the second place. More such instances can be found as we go along the expansion. For example at 500 terms we get:
3.141592653589793238
3.139592655589783238

This is correct to 18 places except for 4 wrong places. Late J. Borwein and colleagues have reported an occurrence of this phenomenon even in a calculation of 5 \times 10^6 terms of this series.

Thus, the double angle series based on \tau fares way better than the basic single angle Mādhava series for \pi. Of course, Mādhava was well aware that it converged very slowly. Hence, he and others in his school like the Nampūtiri-s, the great Nilakaṇṭha Somayājin, Jyeṣṭadeva and Citrabhānu, devised some terminal correction terms to derive alternative series to speed up convergence and obtained approximations of \pi that had good accuracy for those times. Two of their series which were mediocre in convergence speed are in modern notation:

\pi=\displaystyle 4\left(\dfrac{3}{4}+\sum_{n=1}^\infty \dfrac{-1^{n+1}}{(2n+1)^3-(2n+1)}\right)

This sequence produces \pi \approx 3.1415926 after 200 terms.

\pi=\displaystyle\sum_{n=1}^\infty \dfrac{-1^{n+1} \cdot 16}{(2n-1)^5+4(2n-1)}

This one works better than the above and produces \pi \approx 3.141592653589 after 200 terms

The third uses \tan\left(\tfrac{\pi}{3}\right) as a multiplicand:

\pi=\displaystyle\sum_{n=1}^\infty \dfrac{-1^{x-1}\cdot 2\sqrt{3}}{3^{n-1}(2n-1)}

This series fares much better and produces 97 correct digits after the decimal point with 200 terms. This is quite impressive because it outdoes the above double angle series based on \tau, \tau^2. It is quite likely that Mādhava and Citrabhānu used this series for around 20-25 terms to obtain approximations such as the below (expressed in bhūta-saṃkhya):

vibudha-netra-gajāhi-hutāśana-triguṇa-veda-bha-vāraṇa-bāhavaḥ |
nava-nikharva-mite vṛtti-vistare paridhimānam idaṃ jagadur budhāḥ ||
The gods (33), eyes (2), elephants (8), snakes (8), the fires thrice (333), the veda-s (4), the asterisms (27), the elephants (8), the hands (2) is the measure of the circumference of a circle with diameter of 9 \times 10^{11} , so had stated the mathematicians:

\pi \approx \dfrac{2827433388233}{900000000000} \approx 3.14159265359

In addition to approximations of \pi derived from studies on \arctan(x) series we also see that Mādhava’s successors, if not himself, were also using a sequence of continued fraction convergents of \pi. These were probably inspired by the ability to initially calculate good approximations using series derived from the \arctan(x) series such as the above. Of these a large one is explicitly stated by Śankara-vāriyar and Nārāyaṇa Nampūtiri in their work the Kriyākramakarī:

vṛtta-vyāse hate nāga-veda-vahny-abdhi-khendubhiḥ |
tithy-aśvi-vibudhair bhakte susūkṣmaḥ paridhir bhavet ||
Multiplying the diameter of a circle by snakes(8), veda-s(4), the fires (3), the oceans (4), the space (0), the moon (1) and dividing it by the tithi-s (15), Aśvin-s (2), gods (33) one may get the circumference to good accuracy.

\pi \approx \dfrac{104348}{33215} \approx 3.141592653

The \arctan(x) sequence remained a workhorse for calculating \pi long after the heroics of Mādhava’s school. A particularly famous double angle formula was obtained by Euler with a simple geometric proof:

\arctan\left(\dfrac{1}{2}\right)+\arctan\left(\dfrac{1}{3}\right)=\dfrac{\pi}{4}

Using this formula we get we get a rather good convergence and reach 122 correct places after the decimal point with 200 terms.

Tailpiece: From \phi to \pi via \arcsin(x)
We may conclude by noting that the while \tau relates to \pi via the \arctan(x) function, the Golden ratio \phi relates to \pi via the \arcsin(x) function. This stems from the special relationship between \phi and the sines of the angles \tfrac{3\pi}{10} and \tfrac{\pi}{10}. In the Jyotpatti appendix of his Siddhānta-śiromaṇi’s Bhāskara-II specifically presents the values of the sines of these angles as common knowledge of the pūrvācārya-s. We reproduce below his account of the angles, the closed forms of whose sines were know to them (Note that old Hindus used Rsine instead of modern \sin(x); hence the technical term “trijyā” for R. Originally, Āryabhaṭa had set R=\tfrac{60\times 180}{\pi} \approx 3438', i.e. corresponding approximately to a radian measure in minutes. Below we take it to be 1 to correspond to our modern \sin(x)):

trijyārdhaṃ rāśijyā tat koṭijyā ca ṣaṣṭi-bhāgānām |
trijyā-vargārdha-padaṃ śaravedāṃśa-jyakā bhavati ||
Half the R is the zodiacal sine (i.e. \tfrac{360^o}{12}=30^o). Its cosine (i.e. of \cos(30^o)) will be the sine of 60^o. The square root of half the square of the R becomes the sine of arrows (5) and veda-s (4) degrees ( i.e. \sin(45^o)=\tfrac{1}{\sqrt{2}})

trijyā-kṛti+iṣughātāt trijyā kṛti-varga-paṅcaghātasya |
mūlonād aṣṭa-hṛtān mūlaṃ ṣaṭ-triṃśad-aṃśajyā ||
From arrow (5) times the square of the R subtract the square root of 5 times the R to the power of 4. Divide what remains from above by 8; the square root of that gives \sin(36^o) (i.e. \sin(36^o)=\sqrt{\tfrac{5-\sqrt{5}}{8}})

Bhāskara then goes on to give an approximation for it as fraction:

gaja-haya-gajeṣu nighnī tribhajīvā vā ‘yutena saṃbhaktā |
ṣaṭ-triṃśad-aṃśa-jīvā tat koṭijyā kṛteṣuṇām ||
The R multiplied by elephants (8), horses (7), elephants (8), arrows (5) and divided by 10^4 gives the sine of 36^o. Its cosine is the sine of 4 and arrows (5) (i.e. \sin(54^o)).

With this approximation we get \sin(54^o) \approx 0.80901 correct to 5 places after the decimal point (one would not it is \tfrac{\phi}{2}) and implies that Bhāskara was using an approximation of \pi correct to 4 places after the decimal point.

trijyā-kṛti+iṣu-ghātān mūlaṃ trijyonitaṃ caturbhaktaṃ |
aṣṭadaśa-bhāgānāṃ jīvā spaṣṭā bhavaty evam ||
From the square root of the product of the square of R and the arrows (5) subtract R and divide what is left by 4. This indeed becomes the exact sine of 18^o (i.e. \sin(18^o)=\tfrac{\sqrt{5}-1}{4}).

These basic sines emerge from the first 3 constructible regular polygons: the equilateral triangle yielding \sin(30^o), \sin(60^o); the square yielding \sin(45^o) and finally the pentagon yields \sin(18^o) and its multiples (Figure 6).

Figure 6.

Thus, from the geometry of the regular pentagon (the proof is obvious in Figure 6) it is seen that these values are rather easily obtained and can be expressed in terms of the Golden ratio \phi, which emerges in the diagonal to side ratio (Figure 6). Thus, we have:

\sin\left(\dfrac{\pi}{10}\right)=\dfrac{1}{2\phi}

\sin\left(\dfrac{3\pi}{10}\right)=\dfrac{\phi}{2}

Thus, we can use the first angle in the infinite series for \arcsin(x) to obtain the series for \pi in terms of \phi as:

\pi= 10 \displaystyle \sum_{n=0}^\infty \dfrac{(2n)!}{(2n+1) \cdot 2^{4n+1}(n!)^2\phi^{2n+1}}

This series fares excellently in computing \pi — with the same 200 terms as used in the above experiments we get the value correct 208 places after the decimal point.

Now, instead of \phi if we resort to the angle \tfrac{\pi}{6} from the geometry of the equilateral triangle, we get the below infinite series:

\pi=6 \displaystyle \sum_{n=0}^\infty \dfrac{(2n)!}{(2n+1)\cdot 2^{4n+1}(n!)^2}

This is obviously worse than the above series with \phi and yields 124 correct places after the point with 200 terms. Thus, it is only marginally better than the Eulerian double angle \arctan series in its performance.

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Discovering bronze in the characteristic ellipse of right triangles

The arithmetic mean square of a right triangle
An entire family of right triangles that includes all the different forms of right triangles defined in terms of the proportion of their legs can be obtained by setting their altitude to a constant a and letting their base x vary. The right-angle vertex of these triangles, i.e. that which contains the right angle is common to both legs and constant across the family. By the above definition the altitude vertex, i.e. that defined by the terminus of the altitude, is also constant. The base vertex, i.e. that defined by the terminus of the base can be seen as continuously varying as it moves along the line perpendicular to the altitude.

arithmetic_mean_sqr_righttriangle_Fig1Figure 1

Consider Figure 1:
For a given right triangle ( \triangle{ABC}) the half-hypotenuse square is defined as the square erected on half of its hypotenuse ( \square{BDEF}). Then we have the following:

● The feet of the perpendiculars dropped from the vertex of the half-hypotenuse (E) square diagonally opposite to the base vertex to the base and the altitude of the given right triangle, together with its right-angle vertex defines a square (\square{CMEN}).

● The area of this square is the sum of the areas of the given right triangle and the half-hypotenuse square.

● The side of this square is the arithmetic mean of the two legs (\overline{CA}=a, \overline{CB}=x) of the given right triangle.

Hence, the above-defined square (\square{CMEN}) is termed the arithmetic mean square of a give right triangle.

By examining Figure 1, registering all the self-evident congruences and performing the dissections indicated by the colored and marked areas it is possible to obtain “wordless” proofs of statements 1 and 2 _{...\blacksquare}

By bhujā-koṭi-karṇa-nyāya the area of the half-hypotenuse square is \tfrac{a^2+x^2}{4}. Thus,

A(\square{CMEN})=A(\square{BDEF}) + A(\triangle{ABC}) = \tfrac{a^2+x^2}{4} +\tfrac{ax}{2}=\left(\tfrac{a+x}{2}\right)^2

Thus, the side of \square{CMEN} is the arithmetic mean of the legs of the given right triangle A(\triangle{ABC}) _{...\blacksquare}. Thus we term this square the arithmetic mean square.

The characteristic ellipse of right triangles
We also note that the shared vertex of the half-hypotenuse square and the arithmetic mean square of a right triangle (point E) is the vertex of the semicircle erected on the hypotenuse of the right triangle (Figure 1). Thus, the segment \overline{EF} will be the diameter of the maximal circle that can be inscribed in the above semicircle (Figure 2). Now, if we draw a perpendicular to the base of the right triangle at the base vertex B then the perpendicular will cut the inscribed maximal circle at two points G, H (Figure 2). What will be the locus of G, H as the base of the triangle \overline{CB}=x grows in size? In what range of x in terms of the altitude a will this intersection will happen (i.e. for what range of x in a units will the locus exist)?

triangle_ellipse_problem_Fig2

Figure 2

To answer the above let us place point C at origin. Then, given the above-described arithmetic mean square of a right triangle we have the coordinates of its vertex and that of the semicircle erected on the hypotenuse to be E=(\tfrac{a+x}{2},\tfrac{a+x}{2}). This is also one end of a diameter of the maximal circle inscribed in the semicircle. The second end of this diameter is the midpoint of the hypotenuse F. Therefore, the coordinates of F=(\tfrac{x}{2}, \tfrac{a}{2}). Thus, we can get the midpoint of the the segment \overline{EF} to be I=(\tfrac{2x+a}{4},\tfrac{x+2a}{4}). The coordinates of point G (or of point F), which lies on the desired locus will be G=(x,y). From circle \circ I using the equivalence of its two radii we can write the following:
\overline{FI}^2=\overline{GI}^2

But \overline{FI}^2=\tfrac{1}{4}\overline{AB}^2=\tfrac{x^2+a^2}{4}

\therefore (\tfrac{a+2x}{4}-x)^2+(\tfrac{x+2a}{4}-y)^2=\tfrac{x^2+a^2}{4}\\[7pt] (\tfrac{a+2x-4x}{4})^2+(\tfrac{x+2a-4y}{4})^2=\tfrac{x^2+a^2}{4}\\[7pt] (a-2x)^2+(x+2a-4y)^2=x^2+a^2\\[7pt] x^2-2xy+4y^2-4ay+a^2=0

The above equation of the locus is a quadratic equation in two variables. Hence, it is a conic. Now given a general conic Ax^2+Bxy+Cy^2+Dx+Ey+F=0, its shape is determined by the conic discriminant:

\delta_c=-4\Delta=-4 \begin{vmatrix} A & B \big /2\\ B \big /2& C \end{vmatrix} =B^2-4AC

If \delta_c<0 then we have an ellipse. In the special case of A=C, B=0 it reduces to a circle. If \delta_c=0 we have a parabola. If \delta_c>0 we have a hyperbola. In our case \delta_c=4-4 \times 1 \times 4=-12. Thus, the desired locus is an ellipse (Figure 3).

triangle_ellipse_problem_Fig3

Figure 3

We can now rewrite the equation of the ellipse explicitly as:
\dfrac{x+2a \pm \sqrt{4ax-3x^2}}{4}

The discriminant of this equation is:
\Delta=\sqrt{4ax-3x^2}; it will be real only when: 0 \le x \le \tfrac{4a}{3}

From this we can see that the ellipse will exist only in the interval:
\left(0,\tfrac{a}{2}\right)..\left(\tfrac{4a}{3}, \tfrac{5a}{6}\right)

Thus, at those two points G, H become coincident. Thus, the altitude of the right triangles is a tangent to this ellipse at \left(0,\tfrac{a}{2}\right) (Figure 3). Further, since we have the coordinates of point I, the center of the maximal circle inscribed in the hypotenuse semicircle, we can show that as x increases I moves along the line y=\tfrac{1}{2}x+\tfrac{3a}{8}. Thus, outside the above interval we see that the right triangle’s base moves beyond the maximal inscribed circle and the perpendicular to the base at B will never touch this circle.

A shape of a triangle is characterized by the relative proportion of its 3 sides. For a right triangle we have further constraint; hence its shape is defined by the relative proportion of its legs. Thus, a right triangle with legs in proportion a_1:b_1 is the same as that with b_1:a_1 — just that they are rotated with respect to each other. Using this concept we, can see that various landmarks of the characteristic ellipse correspond to notable underlying right triangles (Figure 4). The extreme left bound of the ellipse corresponds to the degenerate case where one leg is 0 and the hypotenuse and the other leg coincide. As we determined above, the right bound of the ellipse corresponds to x=\tfrac{4a}{3}. Thus, this corresponds to the famous 3,4,5 triangle. It is remarkable how 3,4,5 triangle, which represents the most basic bhujā-koṭi-karṇa triplet, comes up as a natural feature associated with the characteristic ellipse.

special_triangles_Fig4Figure 4

Now that we have its equation we can also see that the ellipse will have a maximum at the point (a,a) and a minimum at the point \left(\tfrac{a}{3}, \tfrac{a}{3}\right). The maximum corresponds to isosceles right triangle (Figure 4). The minimum of the ellipse corresponds the 1,3,\sqrt{10} triangle (Figure 4). At both these positions the separation between the ellipse-generating points G, H is \tfrac{a}{2}.

When x=\tfrac{2a}{3}, the ellipse-generating line passes through the center of the ellipse, point C_e=\left(\tfrac{2a}{3}, \tfrac{2a}{3}\right) (Figure 4). This results in the maximal separation attained between the ellipse-generating points G, H of \tfrac{a}{\sqrt{3}}. This corresponds to the 2,3,\sqrt{13} triangle. In this triangle the ratio of the sum of the long and middle sides to the short side is the Bronze ratio, which we shall encounter again discuss in greater detail below.

When x=\tfrac{a}{2} we have the underlying 1,2,\sqrt{5} triangle (Figure 4). As is obvious, the ratio of the sum of the short and the long side of this triangle to the middle side is the Golden ratio (\phi). Thus, not surprisingly, the separation between the ellipse-generating points at this value of x is \tfrac{2\phi-1}{4}. Another notable feature of this case is that only in this configuration the lines drawn parallel to the base of the triangle at the ellipse-generating points G,H are tangents to the ellipse-generating circle.

Between \tfrac{3a}{4} \le x \le \tfrac{4a}{3} (colored section of ellipse in Figure 4), we get similar triangles on either side of x=a. However, to the left of that region every triangle has a unique shape.

The bronze ratio and the characteristic ellipse
Given the general conic equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0 one can calculate the eccentricity of the corresponding ellipse as:
e_e=\sqrt{\dfrac{2\sqrt{(A-C)^2+B^2}}{A+C+\sqrt{(A-C)^2+B^2}}}

By substituting the requisite values, we get
e_e=\sqrt{\dfrac{5\sqrt{13}-13}{6}} \approx 0.9154012

Now, we consider the Bronze ratio \beta=\tfrac{3+\sqrt{13}}{2} \approx 3.302776. We observe that \beta'=\tfrac{1}{\beta}=\beta-3 \approx 0.302776. Thus, \beta, -\beta' are conjugate roots of the quadratic equation x^2-3x-1=0. One immediately notes the parallel to the Golden ratio \phi: \phi' =\tfrac{1}{\phi}=\phi-1 and likewise \phi, -phi' are the roots of the equation x^2-x-1=0 (see appendix for more). Thus, we can write the eccentricity of the characteristic ellipse as:
\sqrt{\dfrac{5\beta'+1}{3}}

With the eccentricity we can determine the axes of the ellipse; we see that:
a_{smin}=\dfrac{a\sqrt{1+\beta}}{3} …semi-major axis

a_{smaj}=\dfrac{a}{\sqrt{3(1+\beta)}} …semi-minor axis

As an aside the area of the characteristic ellipse is therefore A=\tfrac{\pi a^2}{3\sqrt{3}}. We also get the aspect ratio of the ellipse to be:
AR=\dfrac{a_{smaj}}{a_{smin}}=\dfrac{1+\beta}{\sqrt{3}}

Interestingly, the connection of the characteristic ellipse to the bronze ratio does not end with the parameters of the ellipse itself. Its inclination, i.e. the slope of the line joining the two foci of the ellipse (i.e. the tangent of the angle made by this focal line with the base of the right triangles) is \beta'. Thus, the equation of the local line of the characteristic ellipse is:
y=\beta'x+\dfrac{2a}{1+\beta}

triangle_ellipse_problem_Fig5Figure 5

Interestingly, when x=\beta', it is only time the diameter of ellipse-generating circle perpendicular to the hypotenuse of the right triangle (i.e. \overline{EF}) becomes parallel to the focal line of the ellipse and acquires the equation:
y=\beta'(x+\dfrac{3a}{2}) (Figure 5).

Thus, by doing some algebra we can see that the difference in the intercepts is \tfrac{a}{26\beta+8}

This is one of few rare occurrences of the Bronze ratio as persistent feature in geometry and to our knowledge is described here for the first time.

Appendix: Bronze ratio
A simple geometric interpretation of the Golden ratio \phi goes thus: If we construct a rectangle such that upon removing a square whose side is equal to the smaller side of the said rectangle we are left with a remainder rectangle whose aspect ratio is the same as the original rectangle then the aspect ratio of these rectangles is \phi (Figure 6). Likewise, if we get a similarly-shaped remainder rectangle upon removing two equal squares whose sides are equal to the shorter side of the starting rectangle then their aspect ratio is the Silver ratio \sigma=1+\sqrt{2} (Figure 6). These two ratios are also associated with diagonals of a pentagon and an octagon respectively. If upon removing 3 squares we have similarly proportioned remainder rectangle as the original one then their aspect ratio is the Bronze ratio \beta. This is not associated with any polygon.

gold_Silv_bron_Fig6Figure 6

All these “metallic” ratios, their reciprocals, and negative conjugates have a common feature in that they can be described by the general double hyperbolic equation with asymptotes as x-axis and the lines y=x; y=-x (Figure 7),
y^4-x^2y^2-2y^2+1=0

Metallic_ratio_curve_Fig7Figure 7

They emerge as the solutions m_n of the biquadratic equation in y for the integer values of x:
m_n=\dfrac{\pm n \pm \sqrt{n^2+4}}{2}

Thus, n=1 \rightarrow \phi; n=2 \rightarrow \sigma; n=3 \rightarrow \beta .... This also shows that they all follow the conjugate principle with respect to their conjugates, i.e. m_n-n=\tfrac{1}{m_n}; m_n^2=1+nm_n. As the double hyperbola converges to its asymptotes the fractional parts of the ratios converge to 0. Thus, it only the first few metallic ratios are likely to be “interesting”. Indeed the Golden ratio is a common encountered in all manner of mathematical situations. The Silver ratio is seen to a lesser extant and the Bronze ratio is even rarer in it occurrence (conversely to the metals after which they are named). Beyond the Bronze ratio these ratios are mostly “uninteresting” in that they are rarely found in mathematical entities.

The above geometric operation of rectangle dissection is equivalent to the abstract string substitution operation under the rules 1\rightarrow 1^k0, 0 \rightarrow 1. Here the 1^k means k repetitions of 1. For k=1 a string initiated with 1 grows thus:
1\rightarrow 10\rightarrow 101\rightarrow 10110\rightarrow 10110101\rightarrow ...
One observes that ratio of 1s:0s in these strings converges to \phi. Likewise, the ratio of the length of the string n to that of string n-1 also converges to \phi.

For k=2 the string grows thus:
1\rightarrow 110\rightarrow 1101101\rightarrow 11011011101101110\rightarrow 11011011101101110110110111011011101101101\rightarrow ...

Here the ratio of 1s:0s and the ratio of the length of the string n length of string n-1 converge to the Silver ratio \sigma.

For k=3 the string grows thus:
1\rightarrow 1110\rightarrow 1110111011101\rightarrow\\ 1110111011101111011101110111101110111011110\rightarrow\\ 11101110111011110111011101111011101110111101110111011101111011101110111101110\\ 11101111011101110111011110111011101111011101110111101110111011101\rightarrow ...
The corresponding convergents for this case is the Bronze ratio \beta.

As we have noted before these all these strings are aperiodic but have an intricate fractal structure. The growth of the length of these strings can be expressed by the two-seeded sequence f[n]=kf[n-1]+f[n-2] whose starting elements are f[1]=1, f[2]=f[1]+k. When k=1 we get the famous Meru sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… with the \phi as its convergent. When k=2 we get: 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363… with \sigma as its convergent. When k=3 we get:1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807… with \beta as its convergent. This last sequence and related sequences with \beta as their convergent has an intimate connection to the function y=\tfrac{2x-1}{x^2+3x-1}, which is at the root of some remarkable behavior which we shall explore in the next article.

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Matters of religion: the Vāyavīyaṃ Kaubera-vratam

The bus came to halt at the underground stop. Vrishchika of pretty tresses got off the bus and walked up the stairs to reach the over-bridge. She reached the middle of the bridge and stood there looking out expectantly at the road. It was a remarkably clear day and she could see the yonder horizon where the azure heights of father Dyaus met his lover mother Pṛthivī clad in the verdant vesture of the distant woods. Vrishchika passed into a reverie at that sight and remained oblivious to the throng of people who criss-crossed the bridge in either direction. Suddenly, an ugly man sidled up beside her and before she could react to his presence he said: “Ma’am do you visit the Hindu temple at Śukramārga?” Vrishchika: “On rare occasions.” The man responded: “We need wealthy patrons like you to visit the temple more often and make donations. Please do so devī”. So saying he handed a pamphlet to her. Vrishchika wondered: “A soothsayer at the temple of the Vāyu-born vānara had once told my family that I would be the wealthiest of all my sisters. But I’m hardly in that state now. Yet, why does he even think I’m wealthy?” She snapped out of that train of thought and waved the ugly man away saying “śubham astu! It is far way from my place but I’ll try.”

She continued to look out at the road and not seeing what she wanted to see, she turned to the pamphlet. It was crowded with temple events. There was a Rudra-pūjā for the 13th pradoṣa and a Viṣṇu-pūjā for the ekādaśī. There was another event for Vināyaka, for which there was a call for people to join in a mass-recitation of the Gaṇeśopaniṣat. Yet another corner mentioned a Vaṭakini-mahotsava for the fierce ape. Then there was some announcements of secular events. A professor of Indian origin, who was popular with the people at the local chapter of the “right-wing debate club”, with a penchant for peddling false etymologies and outrageous counter-factual theories was giving a lecture series covering issues such as the autochthonous origin of Aryans, reading Vedic mantra-s in the Harappan inscriptions, and situating the Mahābhārata at earlier than 5000 years in India. Vrishchika said to herself: “I’m sure the puruṣa would find this buffoon quite amusing where it not for the pernicious consequences it has on our people.” Then there was a call for people to attend classes for language acquisition or learning the recitation of five sūkta-s from the Kṛṣṇa-yajurveda. She noticed that the latter was organized by a bābā-crazed young lady who had been her patient.

Just as her eyes wandered to the next item on the pamphlet the corner of her vision lit up with the sight of a bus halting on the road below and Indrasena saunter out of it. Upon meeting midway at the bridge the two walked ahead to towards the station to board the train for a long ride back home. Vrishchika: “Dear Ātreya, when I was waiting on the bridge, I received this temple pamphlet from a man who, if my vague recollection is correct, evidently supplements his income by playing a horoscopist and palm-reader at the temple on Śukramārga. What is interesting is that just as I caught sight of you I was reading this little announcement here.” Saying so she handed it over to Indrasena. Even as he was looking at it she articulated it: “It looks as though this banker Atigupta Sādhu is commissioning the consecration of an image of the Yakṣarāṭ in the northern niche of the temple on the ekadaśī of next month. It reminded me that you have yet to impart to me the rahasya of the ekadaśī-vrata that you had mentioned long ago in your hometown.” Indrasena: “That’s quite remarkable. There is actually more than what meets the eye here. Indeed, it was in my mind too that I convey the vidyā to you, O daughter of an Aṅgiras, for after all I’d need a dīkṣita-patnī when I impart it to the ācārya who will be doing that sthāpanā.”

Vrishchika: “That’s interesting. So you were aware of this?”
Indrasena: “Yes Alini. There is a long story here that I’ve not had the chance to tell you.”
Vrishchika: “I’m now very curious to hear it all.”
Indrasena: “Atigupta Sādhu was the student of a charismatic mantravādinī from the regions of Mahākāla. He had received a magic square of Rājarāja from her and consequently his wealth greatly expanded as befits a V3. He wanted to install an image of Dhanada at the temple at Śukramārga. He employed none other than the initial teacher of mine the vaṅgasiṃha, for whom there is no “tataḥ kim?” after “dhanaṃ meru-tulyam” and he was almost half an Uesugi Kenshin of the prācya-s. But, as I’ve told you before, he only possessed incomplete vidyā-s. Thus, his sthāpanā failed like that of Skanda by the marāṭhā chief Raghunātharāu and the image was damaged when lightning struck the temple. The V3 was distraught and went to his teacher back in Mālava. She happened have studied some śāstra-siddhānta under Lootika. She then consulted your sister, who not wanting to give out my identity publicly told her that if Vitteśa were to be pleased with the said V3 he would find the right V1 who will solve his samasyā. The vaṇij went around for a while until he circuitously got introduced to me by Somakhya. I told the V3 that I could not be a sthāpakācārya at a temple but I could impart a key vrata to the vaiṣṇava sthāpaka at the temple who would then achieve success with the endeavor upon performing the vrata. This, my pretty girl, is the whole story. The vrata is none other than the one of the ekādaśī that I’ll expound to you once we are home.”

Back home, taking in the good night air and the dark canopy of the nakṣatra-s, Indrasena expounded to his wife the rahasya-s of the Vāyavīyaṃ Kaubera-vratam: “Dear Gautamī, first you must know why it is called the Vāyavīyaṃ Kaubera-vratam and why the ekādaśī is considered the day of the yakṣa lord. In this regard the following has been narrated by the brāhmaṇa Mahātapās to king Prajāpāla in the Varāhapurāṇa:

mahātapā uvāca:
śṛṇu cānyāṃ vasupater utpattiṃ pāpanāśinīm|
yathā vāyu-śarīrastho dhanadaḥ sambabhūva ha ||
ādyaṃ śarīraṃ yat tasmin vāyur antaḥsthito ‘bhavat |
prayojanān mūrtimattvam ātiṣṭhat kṣetra-devatā ||
tatra mūrtasya vāyos tu utpattiḥ kīrtitā mayā |
tāṃ śṛṇuṣva mahābhāga kathyamānāṃ mayā +anagha ||
brahmaṇaḥ sṛṣṭi-kāmasya mukhād vāyur viniryayau |
pracaṇḍa-śarkarā-varṣī taṃ brahmā pratyaṣedhayat ||
mūrto bhavasva śāntaś ca tenokto mūrtimān-bhavat |
sarveṣām eva devānāṃ yad vittaṃ phalam eva ca |
tat sarvaṃ pāhi yenoktaṃ tasmād dhanapatir bhavān ||
tasya brahmā dadau tuṣṭas tithim ekādaśīṃ prabhuḥ |
tasyāṃ anagni-pakvāśī yo bhaven niyataṃ śuciḥ |
tasyāpi dhanado devas tuṣtaḥ sarvaṃ prayacchati ||
eṣā dhanapater mūrtiḥ sarva-kilbiṣa-nāśanī |
ya etāṃ śṛṇuyād bhaktyā puruṣaḥ paṭhate ‘pi vā |
sarva-kāmān avāpnoti svarga-lokañ ca gacchati ||

The sage Mahātapas narrated this legend to king Prajāpāla:
Now, listen to that other sin-destroying tale of the origin of the lord of wealth, regarding how the wealth-giver arose from the body of the god Vāyu. Within what was the primordial body, Vāyu came into being situated in its interior. For the purpose of assuming a discrete form the deity of the regions [Vāyu] emerged forth. Now, the origin of that form of Vāyu will be made known by me. O sinless and most fortunate you ought listen that narrated by me. Due to the god Brahmā’s creative desire Vāyu issued forth from his face. [He emerged] with a tremendous shower of grit/gravel. Brahmā restrained him by saying: “You should take on a pacific discrete form.” He took the form as stated by Brahmā. [Brahmā said]: “all this wealth of the gods, whatever they produce, you must indeed protect it”. Thus, due to what was said by Brahmā he became Dhanapati. The pleased lord Brahmā gave the ekādaśī [as his] lunar day. On that day, if one with discipline purifies himself and only eats food which is not cooked by fire, then the god Dhanada is pleased and gives him everything. This form of Dhanapati destroys all that is ill. If one verily listens to this narrative with devotion or the person recites it, then he attains all desires and goes to the heavenly realm.

Now, Vrishchika tell me what you think of this narrative?”

V: “I believe it had its roots in the ancient Indo-Iranian system of the supreme Vāyu, with a Prājāpatya overlay atop that. Indeed Kubera and Vāyu simultaneously figure as mighty deities in the backdrop of the Rāmāyaṇa and to some extent also in the Mahābhārata. Hence, their connection does not seem unexpected. Moreover, along with the other deity of the same class, Rudra, the three of them adorn the northern face of the dik-maṇḍala — NE-N-NW. The primordial body referred to in this narrative is verily the puruṣa of the śruti. There indeed it has been said ‘prāṇād vāyur ajāyata |‘ and Vāyu was seen as the breath within the puruṣa. In this myth we see an allusion the the role of Vāyu in the creative process. It is amply clear from the śruti that in the old system focused on Vāyu he was conceived as playing a central role in this process. Indeed, in the mantra to Vāyu-Vāta we hear:
‘ātmā devānām bhuvanasya garbho
yathā-vaśaṃ carati deva eṣaḥ |
ghoṣā id asya śṛṇvire na rūpaṃ
tasmai vātāya haviṣā vidhema ||’
The breath of the gods, the germ of the universe,
this god wanders as he wishes.
Only his sounds are heard, not [seen is] his form.
For him, for Vāta, we would do honor with our oblation.

We see here that the deity is presented as the germ of the universe (reflective of the creative process) and his formlessness which is alluded to in the myth you narrated is also a central element of this mantra. Tellingly, the very first mantra of the sūkta captures aspects of his storminess mentioned in this myth as he emerged from the face of Brahmā with a shower of grit.:
‘vātasya nu mahimānaṃ rathasya
rujann eti stanayann asya ghoṣaḥ |
divispṛg yāty aruṇāni kṛṇvann
uto eti pṛthivyā reṇum asyan ||’
Now for the greatness of Vāta and his chariot:
smashing he proceeds; thundering is his noise.
Touching heaven as he drives, making things red,
also blowing up dust from the earth he goes.

Thus, I would conclude that at the foundation of the Varāha-purāṇa narrative are clear motifs coming from the old Vāyu-centric system such as those expressed in the sūkta I just mentioned.”

I.s: “That is good, Vrishichika. It touches on something which will come up again as I expound the ritual itself. One may also note the this idea of the Vāyu within the primordial body might have been behind another somewhat mysterious name of the god — Mātariśvan — he who grows within the mother or womb. Thus, it is likely that what was meant in this name was Vāyu filling up the interior of the ‘primordial womb’. His emergence from the mouth of the protogonic deity is something that evidently had a deep history: While in the Puruṣa-sūkta it is Indra and Agni who emerge from the Puruṣa’s mouth, we have a bauddha text the Avalokiteśvara-guṇa-karaṇḍavyūha where the nāstika-s have appropriated the mythologem of the puruṣa for their invention, the Avalokiteśvara. In that text we hear of the various gods emerging from the different organs of the macranthropic Avalokiteśvara and Vāyu is specifically mentioned as emerging from his mouth. This suggests the existence of an old āstika source with such an imagery from which this concept was borrowed by the bauddhas.

Now further, while in the Varāha-purāṇa myth the motifs acquired from the Vāyu-focused system may have been placed in a Prājāpatya context typical of the earlier paurāṇika narratives, we must not miss the fact that right from the beginning Vāyu had an intimate connection with the protogonic deity with roots likely going back to the para-Vedic systems of the Indo-Iranian period with evolutes lasting until much latter on both the Indian and Iranian sides. This is already apparent in the prājāpatya portions of the late śruti. In the Śatapatha-brāhmaṇa we hear: ‘sa eṣa vāyuḥ prajapatiḥ |‘ (This Vāyu is Prajāpati). It is further expounded in the Kauśītaki-brāhmaṇa that the manifest form of Prajāpati is Vāyu: ‘prajāpateḥ pratyaksaṃ rupaṃ yad vāyur iti|‘. The Śatapatha-brāhmaṇa goes on to even present Prajāpati as a composite of himself and Vāyu: ‘ardhaṃ ha prajāpater vayuḥ | ardham prajapatiḥ | Regarding this intimate connection between the two deities, I could point to a rather late survival of this idea in the Garuḍa-purāṇa (mokṣa-kāṇḍa) in a rather pristine form albeit lodged within a larger, quite unwholesome Vaiṣṇava section, which tries to establish the superiority of Viṣṇu over the other gods:
‘annābhimānaṃ brahma cāhur murāriṃ
jīvābhimānaṃ vāyum āhur mahāntaḥ |
na śakto ‘sau brahma-devo vivastuṃ
vāyuṃ vinā saṃsṛtāv eva nityam ||
na taṃ vinā mātariśvā ca vastum
anyonyam āptiḥ kālato nyūnatā ca |
yadā mahat-tattvani yāmako bhūd
brahmāṇḍānta-sthūla-sṛṣṭau mahātmā ||
tadā vāyur nāśakadvai mahātmā
bāhye sṛṣṭau kālabhedena cāsti |’
The great sages have declared that Brahmā and Murāri (Viṣṇu) are the nutritive principle, [whereas] Vāyu is the metabolic principle. Verily, this god Brahma does not have ever the ability to emit the universe without Vāyu. Without him (Brahmā) Mātariśvan (Vāyu) does not have place to dwell. (O great soul), their relative strengths and wants depends on the course of time. When [Brahmā] restrained of the principles [evolving from] mahat in the [primordial] world-egg from manifesting as material creation Vāyu could not set it motion. The evolution of the manifest universe [by Vāyu] happens only with passage of time.

Thus, Alini, in this system Vāyu and Brahmā are close and almost equal partners in the manifestation of the material universe. Indeed, it appears that these ideas were preserved by a group of late-surviving Vāyu-focused thinkers who later converted to the Vaiṣṇava-mata in southern India. This becomes apparent from a discussion occurring earlier in this section of the Garuḍa-purāṇa wherein the votaries are asked to set up an image of Vāyu and worship it. However, it is then clarified that Vāyu is merely a symbol of Viṣṇu and that Vāyu should only be offered the ucchiṣṭa of Viṣṇu. Notably, it warns votaries who offer fresh flowers, sandalwood and incense to Vāyu and says that they would eternally suffer unless they meditate on Viṣṇu as an expiation. The last statement clinches their Vāyu-focality prior to their conversion. Their unconverted versions still exist, albeit in a degenerate form, in the Lāṭa country — a point we will revisit in the context of the paramparā of our ritual.”

V: “Ah! Having sat in on a discussion Lootika and Somakhya once had, I do see the parallels to the comparable Iranic systems. In the Zoroastrian stream of the Iranic tradition, despite all the demonization of the ancestral deva-s, Vāyu still retains his primacy. In the yasht of Rama Hvastra even Ahura Mazdāh worships Vāyu to aid him in creation by countering Angra Mainyu. In the para-Zoroastrian system of Zurvān (Avestan: Zrva, cognate of Sanskrit sarva) he comes as an integral partner of the great time deity Zurvān who has a fundamentally protogonic character. Thus, Zurvān, associated with time and Vāyu associated with space are the heart of the genesis of this Iranic system. The integral association of Zurvān and Vāyu is already seen in the Zoroastrian Avesta where they are invoked together and are supplied with the epithet ‘deregho khvadhāta (dīrgha-svadhātṛ in our tongue)’. i.e. those who have long-standing law of their own. That Zurvān was an old deity with his own system is seen from the occurrence of his name close to the Avestan form in the Nuzi documents from Iraq dating to around 3200-3400 years ago where we find some of the earliest attestations of the Iranians in archaeology. This association is also recorded by Eudemus of Rhodes, the student of Aristotle, who notes that the Iranians regard Time (Zurvān) and Space (Vāyu) as the foundational principles of the universe. That he was referring to the para-Zoroastrian Zurvān worshipers becomes apparent from a comparable system presented in Zurvānistic Iranic traditions preserved by the Zoroastrians in their Denkart, where Zurvan as time is the generative principle and Vāyu provides the space for the entities that come into being due to Zurvan. There, Vāyu is also the one who conveys the good things bestowed by Zurvān to men — this perhaps captures a bit of his nature as the messenger, which we see in his Mātariśvan form in our tradition. Thus, his association with the Iranic protogonic deities, be he Ahura Mazdāh or Zrva, in the generative process is indeed a parallel to his association with Brahmā in our tradition.”

I.s: “As we move from Vāyu, who was specifically hymned by your ancestor Vāmadeva, to Kubera, we again find these parallels between Indo-Aryan and Iranian tradition as illuminating their association. Clearly in our tradition Vāyu-Vāta has both a benign and malignant nature. In the mantra you mentioned he is described as `’rujan’ i.e. smashing. This nature is clearly brought out in his earthly epic manifestations, the ape Hanūmat and the Pāṇḍava Bhīma. On the other hand, as you correctly mentioned, in the Iranic tradition Vāyu conveys the good things of Zurvan to men. This is his benign aspect which comes up in the second such Vāta-sūkta in the RV:
vāta ā vātu bheṣajaṃ śambhu mayobhu no hṛde |
pra ṇa āyūṃṣi tāriṣat ||
uta vāta pitāsi na uta bhrātota naḥ sakhā |
sa no jīvātave kṛdhi ||
yad ado vāta te gṛhe ‘mṛtasya nidhir hitaḥ |
tato no dehi jīvase ||’
May Vāta blow towards us a medicine, which is auspicious and conferring weal to our heart. May he lengthen our lives.
O Vāta, you are father to us, and brother, and also a friend to us. Make us live long.
O Vāta, what wealth of ambrosia has been placed yonder in your house, give us of that for us to live.
Here, we see that the benign aspect of Vāyu is emphasized with him being a bearer of medicines (note a similar role played by Hanūmat in the Rāmāyaṇa), fellowship and the ambrosia. Importantly, note the amṛtasya nidhiḥ, which becomes important for the connection with the Mahārāja. On the Iranian side Vāyu again is invoked for benign gifts such as protection from yātudhāna-s, a long life, offspring, and by women for a good husband. On the other-hand, he is also the fierce ‘pitiless Vāyu’, the bone-breaker (Astovidhātu) who slays the man (the bone-crushing facet is well represented in the earthly emanations of our Vāyu, Hanūmat and Bhīma, who more generally also have a certain malignant side to them, which, O Gautami, your ancestors are said to have faced in the case of the former). Thus, in later Iranic tradition we hear of two Vāyu-s, the Vāy i veh the good Vāyu and the Vāy i vattar, the evil Vāyu, who in the Iranic text known as the greater Bundahishn is explicitly termed Astovidhātu who causes death.

Thus, even in our origin myth of Kubera, the two facets of Vāyu are clearly laid out. He originally emerges in what is evidently his malefic form showering gravel/grit and then assumes a benefic form which becomes Kubera. Notably, Kubera, like Vāyu in the RV mantra, also holds the nidhi of amṛta, which is described as the honey of Jambhala, known to the V1s who have mastered the Kaubera lore, in the Mahābhārata.”

V: “Dear Ātreya, may be in the regard I should add that this benign form of Vāyu, which becomes Vaiśravaṇa might relate to the magical concealment power of the yakṣa-s, tirodhā that is alluded to multiple times in the śruti. On one hand it connects Kubera with the primordial goddess Virāj. The Puruṣa hymn, echoing the cyclic generation of the primordial god Dākṣa and goddess Aditi from each other, states that the goddess Virāj arose from the Puruṣa and that conversely the Puruṣa arose from Virāj. The Atharvāṅgirasa-śruti adds that Kubera was a calf of this goddess Virāj in the form of Tirodhā which conferred the concealment (tirodhā) capacity on his folk (i.e. the yakṣa-s). This is again mentioned and worshipped in his ritual in the Taittirīya-śruti. In the AV it is said that he who knows this thing about Kubera causes his pāpaman to vanish, whereas in the TA it makes his enemies disappear, like the prācya Uesugi Kenshin experienced with his devotions. This again reflects the beneficent nature of this deity whose origin is from the primordial being. Perhaps the tirodhā power, which the Taittirīyaka-s describe as coming from the waters represents, the benefic clouds brought by Vāyu as opposed to the destructive sand storm.

Finally, before we go to the ritual, O Atri, I would like to to ask you if you can think of any parallels to the connection between Vāyu and Vaiśravaṇa which might exist in the Iranian world?”

I.s.: “Firstly, O Āṅgirasī, it is pleasing to see that you have thoroughly acquired the śrauta-siddhānta of Puṇyajaneśa. Regarding your question — I am aware of no direct cognate of Kubera in the early Iranic world. However, we may note the following: interestingly, even in the Avesta the benign Vāyu is described as being associated with gold, a characteristic of the Yakṣarāṭ. Subsequently, among the non-Zoroastrian and para-Zoroastrian Iranics of the Kuṣāṇa empire it appears that the personification of royal power, Avestan Khvarena (Kuṣāṇa: Pharro), came to identified with Vaiśravaṇa. This is not surprising given both their association with royal power. This led to syncretic depictions in the Kuṣaṇa age that appear to have eventually reached the eastern steppes whose new masters were the Turkic and Mongolic people. This greatly influenced the development of the Kubera-like deity Kut-Tengri among the Uighurs and other central Asian Turks.”

V: “Now, O Atricandra, expound to me details of the ritual — the observances, actions and the mantra-s.”
I.s: “Sure. The vrata, as has been specified, should be done on an ekādaśī, either in the afternoon or in the evening. After having had a bath the ritualists should observe purity and abstain from sexual activity for the entire day. They should eat air-dried fruits and nuts. They should abstain from consuming milk products but they may drink fresh fruit juices. In old Bhārata the ideal spot for the ritual was said to be under an aśvattha or a vaṭa tree but these days such spots are difficult to come by.

First the ritualists should offer tarpaṇa to the key teachers of the lineage: the founder was a sage of the Vaikhānasa clan. That knowledge then passed on to the queen Vāsavadattā. Her son the renowned Naravāhanadatta was next. His minister Gomukha was next. These were the teachers of the lore at Kauśāmbī. Then the lore was taught at the holy city of Mathurā, where all the great dharma-s were expounded by the mantra-vit-s. As you know both Somakhya and I trace our lineages to gharāṃnāya-s of that holy city. There it was taught in the center known as the Yakṣa-guhā, which is today Maholi near the Mathurā city. Here the teachers were Gomitraka and Kunika. From there the tradition moved to Gopagiri (modern Gwalior), where it was taught by your old clansmen, the Gotama Rudradāsa and his son Śivatrāta. From them it was transmitted to Śivanemi, Śivabhadra and Kubhaka.” From there it was transmitted along the Candravati river (modern West Banas) to the V1 Maṇibhadra who lived in Vāyvagrahāra (a little hamlet near modern Banaskantha) in the Lāṭānarta country. He transmitted it to his students Kuberadatta and Yakṣamitra who lived at Vāyuvaṭa (modern Vayad). All these V1s were special worshipers of Vāyu at his prāsāda-s that were once prominent in the region and the rite we perform takes its form in their teachings.
V: “It is indeed rather remarkable that even today we have nearly forgotten remnants of the old cult at these key cultic centers, like the Vāyu temples in Lāṭa and the Vitteśa near a step-well in their vicinity, or the images of Nalakūbara at Gopagiri and the fragment of the old Yakṣarāṭ-prāsāda at Kauśāmbī.”

I.s: “If the Yakṣa is good to us, may be some day we shall visit those cultic centers to revive our connection. To continue, the ritualist then salutes his teachers: ‘oṃ gurubhyo namaḥ |‘. The spot of the rite should be to the north of the trunk of the said tree for indeed the northern line is where Vāyu, Vitteśa and Bhava, all deities of the rite, are stationed. The ritualist begins by uttering the yajuṣ:
iṣé tvorjé tvā vāyávaḥ sthopāyávaḥ stha devó vaḥ savitā́ prā́rpayatu śréṣṭhatamāya kármaṇe |
For vigor you; for nourishment you; the Vāyu-s abide, the approachers abide. Let the god Savitṛ set you in motion for the most excellent ritual.

Vrishchika, here you may note the multiplicity of the Vāyu-s reminiscent of the Aeolian plurality seen among our yavana cousins. Then he stands up and recites the following ṛk-s:
stóma-trayastriṁśe bhúvanasya patni vívasvad vāte abhí no gṛṇāhi |
ghṛtávatī savitar ā́dhipatyaiḥ páyasvatī rántir ā́śā no astu ||
dhruvā́ diśā́ṃ viṣṇupatny ághorāsyéśānā sáhaso yā́ manótā |
bṛhaspátir mātaríśvotá vāyúḥ saṃdhuvānā́ vā́tā abhí no gṛṇantu ||
viṣṭambhó divó dharúṇaḥ pṛthivyā́ asyéśānā jágato víṣṇupatnī |
viśvávyacā iṣáyantī súbhūtiḥ śivā́ no astv áditir upásthe ||
O you of the 33-fold stoma, lady of the world, with winds set in motion by Vivāsvān, may you be gracious to us.
Rich in ghee, O Savitṛ, through your overlordship, may the generous space be rich in milk for us.
Firm among the quarters, wife of Viṣṇu, the benign [lady], the strong queen of all, the desirable one;
Bṛhaspati, Matariśvan, Vāyu, the Vāta-s blowing in unison, may all be gracious to us.
Stabilizer of the sky, supporter of the earth, ruling this world, wife of Viṣṇu.
Encompassing all space, invigorating all, prosperous, may the goddess Aditi be auspicious to us in her lap.

He then marks out a square area on the ground with the side equal to his full arm’s length and sweeps it clean. Within it he marks out another square region with a side equal to his span. He does this with the sphya (the wooden sword) while chanting the formula:
ástabhnād dyā́m ṛṣabhó antárikṣam ámimīta varimā́ṇam pṛthivyā́ ā́sīdad viśvā bhúvanāni samrā́ḍ víśvet tā́ni váruṇasya vratā́ni ||
The bull has stabilized the sky [and] the atmosphere; he has measured the circumference of the earth; he has set him in all worlds as the emperor. All these are Varuṇa’s laws.

He then digs out a cuboidal excavation corresponding to the central smaller square. While doing so he recites the yajuṣ:
víṣṇoḥ krámo ‘si víṣṇoḥ krāntám asi víṣṇor víkrāntam asi | námo mātré pṛthivyái mā́hán mātáram pṛthivī́m̐ him̐siṣam mā́ | mā́ṃ mātā́ pṛthivī́ him̐sīt ||
You are the stepping of Viṣṇu; you are the step of Viṣṇu; you are the stride of Viṣṇu. Salutation to mother Earth; may I not harm mother Earth; may mother Earth not harm me.

The he fills the sruc with ghee using the the sruva and pours out the ghee into excavation uttering the yajuṣ:
vāyáve tvā váruṇāya tvā nírṛtyai tvā rudrā́ya tvā svāhā | vā́yo vī́hi stokā́nām ||
For you Vāyu; for you Varuṇa; for you Nirṛti; for you Rudra, hail! O Vāyu, taste the drops!

He then fills the sruc again with milk pours it into the excavation uttering the incantation:
vásur vásupatir híkam ási| kṣatrā́ṇāṃ kṣatrápatir asi | áti divás pāhi | vaiśravaṇāya svāhā||
You are indeed rich, the lord of riches. You are the the king of the kings. From the heavenly realm protect us. To Vaiśravaṇa hail.”

Indrasena then brought out a bronze image of a chameleon and showed it to Vrishchika and explained further: “The ritualist should have such a bronze image of a chameleon, which as you know is the special animal of the Yakṣapati. Alternatively, he may have a metal plate with the image of a chameleon etched on it. He deposits it in the excavation for the rite and may reuse it over and over, each time he performs the rite. He takes up the image reciting the following incantation:
iyaṃ varṇagodhā vaiśravaṇa-rūpā bahuvarṇikā | tirodhā ‘si | hiraṇyavaktro ‘si vaiśravaṇo ‘si | kṣatraṃ kṣatraṃ vaiśravaṇaḥ | mahārājāya svāhā ||
This chameleon of many colors is the form of Vaiśravaṇa. You are the camouflaged one. You are the golden-cheeked one; you are Vaiśravaṇa. Kingly power, kingly power [is that of] Vaiśravaṇa. Hail to the great king!

He then deposits the chameleon in the excavation and with the last mantra of the above he pours out ghee from the sruva on to image. He then covers up the image by filling the excavation. To the north of the square he sets up a platform with a silk cloth on it. On that he should place an ornate bronze jar with a lid. He should thereafter invoke Vāyu and his wife Śivā in the jar and offers them water for arghya and pādya and incense with the following incantation of Mārkaṇḍeya:
vāyum āvāhayiṣyāmi sarvagaṃ dīptatejasam |
deva vāyo tvam abhy ehi sarva-bhūta-jagat-priya |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke Vāyu of blazing splendor, who can go anywhere. O Vāyu, dear to all beings and the world, you must come here. Please accept this arghya, pādya (water for washing the feet) and incense.

śivām āvāhayiṣyāmi vāyoḥ patnīṃ prabhāvatīm |
śive ‘bhy ehi mahābhāge varade kāmarūpiṇī |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke Śivā of great luster, the lady of Vāyu. O highly distinguished goddess Śivā, who can assume any form, the boon-giver, come here. Please accept…

He then places an image of Kubera inside the jar. The image should ideally be fashioned generally following the teaching in this regard of the Bhārgava Mārkaṇḍeya to Vajra, the great-grandson of Kṛṣṇa, the king of Indraprastha: The image of the wealth-giver Kubera should be painted dark green, i.e., the color of a lotus-leaf. He should be shown as being borne by a naras yakṣa. He should have all kinds of golden ornaments with necklaces resting on his belly. He should be shown wearing a beautiful cloak. He should have four hands and one of his eyes should be yellow. He must have two fangs in his mouth, and a mustache and beard on his face. His head is adorned with a jeweled crown. On his left lap, his wife, the beautiful, boon-giving goddess Ṛddhi should be comfortably seated. She should be shown with two hands. One of them should be shown hugging the Mahārāja and the other should display a jewel-pot. In one of his left hands he should be shown holding a jeweled vessel and a gold-spitting mongoose in the other. In his two right hands Kubera should be shown with a mace and a śakti missile. Beside him there should be his dhvaja with the lion-emblem. A palanquin should be shown near his feet and his treasure-chests should be shown beside him in the form of a conch and a lotus filled with treasures.

If he cannot obtain such an image, any other image that is properly made might be used. However, he must use the above teaching of Mārkaṇḍeya for the dhyāna of the great Yakṣa because if he does so he obtains wealth. Mārkaṇḍeya gives the following symbolism for the image: The lord should be seen as the embodiment of all arthaśāstra-s. It is only by the institutes of arthaśāstra-s does any endeavor of a living-being exist, as was taught by Cāṇakya who devised the overthrow of the barbarous Macedonians. Kubera’s śakti weapon stands for royal power and his gadā represents daṇḍa-nīti. Goddess Ṛddhi represents a full worldly life. The jeweled vessel in her hand is the receptacle of good qualities. The kingdom over which the Yakṣapati rules is the body itself. The lotus and the conch are the reservoirs of all treasures in nature from which wealth perpetually flows. His lion flag is desire. Hence, he is known as Kāmeśvara in the Yajurveda. His two fangs are the reward and the punishment that he metes out. He who mediates thus has Kubera dwell in him.

Having placed the image of the Rājarāja in the jar he invokes him and his retinue and offers them worship with the following incantations:
yākṣān āvāhayiṣyāmi dhānādhyakṣa-samanvitān |
āyāntu varadā yakṣās trailokya-viditā mama |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke the yakṣa-s together with the lord of wealth. Let the yakṣa-s renowned in the 3 worlds come to me giving boons. Please accept this arghya, pādya (water for washing the feet) and incense.

bhadrām āvāhayiṣyāmi dhanādhipati-sundarīṃ |
ehi ṛddhi mahādevi varade kāma-rūpiṇī |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke the lucky one, the beautiful lady of the lord of wealth. Come, O great boon-giving goddess Ṛddhi of much desired form. Please accept…

aham āvāhayiṣyāmi tvām eva nalakūbaram |
samabhy ehi mahābhāga dhanadasya+ātmasaṃbhava |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall verily invoke you O Nalakūbara. May you, O fortunate one born of Dhanada, come here. Please accept…

aham āvāhayiṣyāmi śibikāṃ paramāyatām |
śibike tvam ihā ‘bhy ehi sarva-sattva-sukhaṅkari |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke the wide palanquin of Kubera. O palanquin who generates all good and pleasures may you come here. Please accept…

naram āvāhayiṣyāmi narādhipati-vāhanam |
nara śīghraṃ tvam ‘bhy ehi tathā nṛpatināyaka |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke the Naras yakṣa who is the vehicle of the king [Kubera]. O Naras may you quickly come here, you have the king as your lord [Kubera]. Please accept…

śaṅkham āvāhayiṣyāmi nidhi-pravaram uttamam |
śīghraṃ śaṅkha tvam abhy ehi dhānādhipati-vallabha |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke the conch, the foremost of excellent treasures. May you, O conch, quickly come here, the one dear to the lord of wealth. Please accept…

padmam āvāhayiṣyāmi nidhi-pravaram uttamam |
padmaṃ śīghraṃ tvam abhy ehi mahāvibhava-kāraka |
idam arghyaṃ ca pādyaṃ ca dhūpo’yaṃ pratigṛhyatām ||
I shall invoke the lotus, the foremost of excellent treasures. May you, O lotus, quickly come here producing great prosperity. Please accept…

Then he offers flowers, honey and bhakṣaṇa-s to the image of Kubera with the famous incantation from the Taittirīya-āraṇyaka:
rājādhirājāya prasahyasāhine | namo vayam vaiśravaṇāya kurmahe | sa me kāmān kāmakāmāya mahyam | kāmeśvaro vaiśravaṇo dadātu |
kuberāya vaiśravaṇāya | mahārājāya namaḥ ||

He should thereafter make the triple-arghya offerings as ordained by the gāyatrī-s of Vaikhānasa-s:
rāja-rājāya vidmahe dhanādhyakṣāya dhīmahi | tan no rājā pracodayāt ||
rudra-sakhāya vidmahe vaiśravaṇāya dhīmahi | tan naḥ kuberaḥ pracodayāt ||
yakṣeśvarāya vidmahe gadā-hastāya dhīmahi | tan no yakṣaḥ pracodayāt ||

Then he offers bali to Dhaneśa and Nalakūbara meditating upon them traversing the heavens in the Puṣpaka space-station made by the god Tvaṣtṛ which is alluded to in the mantra. This is as per the Aruṇa-ketuka-vidhi with the incantations:
adbhayas tirodhā’jāyata | tava vaiśravaṇas sadā | tirodhehi sapatnān naḥ | ye apo ‘śnanti kecana | tvāṣṭrīṃ māyāṃ vaiśravaṇaḥ | ratham̐ sahasra-vandhuram | puruś-cakram̐ sahasrāśvam | āsthāyāyāhi no balim | yasmai bhūtāni balim āvahanti | dhanañ-gāvo-hasti-hiraṇyam-aśvān | asāma sumatau yajniyasya | śriyam bibhrato ‘nnamukhīṃ virājam | sudarśane ca krauñce ca | maināge ca mahāgirau | śata-dvāṭṭāragamantā | sam̐hāryan nagaraṃ tava | iti mantrāḥ | kalpo’ta ūrdhvam | yadi balim̐ haret | hiraṇyanābhaye vitudaye kauberāyāyam baliḥ | sarvabhūtadhipataye nama iti | atha balim̐ hṛtvopatiṣṭheta | kṣatraṃ kṣatraṃ vaiśravaṇaḥ | brāhmaṇā vayam̐ smaḥ | namaste astu mā mā him̐sīḥ ||

After that he makes bali offerings to the rakṣa-s, yakṣa-s and yakṣī-s in the entourage of Kubera with:
miśra-vāsasaḥ kauberakā rakṣo-rājena preṣitāḥ |
grāmam̐ sa-jānayo gacchantīchanto paridākṛtān ||
Wearing mixed clothes the agents of Kubera, directed by the king of the rakṣa-s together with the [yakṣī] consorts go as they please to the villages granting protection.

There after he sets up a sthaṇḍila in the square he has marked out. He strews darbha grass with the shoots facing north. He installs the fire therein from his aupāsana or āhavanīya altar or kindles it afresh. He recites this ṛk:
vaiśvānaraṃ kavayo yajñiyāso ‘gniṃ devā ajanayann ajuryam |
nakṣatram pratnam aminac cariṣṇu yakṣasyādhyakṣaṃ taviṣam bṛhantam ||
The gods, the poets of the ritual, generated the unaging Agni Vaiśvānara,
the primordial star, wandering but not violating [the natural law], the mighty and great lord of the yakṣa.

He makes an oblation of ghee with: agnaye vaiśvānarāya svāhā | idam agnaye vaiśvānarāya na mama |

If he is a purohita performing it on behalf of a yajamāna other than himself, like at the temple it would be done with the V3, then the yajamāna must recite the tyāga-mantra ‘idam…na mama‘ as the oblation is offered. Then he recites:
dhātā́ rātíḥ savitédáṃ juṣantām prajā́patir nidhipátir no agníḥ |
tváṣṭā víṣṇuḥ prajáyā saṁrarāṇó yájamānāya dráviṇaṃ dadhātu ||
May Dhatṛ the giver, may Savitṛ, rejoice in this [oblation], Prajāpati, the lord of treasures, and Agni, for us; may Tvaṣtṛ and Viṣṇu generously give wealth with offspring to the ritualist.

He makes an oblation of ghee with: devebhyaḥ svāhā | idam devebhyo na mama |

He then prepares for the core oblations of the yajña by deploying the nigada recitation as in the vaidika rite calling upon the god Agni who is of the brāhmaṇa-s and the Bhārata-s to bring the gods of the rite for receiving the oblations:
oṃ agne mahām̐ asi brāhmaṇa bhārata | deveddho manviddha ghṛtāhavanaḥ praṇīr yajñānāṃ rathīr adhvarāṇām atūrto hotā tūrṇir havyavāṭ | tvaṃ paribhūr asy ā vaha devān yajamānāya || vāyuṃ niyutvantam ā vaha | kuberaṃ vaiśravaṇam ā vaha | rudram paśupatim ā vaha | devām̐ ājyapām̐ ā vaha ||

He then offers in the northern side of the sthaṇḍila with the sruva:
prajāpatye svāhā | idam prajāpatye na mama ||

He utters the call ‘astu śrau3ṣaṭ |‘ and starts filling he sruc and as he does so he recites:
tava vāyav ṛtaspate tvaṣṭur jāmātar adbhuta |
avāṃsy ā vṛṇīmaho3m ||
Vāyu, lord of the natural law, of wondrous form, Tvaṣṭṛ’s son-in-law, your protection we choose.

Then he recites:
pra vāyum acchā bṛhatī manīṣā
bṛhadrayiṃ viśvavāraṃ rathaprām |
dyutadyāmā niyutaḥ patyamānaḥ
kaviḥ kavim iyakṣasi prayajyo3m ||
vau3ṣaṭ | idaṃ vāyave niyutvate na mama ||
[Our] great meditation (goes) forth to Vāyu; with great wealth and all that is excellent, he fills his chariot; on the brilliant path, master of horse-teams, you the poet seeks to reach the poet [i.e. the hymn-composing vipra], you who are worshiped at the forefront of the ritual.

At the vauṣaṭ call he makes the oblation and then utters the tyāga-mantra. He makes the oblation facing whichever direction the wind blows at the time he is reciting the this mantra. He utters the call ‘astu śrau3ṣaṭ |‘ and starts filling he sruc and as he does so he recites:
rāyas poṣāyāyuṣā tvā nidhīśo bhrātṛvyāṇāṃ mahasāṃ cādhipatyo3m |
[I invoke you] for riches and prosperity with a long life. The lord of wealth [is worshiped] for taking over the glory of our rivals and for overlordship.

This peculiarly-formed mantra is that of the old Jambhala-vit Vaikhānasa, the master of divine rituals. Then he recites:
dūre pūrṇena vasati dūra ūnena hīyate |
mahad yakṣaṃ bhuvanasya madhye
tasmai baliṃ rāṣṭrabhṛto bharanto3m ||
vau3ṣaṭ | idaṃ kuberāya vaiśravaṇāya na mama ||
In fullness he dwells in the distance, is left behind in distance on diminishing;
A mighty Yakṣa is in the center of the universe:
to him the rulers of the tributary rulers bring tribute.

At the vauṣaṭ call he makes the oblation and then utters the tyāga-mantra. I should mention that this is the mysterious mantra is of the Bhṛgu-s by knowing which one becomes the complete Jambhala-vit. It was taught to me by Somakhya on a dark new moon night as we were wandering in a beautiful mountainous region. This completed my learning of the tradition and conferred on me the ability to perform this ritual. The thing which dwells in distance in full and then diminishes is an allusion to the moon. This disappearance of the moon is in turn indicative of tirodhā, the famed concealing power of the Yakṣa. He then utters the call ‘astu śrau3ṣaṭ |‘, starts filling he sruc and as he does so he recites:
kad rudrāya pracetase mīḷhuṣṭamāya tavyase |
vocema śantamaṃ hṛdo3m ||
What might we say to Rudra, the wise, the most generous and mighty one; what might we say that is most pleasing to his heart?

upa te stomān paśupā ivākaraṃ
rāsvā pitar marutāṃ sumnam asme |
bhadrā hi te sumatir mṛḷayattamā
‘thā vayam ava it te vṛṇīmaho3m ||
vau3ṣaṭ | idaṃ rudrāya paśupataye na mama ||
Like a cowherd, I have driven these praises close to you.
Grant your favor to us, father of the Marut-s,
for your benevolence is auspicious, most merciful.
It is indeed your aid that we choose. [translation after Geldner]

At the vauṣaṭ call he makes the oblation, then utters the tyāga-mantra and then touches water.

Then with the sruva he utters the following mantra-s and makes an offering of ghee at each svāhā followed by the tyāga-mantra ‘idaṃ na mama’:
indraḥ svāhā | marutaḥ svāhā | sarva-nidhido ratna-dhātumān svāhā | tirodhā bhūḥ svāhā | tirodhā bhuvaḥ svāhā | tirodhā svaḥ svāhā | tirodhā bhūr-bhuvaḥ-svaḥ svāhā | kāmeśvarāya svāhā | kuberāya svāhā | dhanyāya svāhā | vaiśravaṇāya svāhā | yakṣa-rājāya svāhā ||

He then initiates the expiatory sviṣṭakṛt-yāga by reciting the mantra of my ancestor Vasuśruta Ātreya:
oṃ juṣṭo damūnā atithir duroṇa
imaṃ no yajñam upa yāhi vidvān |
viśvā agne abhiyujo vihatyā
śatrūyatām ā bharā bhojanāno3m ||
As the satisfied lord of the house and the guest in the home,
journey to this our yajña, the learned god,
Having demolished all assaults, O Agni,
bring here the sustenance of our enemies.

He utters the call ‘astu śrau3ṣaṭ |‘ and fills the sruc. He then recites:
agniṃ sviṣṭakṛtam ayāḍ vāyor niyutvataḥ priyā dhāmāny ayāṭ kuberasya vaiśravaṇasya priyā dhāmāny ayāḍ rudrasya paśupateḥ priyā dhāmāny ayāḍ devānām ājyapānāṃ priyā dhāmāni yakṣad agner hotuḥ priyā dhāmāni yakṣat svaṃ mahimānam āyajatāmejyā iṣaḥ kṛṇotu so adhvarā jātavedā juṣtām̐ havir mārjālyo mṛjyate sve damūnāḥ kavi-praśasto atithiḥ śivo naḥ | sahasra-śṛṅgo vṛṣabhas tad ojā viśvām̐ agne sahasā prāsy anyān vau3ṣaṭ ||

At the vauṣaṭ call he makes the offering from the sruc and utters: ‘idam agnaye sviṣṭakṛte na mama|‘ The terminal mantra of the sviṣṭakṛt-yāga is also composed by my ancestors Budha Ātreya and Gaviṣṭhira Ātreya:
Fit to be kindled, he is kindled in his own (house) as master of the
house, praised by poets, our auspicious guest.
A thousand-horned bull having its virility,
O Agni, in might you excel all others.

Then he offers a homa to the Vināyaka-s, Skanda, and other anuyāyin-s of Rudra which include the Yakṣapati himself so that the yajamāna-s rite might be successful and he and his family not be harmed. He utters the tyāga-mantra ‘idaṃ na mama’ after each svāhā call:
Om̐ śālakaṭaṅkaṭāya vināyakāya svāhā | kuṣmāṇḍarājaputrāya vināyakāya svāhā | usmitāya vināyakāya svāhā | devayajanāya vināyakāya svāhā | vimukhāya svāhā | śyenāya svāhā | bakāya svāhā | yajñāya svāhā | kalahāya svāhā | bhīrave svāhā | yajñavikṣepiṇe svāhā | kulaṅgāpamārāya svāhā | yūpakeśyai svāhā | sūkarakroḍyai svāhā | umāyai haimavatyai svāhā | jambhakāya svāhā | vīrūpākṣāya svāhā | lohitākṣāya svāhā | vaiśravaṇāya svāhā | mahārājāya svāhā | mahāsenāya svāhā | kumārāya svāhā | viśākhāya svāhā | śākhāya svāhā | nejameṣāya svāhā | ṣaṣṭhyai svāhā | rudrāya mahādevāya svāhā ||

He then touches water. Thereafter he takes up some grain and throws it in the south-west direction for the rakṣa-s uttering: ‘rakṣasām bhago ‘si |‘ Then he touches water again and takes up the darbha grass and casts it into the fire reciting:
Om̐ sáṃ barhír aṅktām̐ havíṣā ghṛténa sám ādityáir vásubhiḥ sáṃ marúdbhiḥ sám índro viśvádevebhir aṅktāṃ divyáṃ nábho gacchatu yát svā́hā || idaṃ divyāya nabhase na mama | rudrāya paśupatye svāhā | idaṃ rudrāya paśupataye na mama||
Auspicious [is] the grass smeared with offerings and ghee. Let Indra together with the Āditya-s, the Vasu-s, Marut-s, and the Viśvedeva-s go (having been honored). Let svāhā-offerings rise to the heavenly ether.

Then he recites the following to conclude the yajña with the recitation of Agastya:
agne naya supathā rāye asmān
viśvāni deva vayunāni vidvān |
yuyodhy asmaj juhurāṇam eno
bhūyiṣṭhāṃ te nama-uktiṃ vidhema ||
O Agni, lead us to wealth by an easy path:
you know all the rituals, O god.
Keep us from ritual transgression.
May we offer you the greatest reverence.

He finally recites the śānti-mantra-s: ‘tac chamyor…‘ and ‘namo vāce yā coditā…‘ He then goes to the jar with the image of Kubera and utters the formulae: ‘vāyuḥ suprītaḥ suprasanno yathā sthānaṃ tiṣṭhatu | vaiśravaṇo rājā suprītaḥ suprasanno yathā sthānaṃ tiṣṭhatu |‘ He and his family drink some of the honey offered to the yakṣa and he distributes the bhakṣaṇa-s to the beholders of the ritual. He gives a toothbrush, sandals and umbrella along with a hefty ritual fee if he is a yajāmāna who has employed a purohita to do the rite for him.”

V: “I must remark that the final homa has a set of rather remarkable deities. Perhaps we are seeing the worship of Vārāhī for the first time in Sūkarakroḍī. I also wonder if Yūpakeśī is some early version of Cāmuṇḍā. Beyond that I’m most inspired by this account of the rite and feel that I have become a Jambhala-jñā”

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The Platonic culmination of Euclid and the pentagon-hexagon-decagon identity

Why did great sage Pāṇini compose the Aṣṭādhyāyī? There were probably multiple reasons but often you hear people say that he wanted to give a complete description of the Sanskrit language. That was probably one of his reasons but was it the central reason or the most important driver of his endeavor. From his own language and the evolution of the language to his times we can see that the Vedic register of Sanskrit was well known but already largely a thing of history in terms of being a spoke language. Yet, Pāṇini devotes an enormous amount of effort in describing it accurately and it still forms the foundation of our effort to understand the earliest words of our ārya ancestors. Thus, understanding the Vedic language so that we can properly understand the religion of our ārya ancestors and perform rituals as enjoined in the śruti was the major reason for Pāṇini-s effort. This indeed is acknowledged by his great commentator Patañjali. One who is well-acquainted with the śruti also realizes that the ārya-s conceived this linguistic framework for understanding the śruti and performing the rituals enjoined by it as having a deeper significance — it offered an means of apprehending the nature of the deva-s and understanding the universe. Hence, the great Aṅgiras Dirghatamas auccāthya states in the śruti:

ṛco akṣare parame vyoman
yasmin devā adhi viśve niṣeduḥ |
yas tan na veda kim ṛcā kariṣyati
ya it tad vidus ta ime sam āsate || RV1.164.39

In the syllable of the ṛc set in the highest world,
therein all the gods have taken residence,
he who doesn’t know that what can he do with the ṛc?
verily only they who know can sit together [in this ritual session].

It was with such considerations in mind that the sages Pāṇini and Patañjali composed their works. Thus, their works, which aimed at developing a certain comprehensiveness of the framework from as small as set of foundational axioms and with as much internal consistency as possible, never lost focus on its relationship with the “root”, i.e. the śruti.

Just as Pāṇini is for the ārya-s, Euclid is to their yavana cousins. His geometric system was in many ways parallel to the vyākaraṇa of Pāṇini. Like the work of Pāṇini it served as the framework of knowledge at large among the yavana-s, but what was the central reason for its composition? In the modern Occident which considers Euclid as one of the elements of their intellectual undergirding this is either completely ignored or wrongly stated as being the need to compose a comprehensive secular geometric text. However, we hear from Euclid’s great commentator, a sage in his own right, Proclus, that the reason was to ultimately develop a geometric framework to understand the “roots” (rhizomata) of existence. These were first expounded by Empedocles in his verse:

And first the fourfold root of all things hear!–
White gleaming Zeus, life-bringing Hera, Aidoneus,
And Nestis whose tears bedew mortality.

Empedocles identified the “roots” with the four deities Zeus (the fiery element= Skt: agni), Hera (the gaseous element= Skt:vāyu), Aidoneus (the solid element= Skt: pṛthivī) and Nestis (the liquid element= Skt:ap). One may compare the initial part of the Empedoclean system with the proto-Sāṃkhya system described in the upaniṣat of the Taittirīya-s.

Plato, who came after Empedocles, incorporated these “roots” into his theory of knowledge as “stoicheia”, which literally mean letters of the alphabet. This hints a parallel to the ārya usage that might lurk behind Plato’s stoicheia because indeed the akṣara-s, the Sanskrit equivalent of stoicheia, are at the foundation of the Hindu system as encapsulated by Pānini in his Māheśvara-sūtrāṇi. However, the sense in which Plato constructed his stoicheia was not by using linguistic analogs but by using geometric ones. He chose for this purpose the only regular solids that can be inscribed in a sphere — i.e the 5 Platonic solids. Plato is said to have learned of these from his predecessors like Pythagoras and his mathematical interlocutor Theaetetus. He incorporated them into an elaborate geometric “atomism” wherein the the fiery element was the tetrahedron, the solid element was the cube, the gaseous element was the octahedron, the stuff of the universe as a whole was the dodecahedron and the liquid element was the icosahedron. However, as his polyhedral system had 5 solids his system ultimately had to break away from the 4 element system of the old yavana-s and converge towards the 5 element system of the ārya-s.

Plato’s polyhedra may be seen as “molecules” because beneath them there lay his true atomic entities namely the 30-60-90 and the 45-45-90 right triangles (the 3 angles of this triangle are given in degrees). Two of these combined to give the equilateral triangle and the square which in turn constituted the four elemental polyhedra (sort of like the dvayāṇu theory of Akṣapāda Gautama). So in a process like the boiling of water into the gaseous state under the influence of the fire tetrahedron Plato saw the water icosahedron break down into equilateral triangles and reconstitute gaseous octahedra. In his system the solid cube had to keep out of these of inter-conversions. Aristotle was uncomfortable with this but the later Platonists including Proclus saw this a genuine feature of the solid element. Proclus explained it by stating that the solid element might be subdivided in interaction with the other two but does not ultimately transform.

In any case, for Proclus the foundation of understanding was based in mathematics. Indeed, he says that it “purifies and elevates the soul even as the goddess Athena dispersed mist obscuring the intellectual light of understanding.” Hence, it was critical to have a complete theory from few starting axioms to construct the 5 Platonic solids that form the basis of existence. This, he says, is the ultimate objective of Euclid — a complete theory from axioms to construct the Platonic solids. In support of this understanding one may note that the key constructions of the Platonic solids come in the last book of the Elements of Euclid. As per Proclus, Euclid was a Platonist who had studied with associates and students of Plato such as Philippus and Theaetetus before composing his treatise. Now the tetrahedron and cube are easy to construct from equilateral triangles and squares. The octahedron is slightly more involved by is also quite easily achieved as a equilateral bipyramid on a square base. However, the dodecahedron and the icosahedron are much more of a challenge. It is this context we encounter the “culmination of Euclid” in the form of the famous pentagon-hexagon-decagon identity of Euclidean geometry. Below we shall journey through this identity, and note its proof and how it specifies the icosahedron. Once, we have an icosahedron we can construct the dodecahedron as its dual.

penta_hexa_decaFigure 1

Consider a regular pentagon, hexagon and decagon inscribed within the same circle (Figure 1). Let their sides be P, H, D. Then, what is the relationship between their sides? The answer to this is provide by Euclid in his Elements, Book 13, Proposition 10:

The pentagon-hexagon-decagon identity( P.H.D): If an equilateral pentagon is inscribed in a circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

penta_hexa_deca_BKHFigure 2

Alternatively, if the sides of a regular pentagon, hexagon and decagon respectively form the hypotenuse, greater and smaller legs of a right triangle then their circumcircles are congruent (Figure 2).

Now, the the proof for this as given by Euclid is not entirely obvious and, at least to us, it is unclear if he even gave a complete proof for it. Nevertheless, we can prove it using all the material he has described up to that point and the commentary of Proclus. For the proof one needs to first define that special ratio known as the Golden ratio \phi:
\phi^2=\phi+1
Since this is a quadratic equation it has two roots. We call the larger one of these \phi \approx 1.618... and by definition the smaller one is \tfrac{1}{\phi}=\phi-1. Proclus tells us that it was revealed to the Pythagoreans by the Muse goddess who came to them. He then says that Plato recognized its importance and made conjectures on them which his mathematician student Eudoxus then proved using geometry. It is evidently this material of Eudoxus that is presented by Euclid — for example, he provides a means to construct \phi without calling it that in Book 2 of his Elements. Now this ratio has a special relationship to the isosceles triangle with angles \tfrac{2\pi}{5}, \tfrac{2\pi}{5}, \tfrac{\pi}{5}: in such a triangle the ratio of either of the congruent sides to its base is \phi. This Golden ratio triangle is central to the proof of the P.H.D identity.

penta_hexa_deca.proofFigure 3

Figure 3 depicts the construction required for our proof. We observe that the said pentagon of our identity is in green, the hexagon in blue dotted segments and the decagon in red. The radius of their circumcircle is the same as the side of the hexagon H. From the construction we see two instances of the above-mentioned Golden ratio triangles: One with sides H, H, D and another with sides D, D, 2x. From those triangles we can write the below:

\dfrac{P^2}{4}=D^2-x^2\\[10 pt] \dfrac{P^2}{4}=H^2-(H-x)^2=2Hx-x^2\\[10 pt] \dfrac{P^2}{2}=D^2+2Hx-2x^2\\[10 pt] \dfrac{H}{D}=\dfrac{D}{2x}=\phi \\[10 pt] x=\dfrac{D^2}{2H}\\[10 pt] P^2=2D^2+4Hx-4x^2 = 4D^2-\dfrac{D^4}{H^2}\\[10 pt] P^2=D^2(4-\dfrac{1}{\phi^2})=D^2\left(4-(\phi-1)^2\right)\\[10 pt] P^2=D^2\left(4-\phi^2+2\phi-1 \right)=D^2\left(3-1-\phi+2\phi \right)\\[10 pt] P^2=D^2\left(2+\phi \right)=D^2\left(1+1+\phi \right)=D^2\left(1+\phi^2 \right)\\[10 pt] P^2=D^2+\phi^2D^2=D^2+H^2 \quad _{\blacksquare}

If one were to assume the theorem of Brahmagupta or Ptolemaios regarding the diagonals and sides of a cyclic quadrilaterals one can prove it by an alternative path. We leave this for the geometrically inclined reader to work out.

icosahedron_P_H_DFigure 4

The remarkable discovery of the yavana-s was the relationship of this P.H.D identity to the icosahedron. Consider the icosahedron in Figure 4. By definition all edges of the icosahedron are equal and all faces are equilateral triangles. Now, the edges of the icosahedron are congruent to the sides of the pentagon P in the P.H.D identity. Thus, if one constructs the circle as in Figure 4 we get the circumcircle as in Figure 3. With this circle we can construct the sides of the decagon D as above, e.g. \overline{PQ}=D in Figure 4. Then, the relationship it has to the icosahedron is that \overline{AB}=\overline{PQ}=D. The 5 faces of an icosahedron each of which share a common edge constitute a pentagonal pyramid. The height of this pyramid is D (Figure 4). Now, by construction the radius of the circumcircle of the pentagonal bases of such pyramids in the icosahedron is of length H, i.e. the side of the hexagon in the P.H.D identity (Figure 4; \overline{BC}=H). Interestingly, if we visualize the icosahedron as being made up of two pentagonal pyramids stuck to a central band of 10 facets then the distance between the pentagonal bases of the two pyramids is H (Figure 4; \overline{QR}=H). Thus, the planar P.H.D identity is able to describe the icosahedron in 3D space and we can use it to construct this non-obvious Platonic solid. In order to prove this relationship between the P.H.D identity and the above landmarks of the icosahedron we have to effectively prove that \triangle{ABC} \cong \triangle{PQR} (Figure 4). They are right triangles and their hypotenuse and legs are respectively P, H, D.

penta_hexa_deca.GRD_icosaFigure 5

In order to prove that we have first construct rectangles inside the icosahedron like the one shown in Figure 5. From the above Golden triangle (Figure 3) and its relationship to the pentagon (used in the compass and straight-edge construction of the pentagon; see this note) we can see that such an internal rectangle of the icosahedron will be a Golden rectangle, i.e. ratio of its non-equal sides would be \phi.

penta_hexa_deca_icosahedroFigure 6

We then use this rectangle to make an orthographic projection of the icosahedral construction in Figure 4 on to a plane (Figure 6). In the right panel of Figure 6 we see how this Golden rectangle is placed with respect to key landmarks of the icosahedron. In the left panel we see the actual projection of icosahedral construction in Figure 4. This preserves the mutual relationships of \triangle{ABC}, \triangle{PQR} between 3D construction and the planar projection. We construct \overline{XY}. By examining this figure we observe \overline{XY} || \overline{AC}. Thus, \angle{ACB} \cong \angle{XYP}=\theta. From this, one can show that \angle{PRQ}=\theta. By the nature of the projection we preserve the equivalence of the edges of the icosahedron such that \overline{AC} \cong \overline{PR}. Thus, by Side-Angle-Angle test we can show \triangle{ABC} \cong \triangle{PQR} \quad _\blacksquare

Thus, we complete our journey through one of the profound aspects of Euclidean space — a thread passing through its defining feature the bhujā-koṭi-karṇa-nyāya, the three regular polygons and the regular polyhedron, the icosahedron. When we were young, on account of our fascination for viral capsid structures, we closely studied the geometry of polyhedra by practical means, i.e. making them out of paper. It was in course of this we learnt of the P.H.D identity. We were able to practically confirm that for ourselves based on our models. However, it took us some time before we actually apprehended the proof.

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Leaves from the scrapbook-4

As described here these entries are from the scrapbook of the second of the caturbhaginī, Vrishchika.

Entry 1; kāmāturā, year Siddhārthin of the first cycle: Indrasena departed for grad school last week and I am feeling a bit of loneliness of a kind I did not even feel when Lootika left. In the midst of all this I did not even register the fact that I have completed my basic medical degree in good time. But dear my agrajā’s words come to mind that the true prowess of a person becomes apparent only when they prove themselves all on their own against hostile forces. I was also reminded of her admonishing me that I was way too swayed by sentiments of attachment and eros. But sometimes I feel concerned that she might have taken that too far — she has totally lost all contact with Somakhya for an year now. At least now Somakhya has Indrasena beside him but my pretty agrajā is all by herself battling hostile mleccha-s in the quest for glory.

Thankfully, due to the favors of the gods I am likely to get any internship I would like, but that is just practice. It goes without saying that I would be pursuing experimental medicine for that was how I always supposed to be doing. In any case, I have got a waiver to already start accompanying and assisting my senior Vidrum on the rounds till my program formally opens. I guess he harbors some edginess towards me right from school days but still tolerates me because of all the curricular help I have rendered him over the years and his friendship with Lootika and Somakhya. One of my reasons for writing this entry was an interesting case that confronted us last week — it brought a closure to me in an unexpected way. A young mleccha man arrived with acute cardiac symptoms. He had a generally culprit-free angiogram but rapidly deteriorated and died a couple of hours after admission. Since it was an unusual case we pursued it further for it might teach us something we did not know. One thing which became apparent was that he did not show any elevation of cardiac-specific troponins in the blood. However, his cardiac muscles showed a dramatic fall of calcium ions in the sarcoplasmic lumen a sign that his SERCA calcium pump was compromised. Yet, it was totally normal in sequence.

An interview with his parents who was accompanied by an official of the embassy of the said mleccha country revealed something interesting. They were working for a NGO for the “inculcation of scientific temper among Dalits and tribals”. I wondered what that could mean. They were full of pride on their son and declared that he was a great genius from childhood and a “Do it yourself” scientist. However, they said he had a strong suicidal tendency; hence, due to certain episodes they had brought him over from their country to stay with them for sometime and cool off. They showed me some of his notebooks, which indicated that he was indeed some kind of a polymath with ranging interests and insights. Moreover, he was a DIY molecular biologist. That stirred something in me for it reminded me of none other than myself and my sisters. The scan of his notebook revealed that he had engineered a variant of the SERCA inhibitory peptide phospholamban, which was effective in shutting down SERCA in the heart. A closely analysis of his tissues revealed that it was by means of that peptide he had sent himself to king Vaivasvata — he thought no one would ever figure out that it was a suicide. Via Vidrum I provided a report to his parents and the official and closed the case.

I was struck by the strangeness of all this. His death reminded me that often a great tree grows in a jungle and then one fine day is struck down by the indra-senā or the wrath of the pitiless Vāyu as our Iranian cousins would have said. Thus, indeed this young man had seen many realms of knowledge but no one knew that he had reached so deep for he had fallen like that tree in a lonely boreal forest with no one to hear the crash of its fall and thereby judge its great mass. But the case brought to something more sinister to my mind: once when we were young and had expressed our svābhāvika-siddhi-s for the first time Lootika and I had made the ghost of a brāhmaṇa speak. He prophesied my vīra but at the same time mentioned that a prativīra would also emerge. But the ghost added that we would not be affected by the prativīra for he would attain Vivasvān’s mace-wielding son even before he could engage in a contest with us. I realized that the individual was the prativīra and that he had passed away just in front of my eyes. That also gave me a signal because Indrasena had told me via the kauberī-vidyā that after the prativīra has departed we will face the true contests of our life even as I proceed towards the bus stop at Mlecchadigdvāravṛti.

Entry 2; rakta, year Siddhārthin of the first cycle: Things seemed to precipitate by the day as though drawn by fate inexorably towards a certain preferred unfolding of events things. Or at least that is how the human mind tends to rightly or falsely pick patterns from the noise in the background. I was part of the team attending to a trauma case involving activists of the mandira vahīm̐ banega group who were holding a meeting for mobilization for the reconstruction of the temple for the Ikṣvāku heroes at Ayodhyāpuri. As they were proceeding to their office beside the Vināyaka shrine on the Somamārga for a meeting they were attacked by assailants who had vanished almost without a trace. The eyewitness declared that the shots came from the snaking queue, which was awaiting a darśana at the shrine and that the assailants created some chaos in the line by shooting in the air and took advantage of that to merge into the crowd and escape. Some even insisted that the assailants wore saffron robes and were actually worshipers at the shrine. In any case the mayhem left one of the activists with a bullet lodged in his left clavicle. We were able to quickly stabilize his wound and save his life. However, the secretary of the mandira vahīm̐ banega committee had succumbed to his injuries by the time he was brought to us and the only task we were left with was the autopsy. He had two bullets which had entered his skull through the occipital region — one had exited through the right frontal while the other was lodged in the brain.

There was some discussion after the autopsy which involved one of our seniors the resident Samikaran and Vidrum’s new friend, another resident Marxenga Sen. They both remarked that his death was probably fully deserved because he was there to cause trouble. They added: “After all had the Chief Justice Shashiyabh not declared that such people do not want the country to ever be in peace.” Duly, I did not pass the opportunity to rile them by stating that it was not the mandira vahīm̐ banega group but their mustache-less bearded friends who were the biggest cause of not just the country not but the world not being at peace. They went on like a person on bhaṅga or a song playing in a loop on how the assailants were certainly Hindu terrorists as they were saffron-clad and visiting the temple and that it was an internal gang-fight among the Hindu terrorists. I quickly made my exit wondering when these might join the beards or the well-known socialist terror movement.

Just then my mother had called me and asked me to catch sleep in the residents’ lounge. She did not want me coming back home that night because apparently there was an operation to apprehend Saif ad-din as-Sullami on way between the hospital and home. She said that she would pick me up in the morning. That name sounded vaguely familiar; so, I looked him up and found that he was known to be a peddler of a flavor of music called sufi-pop, which accompanied gyrating dances that were popular with a section of the educated youth with a proclivity towards debauchery. There was a rumor in certain circles that he was also involved more serious stuff like using his musical outlet for love-jihad and running a hitman service for a national and international clientele of criminals. It was seemed that he was behind the shooting that took place earlier in the day and had also played a big role in the UCC riots. I wondered his men were among those who had tried to kidnap me last year during the UCC riots.

As that scary incident flashed in my mind I took out my knife and garala-śaṅkula and placed them on my desk. Jhilleeka had improved the design of the garala-śaṅkula such that it was less-likely to jam as it had done during my encounter with the marūnmatta-s. I even have a third weapon, the śilī which Indrasena had gifted me along with the instructions of its use. It is a small peg which can be used to impinge on particular points that were termed marman-s in the old Hindu medical tradition. Our ancestors held that: “mārayantīti marmāṇyucate ।”; hence, pressing at those points with the śilī could in the least incapacitate the attacker. These points of the old tradition are mostly a secret and mentioned in the old medical literature. In Cerapada there is also a legend that the twin Aśvin-s appeared on earth to teach the secrets to their kṣatriya-s to counter their drāviḍa cousins to the east, while the latter were favored by Bhava. I realized that one could fairly precisely strike these marman-s by locating certain superficial blood vessels, fascia, tendons and aponeuroses, which tradition called sīrā- māṃsa- snāyu- and saṃdhi-marmāṇi. I had a good sense of their location from actual anatomical examination. I also felt I was much better than my dear agrajā in wielding these implements and could actually cause some serious damage even if I were to be completely overwhelmed by a male assailant. But perhaps this is only feeling arising from the human tendency to overestimate oneself.

Entry 3; rasa, year Siddhārthin of the first cycle: On my way to college I pass by a little park, which originally had a small shrine in the middle of it. Several years ago out of curiosity Varoli and I went to see what was there and found it to house the statue of a man called Rasgol-bābājī. There, a woman told me that miracles could be witnessed by praying to the said bābājī. She explained that Rasgol-bābājī had the power of a 1000 Sai-fakīr-bābājī-s, a 100 śrī-śrī-108-bābājī-s and 10 das-ser-jigar-bābājī-s. Even as I was taking in the magnitude of the hepatomegaly of the last of them, she exuberantly mentioned how bābājī used to appear in her dreams to warn her of all manner of impending disasters. She then went on to say that his student Khambhā-vālā-bābājī was now in charge and that he would occasionally visit the park to dispense miraculous medications that could “cure any disease”. Sometime later I noticed that a new shrine had emerged next to that of Rasgol-bābājī. Apparently, his successor Khambhā-vālā-bābājī had joined his master in the abode of the buffalo-rider and was now consigned to a samādhi beside him.

Vidrum had been informing me of the multiple cases of lead and mercury poisoning he has been encountering in the past couple of years. We checked the data an realized that there was indeed a significant spike. We wondered what might be cause for those cases as our state’s environmental agency had given a clean-chit for our water and food sources. The answer dawned on us with a crash yesterday. We had a “god-man” rushed to our care by his throng of devotees. They claimed he was under an abhicāra from his rival śūl-vālī-mātājī. They added that he could cure himself but his symptoms were so severe that they need us to give him some medication so that he could become stable enough to counter the abhicāra or prepare medications for himself. He was not cooperating constantly stating he could cure himself but was clearly in deep agony. His cheeks were red and he was complaining that scorpions were constantly running under his skin. Reading my name on my badge he became delirious and started yelling that the scorpions running under his skin had suddenly manifested a dangerous woman. Before I could even process what happened next, one of his votaries rushed at me and started pulling my hair while another grabbed my hand. Thankfully, Vidrum who was beside me acted quickly and knocked out the guy who was grabbing my hand. That gave me a breather by which time I had drawn my śilī and bore it down on the exposed sīrāmarman on the attacker’s hand. His hand went limp and I freed my hair and struck him again on his skull with śilī thus making him fall. Vidrum and others signaled me to retreat as they handled the case. The scary memory of the murder of a physician near my school by the irate relatives of a patient flashed in my mind but in the adrenaline rush I was feeling elated as I had never felt recently — I had knocked out an attacker successfully with the śilī! In my excitement, I sent off mails to Indrasena and my sisters describing the incident and asked them not to tell my parents.

Later that day, I heard that Vidrum had referred the god-man to the poison-control department. There the god-man’s anuyāyin-s had caused quite a bit of trouble and took their man away after the department told them that they had to stop him from using mercury and lead and that he had to remain in the hospital for an attempt at chelation therapy. Vidrum said that he was considering reporting the man to the poison-control authorities because he believed he might be behind the metal-poisoning cases we had seen. Out of the blue, Marxenga came to me and started haranguing me that “right wing” types like me were the cause of emergence of likes of that god-man. Vidrum for a change shooed her away. He told me that the supposed holy man was a certain Gajpuṭ-bābājī, the student and successor of the late Khambhā-vālā-bābājī. I looked him up and realized that he ran a popular channel that was frequented by many believing in his miraculous “medicines” for whitening the skin, curing malignancies, diabetes and the like. Vidrum asked me regarding his peculiar name. I explained to him that evidently he had taken it up from a traditional kiln known as the gajapuṭa, which among other things was used in baking mercurials. This led me to investigating his channel and I saw that he had a show in which he described the preparation of leaden, golden and silver rasagola-s for people to wear. There he took elemental mercury heated it successively with the juices of garlic, aloe, wood-sorrel, betel leaf and the triphala-s, each for the duration of a week in his famed gajapuṭa that had lent him his name. At the end of it he claimed that he had pacified the mercury and started handling it with his bare hands mixing it with powdered silver. Then he squeezed out the mercury remaining from the amalgamation and continued repeatedly heating it with the pastes of various herbs prepared with gomūtra, ajamūtra, nārīmūtra of a menstruating woman and milk. At the end he arrived at an amalgam globe that he advised his devotees to wear or keep in their mouth as a panacea for all kinds of diseases. With that it was clear how our failed rasa-siddha had landed up in our hospital and had probably also sent many of his votaries over too with his mercurials, irrespective of the powers of his rival śūl-vālī-mātājī. I wait to learn more of his fate and that of his enterprise.

Entry 4; vi, year Siddhārthin of the first cycle: A lecturer in the geology department at the university called on me. He was known to Somakhya and had been friendly with him. It was through him he had a passing acquaintance of Lootika and me. He had with him a visitor from Argentina. He had come to ask me if I could somehow arrange to use the CT scanner in the hospital for scanning a rock with a fossil that he had found at Utnoor in Telangana. I asked him for more details. He told me that the Argentinean was an expert on such extinct reptiles and believed that the skull he had found was notable. The Argentinean told me that he believed it was fossil of utmost importance and a relative of the dinosaurs. I took a closer look and could make out the partial contours of a skull. It looked too primitive to be a dinosauromorph or a pterosaur. I realized it might be the skull of an aphanosaur and remarked so. The Argentinean was surprised and he said that nobody he had encountered in India till now had known about aphanosaurs. I told him that I immediately realized its significance because it promises to be the first complete skull of an aphanosaur and we could test the hypothesis of their being close to the ancestry of the greater dinosaurian lineage. I agreed to help them with arranging the scans on condition that they would give me a chance to study the skull too. They agreed to include me as an author on the study. I am sure Somakhya would be as jealous as he can ever be of me when he hears of my luck in this regard.

 

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