## The Rāmāyaṇa and a para-Rāmāyaṇa in numbers-II: Evolving early Indo-Aryan warfare

This article might be read in as a continuation of this earlier one. The methods/caveats mentioned therein apply here too. Some of the counts mentioned in this article might be approximate but should be generally in the correct range, i.e. errors < 15%.

The Ṛgveda is the oldest extant Indo-European text (This position of ours is contrary to that of most mainstream western academics and their imitators who privilege Hittite texts as the oldest extant IE text. While we hold the view that proto-Anatolian was the first Indo-European language to branch off we do not think that the Hittite texts from West Asia are the earliest extant expressions of the IE people). While the RV's primary concerns are the rituals of the ārya-s directed to their gods it incidentally preserves several features of early Aryan life. We can clearly see that cows, chariots and horses were of enormous importance: An approximate count indicates at least: 354 sentences mentioning the horse by its common name aśva; 586 sentences mentioning the cow by its common name go; 639 sentences mentioning the chariot by its common name ratha. On the other hand common words for a dwelling amount to 167 (gṛha; 97; dama 53; chardis: 17). Place names are very rare, while rivers, seas, pastures, mountains, trees and forests find common mention. Our ancestors offered a seat of grass to the gods, barhiṣ, mentioned at least 155 times, a practice we continue to follow to this date in our rituals. They had continuously lit fires into which oblations were made. These features emphatically indicate that they were a mobile people living in higher latitudes in grasslands with great rivers and gigantic water bodies that qualified as seas. We even hear of fire within the sea. Importantly, they were a war-like people. Common words indicative of conflict occur at least 572 times: the root yudh indicating battle (71); samar- indicating military encounter (28); pṛtanā, battle (97); śatru, enemy (98); different kinds of weapons specific or generic (278). This count of weapons excludes the weapons used by the gods like vajra, didyu and the like.

Figure 1

In order to understand early Indo-Aryan warfare we need a closer look at these weapons. Figure 1 shows a breakdown of various implements of war in the ṛgveda. Of these several are generic references to weapons: āyudha (weapon; 60); heti (missile; 11); vadhar (killer weapon; 11). There are 23 references to senā which could mean a missile or an army. Beyond these there are specific references to particular weapons. The maximum number of references are to bows (dhanus) and arrows (74). The most common words for arrows are: 1) the word of proto-Indoeuropean vintage śara, the cognate of English arrow and iṣu, which is shared by the Greco-Aryan clade of Indo-European. The former may occur in derived forms like śarva or śaryhan. The word cāpa or kārmuka for bow and bāṇa for arrow, which are common in the later dialect of Sanskrit, are respectively nonexistent or found only once (that too in a potentially late mantra). The divine weapons known as astra-s which are common in the itihāsa and purāṇa are not mentioned in the RV, though the term brahma-saṃśita for an arrow does imply the same kind of weapon. A rare word bunda for arrow, probably of non-Aryan origin is used only twice in the whole RV by the Kāṇva-s. The RV does offer us some details about the arrows used by the early ārya-s. For example, we know they used both horn and metal arrow heads; the latter in particular appear to have been combined with poison. The horn arrow heads were lashed to the shafts with leather strips.

After bows and arrows, the next most frequently mentioned weapon is the ṛṣṭi, a lance that was primarily used for jabbing. It seems to have been transferred to Dravidian as īṭī. The śakti was a comparable weapon which was hurled but it is rather infrequent in the RV. Then two types of axes are mentioned the vāśī and the paraśu. The vāśī has been a bit of a mystery. Recently, Finnish indologist Asko Parpola, otherwise known for some outrageous theories on Indian prehistory, proposed rather reasonably that it might be identified with a related set of weapons found at Mohenjo-daro in the Indus valley, Tepe Hissar III in Iran and Zeravshan in Central Asia. He also notes that the word was transmitted early on to Dravidian as vacci – a very plausible hypothesis. Based on these identified we suspect that this weapon was indeed a battle-pick that latter went out of vogue among the Indo-Aryans. In contrast, the word paraśu of PIE provenance seems to have persisted and gained in importance over the earlier vāśī. Another triad of weapons, the cakra (the discus), the aṅkuśa (the hook) and the pāśa (the lasso), find mention in the RV and continued to be used in steppe warfare through the Mongol period. The cakra was widely used in India down to early modern times (e.g. mahārāja Ranjit Singh and Rājpūt-s). There are medieval accounts of how skillful cakra wielders like rājpūt-s could slice limbs with it. The latter two were effective in striking at chariot warriors or cavalry from infantry positions. The word varman meaning body-armor indicates that the Indo-Aryans wore protective gear in their battles. The word is sometimes juxtaposed with śarman meaning helmet (homologous “helm”) but this word is not counted here because it is also very frequently used as a metaphor for protection offered by the gods. They also wore a hastaghna (arm-guard) which protected the arm from the released bowstring.

Notably, the words for swords (niṣaṅga and asi) are rare. Indeed some have even questioned if niṣaṅga even originally meant a sword. Another word sṛka went out of vogue in later times but from the context it clear that it had a slashing edge. It could be interpreted as a billhook-like weapon. This rarity of the sword is a clear indicator of a very early age of the RV when most fighting was done from mobile chariot platforms, and probably to a lesser extent from horseback or the foot. While some close contact might have occurred in combat when weapons like the axes might have been used, most fighting focused on deploying projectile weapons and probably lances from the chariot platform while in motion. Thus mobility and volume of fire were one key elements in the RV warfare. On the other hand the text mentions at least 26 times pūr and at least 31 times varūtha meaning forts and fortified positions of both the ārya-s and their enemies. We also believe that the Varūtrī goddesses were guardian deities associated with such fortifications (perhaps leading to the later Durgā). The RV talks of demolition of fortifications of enemies. Elsewhere in the Veda (e.g. the legend of the destruction of the Tripura by the god Rudra) we hear of siege warfare. So a second aspect of early Aryan warfare was defensive use of fortified positions and offensive operations to take such forts. This feature remained an constant feature of Hindu warfare until the destruction of the last Hindu empire by the English, a conflict which featured destruction of ancient forts by the latter.

While across the sphere of their spread the Indo-Europeans settled in their new lands succeeded by their tradition tactics developed on the steppes there were also shifts in their methods of warfare and preferences of weapons. Thus, we witness a mixture of old survivals and new features. In Greece the older element is seen in the form of the importance of the poisoned arrows of Herakles, and skilled archery of Teucer, Philoctetes, who inherited Herakles’ missiles, and Odysseus. However, the heroes of the Trojan war predominantly use javelins and swords as infantry fighters indicating a shift in method. In Rome too archery declined to a degree, which in the end proved to the detriment of the Romans in some of their encounters with the Iranians, where whole legions were crushed. We see similar shifts among the Celts, where in the epic of Cúchulainn we find the rise in importance of the spear, sword, staff and lasso.

To look at the shifts in India we turn from the RV to the Rāmāyaṇa of Vālmīki. It is a great battle epic clearly from a time much later than the RV. By this time the ārya-s had settled firmly in India and had completed the first phase of their expansion across the subcontinent. While to the casual observer the fights of the Rāmāyaṇa with the apes and rakṣas-es might appear fantastic much of the warfare recorded therein has a very conventional Indo-Aryan system to it. It shares with the Greek Iliad the amphibious assault on the enemy position. However, in the Hindu case the assault it self was rather conventional involving a land army and a causeway across the water. In other ways it retains a lot of the elements of the old ārya warfare: the predominant use of archery, the great chariot fight between Rāma and Rāvaṇa, the invasion of the fortified positions of Lankā along with firing of the fortified city (c.f. the god Agni destroying the forts of the dasyu-s for the Pūru warriors in the RV). However, when we look at the actual weapons listed in the Rāmāyaṇa we note the emergence several distinctive new features (Figure 2).

Figure 2

The conservative element of warfare in India and greater Iran is illustrated by some of the old methods remaining strong; this is seen in the dominant role of archers in Rāmāyaṇa. Further, bows and arrows remain the most commonly mentioned weapons although there are interesting changes in the words for them. The word śara remains the most common word for arrow but the old word iṣu which was as frequent in the RV now fades away in the R. Notably, bāṇa rises in frequency to be second most common word for arrow in the R. Similarly, the words cāpa and kārmuka, which are unknown in the RV, become common if not as frequent as the old dhanus in the R. Given that we have established archery to be central to old Indo-Aryan warfare, one may ask regarding the origin of these new words bāṇa, cāpa and kārmuka. We believe that these words were likely acquired by the ārya-s from the earlier inhabitants of India, perhaps the original people of the Sindhu-Sarasvati region. The importance of archery to Indo-Aryan warfare was what probably allowed the survival of these words within Sanskrit during the Aryanization of the Sindhu-Sarasvati region. Interestingly, the Yajurveda saṃhitā-s preserve two peculiar words for bow: dālbhūṣī (Kaṭha-saṃhitā) and drumbhūlī (Maitrāyaṇīya-saṃhitā). Their relationship is evident but they are noticeably different between the two saṃhitā-s and clearly have a non-Aryan origin. This difference in pronunciation in the two related texts indicates that ārya-s had acquired a native word and were trying to render it as closely as possible in Sanskrit. This suggests that the original word might have had a form like d-x-bhū-x-ī. We suspect that this was the word for bow in the naiṣāda language which was spoken by the people of Niśāda chiefs like Guha of R and Naḍa Naiṣidha (the original form of the name Nala Naiṣāda in the itihāsa) in the śatapatha brāhmaṇa. Memetic retentions of these early Aryanization events of tribals might also be seen in the survival of a form of the aindra religion among the Rathva tribes. Indeed, the tribal groups with good archery skills might have been quick to Aryanize as those skills provided them with military employment in the Aryan system.

We also note that the R specifically mentions several different types of arrows such as the: bhalla- an arrow with a large heavy head; ardhacandra- with a crescent head; kṣūra- with a blade-like head; śalya- with a single point head; añjalika- with a broad head; naraca – a short bolt-like arrow. Such heads are also seen in later Indian and Mongol warfare. For example, Timur was slashed using arrows similar to the ardhacandra or the kṣūra which led to his limp. Finally, the we have the frequent mention of the term astra. Beyond meaning a missile it also implied special weapons presided over various Vedic deities and other entities like snakes. It is possible that some of these were ultimately special physical weapons – earlier versions of the incendiary, explosive and poison weapons mentioned in the Mauryan age Arthaśāstra of Kauṭilya. Consistent with the use of such weapons in the Mauryan age we have an account of Apollonius of Tyna recorded by his biographer Flavius Philostratus of Athens. He says that if Alexander had penetrated beyond the Vipāśā river he might have not been able to take the fort of Indians even if had 10000 Achilles-es and 30000 Ajax-es with him. He then narrates a tale of how Herakles of Egypt and Dionysus were defeated by such weapons from this fort during their invasion of India:

“It is related, anyhow, that Herakles of Egypt and Dionysus after they had overrun the Indian people with their arms, at last attacked them in company, and that they constructed engines of war, and tried to take the place by assault; but the sages, instead of taking the field against them, lay quiet and passive, as it seemed to the enemy; but as soon as the latter approached they were driven off by rockets of fire and thunderbolts which were hurled obliquely from above and fell upon their armor. It was on that occasion, they say, that Herakles lost his golden shield, and the sages dedicated it as an offering, partly out of respect for Herakles’ reputation, and partly because of the reliefs upon the shield. For in these Herakles is represented fixing the frontier of the world at Gadira, and using the mountains for pillars, and confining the ocean within its bounds.” 2.33; translated by F.C. Conybeare.

The old lasso (pāśa) still remains prominent and so does the battle axe (paraśu; also paraśvadha) but the vāśī has entirely become otiose. A notable change from the RV in the R is the rise of prominence of the sword now going by at least 3 unambiguous names asi, khaḍga, and nistriṃśa. The sword is used prominently in the conflicts described in the text and there is no doubt it was rising in importance as a weapon. Yet it is clear that it has not attained that exalted position it accorded in the ākhyāna of the sword seen in the Mahābārata. Importantly, together with the sword we see the leather shield (carma), which indicates the classic mode of Indian sword combat involving parrying with the shield and slashing and thrusting with the sword had fallen in place between the days of the RV and R. We also witness the rise of a new weapon paṭṭiśa whose interpretation is confused. Some have taken it to be an axe-like weapon i.e. a halberd which commonly depicted as a weapon of Rudra and Skanda on early Indian coins. Others based on its etymology (paṭṭi band or strap) interpret it to be a sling which was widely used in ancient warfare. Yet others based on the medieval name (daṇḍapaṭṭa) interpret it as a flexible sword. It has indeed been used in this sense by the chroniclers of the great rājan Śivājī. Now the epic and paurāṇika accounts describe warriors cutting heads with the paṭṭiśa. This indicates that it was a sharp-edged weapon. Whereas Cāṇkaya groups it with the axe suggesting its identification with a halberd, we have an unambiguous medieval description:

paṭṭiśaḥ pum-pramāṇas syāt dvidhāras tīkṣṇa-śṛṅgakaḥ ।
hasta-trāṇa-samāyukta-muṣṭiḥ khaḍga-sahodaraḥ ॥
The paṭṭiśa is of length of a man, is double-edged with a sharp tip.
Its handle has a hand-guard [and is] called the brother of the sword.

This description unambiguously indicates that it was indeed seen as no different from the daṇḍapaṭṭa used by the Marāṭhā-s. Hence, we may provisionally identify it with the same.

The next major development in the R with respect to the RV is rise of numerous club-like weapons. The most prominent of these are: the gadā- the mace; parigha- club with round head; musala– pestle; tomara- battle-flail; mudgara- battle-hammer. The gadā, like the cakra also has an Iranian cognate gadhā but is not seen in the RV. This is a notable point that would be discussed further more generally below. Some translators see the vajra of Indra and other gods as a mace (e.g. Jamison and Bereton translation of Ṛgveda). In support of such an identification one may point to the use of the cognate Iranian word gurz

As with clubs the R also shows a proliferation of spear-like weapons. The ṛṣṭi recedes into anonymity while its place is taken by the śūla. The only mention of the śūla in the RV is not as a weapon but as a sacrificial spike. In R its displacement of the ṛṣṭi suggests that it was used in a similar capacity as a jabbing pike. The śakti greatly rises in prominence as a hurled javelin. The new spear-like weapons include the frequently mentioned prāsa which from medieval sources is known to be over 2 meters in length with sharp points at both ends. From the account of the battle fought by Rāvana’s son Narāntaka against the army of apes, who is described as riding his horse like the god Kumāra his peacock, it is clear that the prāsa was used from horseback. The other new weapon finding rare mention in the R is the kunta which from medieval sources is a lance with a multi-flanged head.

Finally, this brings us to two weapons the bhindipāla and the śataghni. The former is described thus in medieval sources:

bhindipālas tu vakrāṅgo namraśīrṣo bṛhac chiraḥ ।
hasta-mātrotsedha-yuktaḥ kara-sammita-maṇḍalaḥ ॥
tribhramaṇam visargaś ca vāmapāda-puras saran ।
pādaghātāt ripuhaṇo dhāryaḥ pādāta-maṇḍalaiḥ ॥
It is an arm’s length and its circular part is a span in diameter.
It is released by whirling three times and placing the left foot forward.
It is held by infantry array and slays the enemy by breaking the foot[soldiers].

From this it is clear that it was a sling or a bullet thrower-like device that hurled stone or metal bullets similar to those that have been found in the Bhita excavations or at Roman battle sites.

The śataghni is somewhat more enigmatic. It already finds mention in the Taittirīya āraṇyaka of the Yajurveda. From some epic descriptions it is a spiked mace – thus the term śata-ghni should interpreted as a killer [weapon] with a hundred [spikes] – a description explicit in the Mahābharata. It is however also mentioned in the context of yantra-s on fortifications right from the Rāmāyaṇa. Further, they are conceived of being of large size as trees. For e.g. in the battle fought by Kumbha the son of Kumbhakarṇa we hear:

abhilakṣyeṇa tīvreṇa kumbhena niśitaiḥ śaraiḥ |
ācitās te drumā rejur yathā ghorāḥ śataghnayaḥ || R 6.63.33 (“critical”)
The trees (hurled by the ape Sugrīva) studded by the sharp arrows aimed and shot by Kumbha shone like terrible śataghni-s.

Descriptions such as these indicate that the śataghni-s also meant large multi-spiked bolts which were hurled from fortifications. This might hence justify the alternative interpretation of the name as śata-ghni, hundred-killer.

The above observations indicate that despite the overall conservative retention of old Aryan military technique and technology by the Indo-Aryans, between the RV and R, there was a clear shift, which in part reflects the peculiar requirements of the conflicts in forest India and the island stronghold of Rāvaṇa, and in part actual changes in military technique. Thus, the philological shifts which we detect in the R reflect both changes on the “on the ground” as well as the new environment of the ārya-s. This is seen in the form of the greater use of certain words for weapons of Indo-European or Indo-Iranian vintage as well as adoption of non-Indo-european terminology. How do we interpret these changes?

First, we know from the historical period that Greeks, Iranics and Indians served are mercenary fighters in each others armies often even against their own coethnics. Likewise, during the rise of the first Mongolian Khaganate of the Huns we find interactions across ethnic groups like the Altaics and the Iranics due to hostage taking and apprenticeship. This subsequently manifested as mixed hordes, which we observe among the Huns, combining Altaic and Iranic elements. This continued down to the Chingizid times where the great Kha’khan amalgamated armies of Turkic, Tungoosic, and remnant Iranic groups with his own Mongol hordes. These phenomena can hence be extrapolated back to the early historic/prehistoric times among the Indo-Europeans. Thus the deployment of multi-ethnic armies would explain how several words for weapons from non-IE sources were absorbed into Sanskrit. While some of these might have happened in the steppes and their border zones, as posited above the remaining likely happened as the ārya-s absorbed the older Sindhu-Sarasvati and niṣāda peoples in India. This we suspect explains the rise in prominence of words like bāṇa, cāpa and kārmuka.

Second, in the historical period India witnessed the invasions of Iranic Shakas, Pahlavas and Kushanas who were linguistic and religious cousins of the Indo-Aryans. Even earlier there was the invasion of the yavana-s and later the Huns. Most of these groups had a degree of religious and linguistic relationship to the Indo-Aryans. Thus, we can also extrapolate this scenario to prehistory where multiple religiously and linguistically related groups invaded India in separate waves. We observe faint linguistic echoes of this in the form of the precariously poised Kalasha who were distinct branch within the Indo-Iranian clade. The kentum substratum in Bangani hints at other such more distant waves. Finally, philological evidence in the form the śalva-s and the Pāṇḍava-s, and the rise of the kaumāra religion also indicate that the Ṛgveda ārya-s where not the only Indo-Aryan group to invade India. There were others that followed which, while Indo-Aryan, appear to have likely retained a closer proximity to the Iranic branch. A legendary motif also supports such a link: The greatest hero of the old Zoroastrian tradition in the Avesta was Keresāspa who likely had a high position even in the greater Iranic world. He was a warrior priest associated with what it is today Afghanistan who is said to have killed a gigantic Śuśna-like demon as well as the gandarewa (Iranic cognate of gandharva). Notably, his Indo-Aryan cognate Kṛśāśva does not appear in the Veda which is temporally closer to the Avesta. However, he appears in Rāmāyaṇa as a major even if only fleetingly mentioned figure. In the R he is described as the father of all weapons which he gave to Viśvāmitra who in turn gave them to Rāma. Interestingly, on the Iranian side too Keresāspa possesses special mighty weapons. Later Iranian tradition remembers their mighty hero Raosta-takhma (later Iranian Rostam/Rustam) receiving the weapons of Keresāspa with which he performs great deeds even as Rāma in India. We posit that this motif was carried by a secondary wave of Indo-Iranians entering India after the earlier ārya-s of the RV. It is the military influence of these later waves that likely caused the dominance of the use of śara over iṣu and also made the gadā a prominent weapon (incidentally also the favored weapon of Keresāspa). It is of interest to see if molecular evidence from ancient DNA might in anyway corroborate the multiple invasion model. In the least it does place support in favor of a relatively early entry of the first Vaidika ārya-s into India.

Lastly, we may note that, like with agricultural terms, there is hardly any evidence for loans from early Dravidian into early Indo-Aryan. However, we do find them in the opposite direction. This does strain the view that the Dravidians were the pre-Aryan occupants of the Sindhu-Sarasvati-Ganga region. Rather the Dravidians either invaded later and separately via a southern route or always occupied a southern position and came in contact with the ārya-s only later. Whatever the case, the weight of the evidence suggests that the early Dravidians upon contact with the ārya-s were quick to adopt similar military and organizational strategies. It even appears that they too might have been pastoralists with some proclivity for mobile archery. Thus, they seem to have become part of the “Aryan-system” in the subcontinent right from the inception, which in part allowed them retain their distinctness unlike the original SSV peoples. However, this is necessarily an indirect inference because the Dravidian sources themselves are of much later provenance in the form of the earliest Tamil poems like the Puranānūru.

Posted in Heathen thought, History |

## The square root spiral and the Gamma function: entwined analogies

The topic discussed here is something on which considerable serious mathematical literature has published by P.J Davis, W. Gautschi and others. This partly historical narration is just a personal account of our journey through the same as a non-mathematician. As for the detour on the Gamma function its allure has been so much that everyone from the ordinary college lecturer to the Fields medalist Manjul Bhargava has written something about it. Its invisible hand is felt in science especially in the form of various statistical distributions that show up in as disparate phenomena as the distribution of the number of molecules of widely expressed proteins or the distribution of positions in a protein’s sequence evolving at different rates. Thus, we could not pass up the opportunity for a little mention of it.

The story begins with a problem which one might have encountered in the context of elementary geometric puzzles that one is asked to solve as a youth: If you have a circle of radius $R$ then how do you dissect it into $n$ concentric circular rings of equal area using just a compass and straight edge? Of course the first “ring” would be disc rather than a ring. It is achieved by means of the below construction (Figure 1):

Figure 1

1) Let point A be the center of the circle of radius $R$ which we seek to dissect into $n$ rings.
2) Construct a unit of length of $r=\tfrac{R}{\sqrt{n}}$.
3) Draw $\overline{AB}=r$. Then draw a segment of the same unit length $r$ perpendicular to $\overline{AB}$ at point B. Join A to the free end B1 of this new segment.
4) Draw a further segment of length $r$ perpendicular to $\overline{AB1}$ and join its free end to A.
5) Repeat this procedure $n$ times; $n=17$ in figure 1. Draw circles with A as center with each of the segments beginning with A as their respective radii. This completes the required dissection.

One can see from the figure that the construction results in a sequence of side-sharing right triangles whose hypotenuses increase successively as the square roots of the natural numbers. Their legs of unit length trace out a spiral path. Hence, this figure might be termed the square root spiral (SRS). One may also see that as the number of dissections $n\rightarrow \infty$ we are left with thinner and thinner rings that are essentially the circumference of the circle: $\tfrac{\text{d}A}{\text{d}r}=2\pi r$.

In the above construction we used $n=17$ because adding one more unit results in the spiral going past the first turn (Figure 2). Thus, 17 is the whole number in turn 1; 54 in turn 2; 110 in turn 3. Now, based on this number 17 it has been claimed on flimsy evidence that this figure might have been constructed by Plato’s teacher Theodorus in Greek antiquity. Plato, with his characteristic eye for interesting mathematics, records the following conversation (at ~360 BCE) between Socrates and his friend the mathematician Theaetetus who laid the foundations of some key aspects of Greek mathematics. Here Theaetetus is telling Socrates regarding his study of square roots with another mathematician Theodorus (Theaetetus, Jowett translation):

Theaet.: Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit: he selected other examples up to seventeen — there he stopped. Now as there are innumerable roots, the notion occurred to us of attempting to include them all under one name or class.
Soc.: And did you find such a class?
Theaet.: I think that we did; but I should like to have your opinion.
Soc.: Let me hear.
Theaet.: We divided all numbers into two classes: those which are made up of equal factors multiplying into one another, which we compared to square figures and called square or equilateral numbers — that was one class.
Soc.: Very good.
Theaet.: The intermediate numbers, such as three and five, and every other number which is made up of unequal factors, either of a greater multiplied by a less, or of a less multiplied by a greater, and when regarded as a figure, is contained in unequal sides;-all these we compared to oblong figures, and called them oblong numbers.
Soc.: Capital;…

It is clear from what Theaetetus tells Socrates that they were talking about dividing all natural numbers into perfect squares like 1, 4, 9, 16, … which yield a root which is measurable as units and all other numbers in between which do not. These Theaetetus considered oblong numbers for they needed unequal factors to constitute them. The question is when Theodorus wrote out those numbers to illustrate the non-perfect squares why did he stop at 17? It has been speculated that in establishing their being “incommensurable by the unit” Theodorus had made use of a construction as in figure 1 and he stopped at 17 because he had reached maximum number of natural numbers that can be accommodated in a single turn of the spiral. Of course there is no evidence that this was the real reason for his choice of 17 (17 had some importance in Indo-European tradition so there could be other reasons).

Figure 2

Interestingly, while we drew out this figure of the SRS for ourselves as part of the geometric constructions we indulged in in our youth, it has sparked much mathematical activity only in the last 3 decades. As is clear from figure 2 this spiral is a discrete spiral in that the radius increases in jumps of the square roots of the natural numbers. The mathematician Davis (also a historian of Euler’s and Gauss’ studies on special functions) asked an interesting question: Can one find a smooth, analytic curve that describes the square root spiral and interpolates all the intermediate values between the discrete square root radii of the discrete spiral? Davis the solved this question rather remarkably following the footsteps of Leonhard Euler.

Before we get to that we shall first create a further generalization of the discrete SRS: After having constructed the initial discrete SRS as described above, reflect (invert) the point B which initiates the spiral on the $\sqrt{2}$ hypotenuse to get a new point. Then reflect the outer end of the $\sqrt{2}$ hypotenuse on the $\sqrt{3}$ hypotenuse; then reflect the outer end of the $\sqrt{3}$ hypotenuse on the 2 hypotenuse, so on. This yields the second branch of the discrete SRS (in red in figure 2). So the Davis problem in its more general form requires one to interpolate a smooth curve through both the branches of of the discrete SRS.

The approach taken by Davis to solve it along with the solution has some striking parallels the story one of the great problems in the history of modern mathematics (described by Davis himself). Hence, we shall take detour to look a bit at that famous problem. The factorial function was originally discovered by Hindu mathematicians. For instance, it is clearly provided by the Kashmirian polymath finance minister of the Seuna Yādava rulers Śārṅgadeva in his work on the theory of Hindu music the Saṃgīta-Ratnākara in 1225 CE. This original form of the factorial function is organically described as the serial product of natural numbers: $n!=1\times 2\times...(n-1)\times n$. In the first half of the 1700s it was noticed that these discrete points of the factorial function seemed to define a curve. But the question was how does one find the intermediate points of the curve like say 2.5!. The interest in this type of interpolation problem was likely initiated by Newton in England and passed on to his junior associate, the Frenchman de Moivre. In course of his study of probability de Moivre discovered the first continuous function that was an approximate fit to the discrete factorials:
$n! \sim k\cdot \sqrt{n}\left(\dfrac{n}{e}\right)^n$; where $k$ is a constant.
His junior associate Stirling after some experimentation refined the value of the constant as $k=\sqrt{2\pi}$ leading to many people wrongly attributing the formula to him instead of de Moivre.

In this context we might note that Srinivasa Ramanujan discovered another close approximation for the factorial function:
$y= \dfrac{\log(\pi)}{2}-n+n\log(n)+\dfrac{\log(n(1+4n(1+2n)))}{6}; \; n! \approx e^y$

This formula of de Moivre indicated how the curve fitting the discrete factorials should approximately look at intermediate values. However, the precise fitting function was a problem baffled all attempts made by Stirling, Daniel Bernoulli and Goldbach. Finally, in 1729 CE Goldbach brought the problem to the 22 year old Leonhard Euler’s attention in his letter from Moscow to the latter at St. Petersburg. Euler, giving a taste of his unrivaled greatness, solved the problem in his letters responding to Goldbach the same year and the next year published a detailed paper on his use of “higher” calculus to solve the problem. Answer to the general interpolation problem was the Gamma function $\Gamma(x)$; In modern notation $n!=\Gamma(n+1)$ and the values of the function at intermediate positions provides the smooth interpolation between the discrete factorial values.

One of Euler’s definitions of $\Gamma(x)$ was the Eulerian integral:
$\Gamma(x)=\displaystyle \int_0^\infty e^{-t}t^{x-1} dt$

While there is no general way of solving this integral one can see the following:
If $x=1$, $\Gamma(1)=\displaystyle \int_0^\infty e^{-t}dt=-e^{-t} \Bigr |_0^\infty=0-(-1)=1$

Now, if we make the substitution $x\rightarrow x+1$ then we get:
$\Gamma(x+1)=\displaystyle \int_0^\infty e^{-t}t^{x} dt$; on which we use integration by parts,
$\int u dv = u\cdot v - \int v du$,
thus for the above we have:
$\displaystyle -\int_0^\infty e^{-t}t^{x} dt=e^{-t}t^{x}\Bigr |_0^\infty- \int_0^\infty e^{-t}xt^{x-1} dt=-e^{-t}t^{x}\Bigr |_0^\infty+x \int_0^\infty e^{-t}t^{x-1}dt =\lim_{t \to \infty} -\dfrac{t^x}{e^t}-0^x\cdot e^{-0}+ x\Gamma(x)$
Since, in the above expression the exponential function in the denominator will always catch with the power function in the numerator that limit will be 0. Thus we get,
$\Gamma(x+1)=x\Gamma(x)$

One will notice right away that this captures the discrete factorial function when $x$ is a natural number. Further, all we need to do is to somehow obtain the values for $\Gamma(x)$ from 0:1 then we can use the above relationship to extend it for other positive intervals. Thus, Euler’s $\Gamma(x)$ provides a function to extrapolate the factorial for the intermediate values but the question remains as the how do we get the values from 0:1. One of those values $x=\tfrac{1}{2}$ can be obtained using relatively straightforward means from the Eulerian integral:
$\Gamma(\tfrac{1}{2})= \displaystyle \int_0^\infty \dfrac{e^{-t}}{\sqrt{t}}dt$
We resort to the substitution $t=u^2 \; \therefore dt=2u du$
$\therefore \Gamma(\tfrac{1}{2})= \displaystyle 2 \int_0^\infty e^{-u^2} du$
Remarkably, the core of above integral is the one specifying half the area under a Gaussian Bell curve; so it evaluates to $\tfrac{\sqrt{\pi}}{2}$
$\therefore \Gamma(\tfrac{1}{2})=2 \dfrac{\sqrt{\pi}}{2}=\sqrt{\pi}$
This is one of those deep results that when you see and imbibe for the first time it produces a profound effect on you — how the problem of generalizing the factorial function leads you to the squaring of the circle and the limit from the famed central limit theorem i.e. the normal distribution. Thus, in the least we can get $\Gamma(1.5), \Gamma(2.5)$ etc trivially using the above recurrence relationship.

However, Euler himself obtained $\Gamma(\tfrac{1}{2})$ by using a product formula. It was such product formulae and other series which Euler, Gauss and their successors used to provide the other values of $\Gamma(x)$. Gauss with his student, the astronomer Friedrich Nicolai, who was a calculating prodigy, prepared tables of the $\Gamma(x)$ which served the mathematical community throughout the 1800s. In his investigations on $\Gamma(x)$ Gauss, building on Euler’s product formula developed the famous product formula:
$\Gamma(x)=\displaystyle \lim_{n \to \infty} \dfrac{n^x n!}{x(x+1)...(x+n)}$
This formula is quite slow in converging and rakes up huge numbers in the calculation: e.g. with $n=100$ we get $\Gamma(0.1) \approx 9.51$ correct to 2 places after the decimal point when rounded.

It is this kind of product formulae that bring us back to our original question of interpolation of the square root spiral. Armed with his deep knowledge of the history of the methods of Euler and Gauss in attacking the Gamma function, Davis solved the SRS problem by producing a product formula, which for a given number $\alpha$, which corresponds to the natural numbers 0,1,2,3… or any value between them, produces the exact coordinates of the SRS in the complex plane. This remarkable formula is:

$T(\alpha)=\displaystyle \prod_{k=1}^{\infty} \dfrac{1+\dfrac{i}{\sqrt{k}}}{1+\dfrac{i}{\sqrt{k+\alpha-1}}}$, where $i=\sqrt{-1}, \alpha \ge 0$
Here $|T(\alpha)|=\sqrt{\alpha}$ This captures the basic square root radius or hypotenuse seen in the discrete SRS.

Strikingly, analogous to the Gamma function the SRS equation also has a recurrence formula:
$T(\alpha+1)=\left(1+\dfrac{i}{\sqrt{\alpha}}\right)T(\alpha)$
From these relationships we get:
$|T(\alpha+1)-T(\alpha)|=|T(\alpha)+\dfrac{iT(\alpha)}{\sqrt{\alpha}}-T(\alpha)|=|i|\dfrac{\sqrt{\alpha}}{\sqrt{\alpha}}=-i\cdot i=1$
This captures the basic relationship of the successive natural numbers in the discrete SRS. Like with $\Gamma(x)$ all we need to do is to calculate the values of $T(\alpha)$ from 0:1. Then using the recurrence relationship we can extend it for all other values.

Like the Gauss product formula for $\Gamma(x)$ the $T(\alpha)$ formula is also slow converging. To get reasonable accuracy we need to calculate at least 20000 terms of the above product. E.g. with this computation we get $T(2)=1.414213562373095$. Thankfully, this can be done quite fast with even your modern laptop and thus we get the continuous SRS in figure 3.

Figure 3

With this in place we can look at a few other things. For example, if one plots the positions of the perfect squares (those which Theaetetus mentions to Socrates) on the SRS we surprisingly find them to lie on a triradiate pattern of gently curving spirals radiating from the number 1. The three arms have respectively:

$\alpha=4, 25, 64, 121...\rightarrow 9k^2+12k+4$
$\alpha=9, 36, 81, 144... \rightarrow 9k^2+18k+9$
$\alpha=16, 49, 100, 169... \rightarrow 9k^2+24k+16$ where $k=0,1,2,3...$

The next question is how does one capture the second branch of the SRS which was obtained in its discrete form via reflection on the subsequent hypotenuse. For this Davis figured out that one could use a generalized reflection formulation which gives $S(\alpha)$ i.e. the position in the complex plane for the second branch:
$S(\alpha)=\dfrac{1+\dfrac{i}{\sqrt{\alpha}}}{1-\dfrac{i}{\sqrt{\alpha}}}T(\alpha)$
Thus once we have $T(\alpha)$ we can now use this reflection formula to get the points on second branch. The thus computed complete continuous SRS is shown in Figure 4.

Figure 4

Here again one can see a parallel to the $\Gamma(x)$: How does one gets its values for negative $x$. This brings out a remarkable connection between the $\Gamma(x)$ and the trigonometric functions in analogous reflection formula:

$\Gamma(-x)=-\dfrac{\pi}{\sin(\pi x)x\Gamma(x)}$

Thus, once we know the $\Gamma(x)$ for positive values we can get the negative ones by this reflection formula. Unlike the positive $x$ for which grows explosively we see oscillations in negative $x$ arising from the trigonometric connection.

Davis then investigated the slope of tangent to the SRS at the point where it cuts the x-axis for the first time after the origin, i.e. the derivative of $T(\alpha)$ at 1, $T'(1)$. Gautschi termed this number $\theta$ in the honor of Theodorus and it is given by the simple but interesting series:
$\theta=\displaystyle \sum_{k=1}^{\infty}\dfrac{1}{k^{3/2}+\sqrt{k}}$

This is an awfully slow converging series: by computing the above sum for 1000000 terms we obtain rounded off to two places after decimal point $\theta=1.86$, which is correct for those two places. There are complicated, faster-converging methods which have been published in computer science studies. One such discovered by Phillips involves getting the sum $\theta_{n-1}$ as above till $k=n-1$ where $n$ is relatively small and then adding a monstrous remainder term derived from the Euler summation formula:

$\theta=\theta_{n-1}+\dfrac{1}{\sqrt{n}}\left(2-\dfrac{1}{6n}+\dfrac{1}{40n^2}+\dfrac{1}{168n^3}-\dfrac{5}{1152n^4}-\dfrac{3}{1408n^5}-\dfrac{303}{66560n^6}\right)$

Plugging $n=15$ we now effortlessly get $\theta= 1.8600250790563$ which is correct to 9 decimal places after the point.

Again there is a parallel to the $\Gamma(x)$. Euler discovered that the slope of the tangent to $\Gamma(x)$ at $x=1$ is $\Gamma'(1)=-\gamma$. This $\gamma$ is the famous Euler’s constant and is similarly given by an infinite sum:

$\gamma =\displaystyle \sum_{k=1}^\infty \dfrac{1}{k}-\log\left(1+\dfrac{1}{k}\right)$

This sum converges more quickly: with 100 terms we get rounded of to 4 places $\gamma=0.5772$ which is correct for that many places after the decimal point. This $\gamma$ is an important constant in science coming up a lot not just in the calculation of $\Gamma(x)$ but also the famed zeta function and various statistical distributions. Euler with his prodigious mental computational abilities attacked it several times in his life computing it to 16 decimal places before his death. Later the Mascheroni computed it to more places. Gauss himself capable of extraordinary numeration found that Mascheroni had made a mistake in the 21st place and asked his student, the astronomer Nicolai, who was a human computer of his age to to verify his result. Nicolai obtained the value correct to 40 decimal places, a record which stood for 50 years. Ramanujan discovered some really dramatic looking formulae relating to $\gamma$, including those with fast convergence; however we shall not talk about them here.

Figure 5

In figure 5 we see that each successive turn of the SRS is greater than the previous one by a value of nearly $\pi$. The SRS turn separation indeed converges towards $\pi$ asymptotically. This leads to the question of whether there is a way to calculate the number of discrete radii that lie within each turn of the SRS. For this Hlawka calculated something called Schneckenkonstante $K = -2.157782$ based on which Kochiemba/Wilson have computed the sequence of the number of discrete radii ( $nr[n]$; OEIS: A072895) contained within the turn $n$ of the SRS thus:
$nr[n]=\text{trunc}\left(\left(n\pi -\dfrac{K}{2}\right)^2 - \dfrac{1}{6}\right)$
This yields the sequence: 0, 17, 54, 110, 186, 281, 396, 532, 686, 861, 1055… as the number of discrete radii per turn. The difference between the square roots of successive terms of this sequence (Figure 6) fluctuates around $\pi$ with the fluctuations decreasing in amplitude as the number of turns increase. The value can come quite close to $\pi$. For the first 100 turns $\sqrt{nr[85]}-\sqrt{nr[84]}=\sqrt{70210}-\sqrt{68555}=3.1415990193488$, which comes within $6.366 \times 10^{-6}$ of $\pi$.

Figure 6

Finally, coming back to where we started, there is a simple a map in the complex plane discovered by Davis that produces discrete points of all manner of spirals including the SRS:
$z_{n+1}=az_n+\dfrac{bz_n}{|zn|}$
Here $a$ and $b$ are constants or vary with each iteration. When $a=1;\; b=i; \; z_0=1$ we get
the discrete points on the SRS (Figure 7). When $|a| \ne 1;\; Im(a) \ne 0; \; b=0$ we get discrete points on the logarithmic spiral discovered by Rene Descartes. When $|a|=1; \; a \ne 1,-1; \; b=k\cdot a; k>0$ we get an Archimedean spiral. When $a=\cos\left(\tfrac{\pi}{4}\right)+i\sin\left(\tfrac{\pi}{4}\right); b=\overline{a}, z_0=1$ we get something that grows like the SRS but has a more complex wave like internal pattern (Figure 7). This is reminiscent of the convergences seen in the Henon map, only that these are divergences. When $a=1.1+.3i; \; b=-a; \; z_0=1.1$ the map shows chaotic behavior but the attractor is localized to 4 concentric circular shells (Figure 7). Finally, when $a=\cos\left(\tfrac{\pi}{4}\right)+i\sin\left(\tfrac{\pi}{4}\right)$ and $b$ varies with each iteration $n$ as $b=\sin(n)+\sin\left(\tfrac{n}{5}\right); \; z_0=1$ we get a complex braided spiral arrangement of the points (Figure 7).

Figure 7

Posted in art, Scientific ramblings |

## Leaves from the scrapbook-2

As described here these entries are from the scrapbook of Somakhya.

Pinakasena was also doing stuff in preparation for that famous mathematics competition that Mandara was studying for. He raised the question of describing all the following curves with single equation in a one parameter space: straight line, circle, parabola, hyperbola, cardioid, lemniscate, tri-lemniscate, tri-hyperbola, 3-flower etc. Because of the hours we had spent pondering about these curves we were able to give him that right away as the following polar equation with a single parameter a:
$\rho=\left(\cos \left(a\cdot \theta \right)\right)^{\frac{1}{a}}$

When a=1 it is a circle; a=-1, a line; a= 1/2, a cardioid; a=-1/2, a parabola; a= 2, a lemniscate; a=-2, a hyperbola; a=3/2, a 3-flower; a=3, a tri-lemniscate; a=-3/2 a tri-hyperbola with $120^o$ asymptotes; a=-3, a tri-hyperbola with $60^o$ asymptotes and so on (the n-gon conics). In general if $a=\frac{p}{q}$ where p and q are mutually prime integers then it is a curve of p lobes or branches. If $\frac{p}{q}$ is negative then it is a diverging curve and if it is positive it closes with maximal radius of a unit. If $\frac{p}{q}; p=1, q>1$ then the curve internally loops with the number of crossovers being the floor of the square root of q.

Both Indrasena and Pinakasena were sort of cursing themselves that it could be so easy. I pointed out that it was indeed the easy part and pulled out the theorem of the intersection of three ellipses with shared foci on a triangle and informed the upātreya that it was the least of the questions that the mahārathin-s were supposed to surmount in such competitions.

Prove the lines passing through the points of intersections of the three ellipses with foci on a triangle are concurrent.

Another was to double the cube as the yavana-s did to Apollo with ellipses. Then I assuaged him that there was no point struggling for such competitions – if one was truly a mahārathin one would know it and competitions should not matter. If one were not a mahārathin one should study such things just for mental entertainment or knowledge acquisition and play those games as a professional of which one is a master.

Indrasena then revealed to me how he had figured out a way to find genes that had really undergone selection in different Hindu populations. We looked at the genes he identified for sometime and thought about what they might imply.

Then people started asking him questions for oracular prognostication. He gave answers and they came to us and other V1s for having the answers translated. Then I asked for a prognostication and he gave me a bad one. Thankfully he pointed to Indrasena with his trident and said he would be there to shore me up when that time comes. Pinakasena asked if he might become a vīra like his brother. He gave a detailed response that after undergoing rudrāveśa he would unite with a dūtī named Shallaki (śallakī) and then he would become one. He further added that they would protect our kula once I, Indrasena, Lootika and Vrishchika die. I found that answer to be very strange in more than one way and wondered if my apprehension of the “pattern” was incorrect – after all other than the caturbhaginī there was never supposed to be another. It hinted at the existence of an orthogonal kula.

Entry 13; mṛtamīna: We returned early next morning to collect packets of bhasma when we saw the electrician still dancing unfazed. We heard that he had continuously danced through the night and that he was going to do so till midday. The sky was overcast and there was moisture in the air but no rain. My mood matched the somber weather as I pondered over the āviṣṭa’s prognostication; I was turning over in my mind the kinds of mantra-prayoga-s that might be able to see me through it and all the vighna-s that could come in the way. I pondered what implications of the absence of Lootika might mean when the time came but I told myself that in the ultimate struggle a man is always all by himself. The ātreya-s seemed to sense my mood from my silence. Indrasena suggested that we go to brāhmaṇāhāraśāla to give ourselves to the bhoga of bhojana. I agreed and somehow convinced my aunt and mother that I would rather not eat at home. As were walking to the āhāraśāla Pinakasena wondered if āveśa was avaidika and whether we should give much attention to it. Indrasena correctly told him that it was entirely within the vaidika circle though not a codified practice performed by brāhmaṇa-s as part of their system. He pointed to the āveśa-s had by brāhmaṇa women during which my great ancient ancestor Kabandha ātharvaṇa spoke using them as the medium.

Thereafter Pinakasena asked about Rudrāveśa and its foundations. I told him of it deep Indo-European provenance. In Indo-Aryan tradition we had the case of Rudra animating the corpse of a brāhmaṇa at the holy town of Kāyāvarohaṇa in Gujarat and then he walked all the way to Ujjaini where he is said to have initiated his student in the Pāśupata doctrine. They say this animated brāhmaṇa was Lakulīśa. The muni in the ṛgveda hints at possession by Rudra as he is flying in the air. This tradition of Rudrāveśa continues down to the Bhairava-tantra-s. Even in the medieval period the brāhmaṇa Appayya Dīkṣita underwent a muni-like possession using dhattura. Among the yavana-s we have the tale of Aristeas of Proconnesus who was dead when Apollo animated him and wandered in that possession to the land of the Central Asian Iranics knows as the arimaspa-s. Then under the possession of Apollo he is said to have appeared as crow in Italy. Indeed, even yavana hero Odysseus’s travels were made known to Demodocus in a possession by Apollo.

Entry 14; mṛtamīna: That evening I and the two ātreya-s went up to the granitic eminence of the Vīrabhadrāśman. Having scaled the rock, for a while we sat in silence stretching out our legs watching the blazing eye of the god Vivasvān climbing down at the western horizon. We pensively felt the weapons in our pockets for in this downward turn of the Kali age one could never say from where a marūnmatta or some other dasyu might dart out like a Malimluca who had been slain in the days of yore by the great vajra-wielding Maghavan. Evidently, Indrasena was thinking of the great god too for he was non-verbally intoning the svara-s of the sūkta of Gṛtsamada śaunahotra. As this is something brāhmaṇa-s do while testing each other on the śruti, I caught the mantra:

dyāvā cid asmai pṛthivī namete
śuṣmāc cid asya parvatā bhayante ।
yaḥ somapā nicito vajrabāhur
yo vajrahastaḥ sa janāsa indraḥ ॥

Even heaven and earth bow to him;
indeed, the mountains fear of his fury.
The soma-drinker who is praised as Vajrabāhu,
he who has the Vajra in is hand, he, O folks, is Indra!

Entry 15; abhijānat: We were at dinner with my clansman. Unlike the carefree and completely informal affairs at my house, that of Indrasena’s or Lootika’s here it was always a dour and serious affair which matched mood created by the culinary productions of my aunt. I have always had some discomfort in their house though they are my kinfolk and at dinnertime it was particularly so. Thankfully, it appears as though subtly more consonant genetic combinations have formed among my other coethnics like the gautamī-s and ātreya-s who, while separated from me by around 7-9 generations, share some genes as lines of descending from a common vipra community. That stressed to me the importance of belonging as part of a larger coethnic community. Mandara and Saumanasa (and apparently I too) were questioned by my uncle on various topics of mathematics and physics as dinner proceeded. While I could have smashed at least some of those questions without batting an eyelid, I remained silent even as the god of the yavana-s Zeus allowed Herakles to win in the wrestling match in the first Olympic games.

The conversation lit up a bit when Mandara announced his intention to study and make robots. My uncle called him an idiot but thankfully did not squelch the conversation as aunt incited it further. My aunt questioned him about the dangers that might arise from it. She told him that ours was a country which had a lot people who earn their living via manual labor. The introduction of robots would destroy their means of livelihood. Hence, it would lead to social instability that could be easily exploited by mleccha-s. For example, there are potters who make pots for the railways; vegetable sellers who sell vegetables from a basket on the street corner; people who man the counters and stock material at the supermarket: a robot could easily take their jobs: then what will they do? Saumanasa argued that taking such a view would only keep those who are in the lower economic rungs of society in the same place rather than giving them a means to explore “better” and more modern jobs. She felt this would only add to the good of the country with more people doing “advanced” jobs than cleaning garbage or dredging drains. Mandara argued that the industrial revolution in England could be used as a model. The Marxists claimed a disaster would occur due to the unskilled laborers being rendered irrelevant by the industrial revolution. However, he argued that the opposite happened in terms of their individual affluence. He said the market-liberalization in India gave glimpses that such a thing could happen and we could cope and prosper from robotization.

There are some key things I do not know precisely enough to make an accurate prognosis. We first need to know what the human capital of those who might be displaced by robots is. It is uneven in our nation. While drain clearance, garbage disposal and the like would certainly benefit from robotization, we do not believe that it is a given that those freed from other forms of manual labor by robots can now automatically perform more advanced tasks that would do greater good for the country. Hence, robotization cannot work out in too humane a manner unless it is carefully phased and goes hand in hand with huge political restructuring that allows management of human capital.

We could look at the country which has had the greatest robotization to date, Japan. People there have had long lives from at least the medieval period and generally have been reported as having a low infection load. There are exaggerated medieval tales transmitted by Europeans of Japanese living hundreds of years! This does suggest that there has been an evolutionary shift in the Japanese population towards a more K-strategy. In contrast, our nation is located partly in the tropics and has at least in the past 700 years (aggravated by Islam and Christianity) has had a much lower life-expectancy. This has selected a r-strategy. Japan as an island is hard to reach from without and conquer. Most major powers, after the initial conquest by East Asians, a possible shaka elite conquest, and before the Americans, failed in doing so; even then other than the first they could not make a serious genetic imprint. Our land being a subcontinent can easily be overrun by large land-bound migrating groups much like the major component of our ancestry from the Eurasian steppes did. This meant a certain type of insularity along with a mostly K strategy was much less likely to develop. While some invasions can be good in infusing new blood those by Abrahamists can be disasters as is clear from our case. We believe these factor have been major determinants of why robotization never caught on despite proto-robotization having emerged among our ancestors well-before the first surviving lines of Japanese literature were put down. This again was felt in our engineering and text production while both displaying high sophistication and magnitude always had a huge manual/animal labor component (especially so with respect to text production compared with the East Asian locus). This is also potentially affects commitment to accuracy and maintaining time (say in railways and airways even during good weather) in the later Indian labor force. Moreover, given the likely genetic component of the profound differentiation in the people of Japan and India, robotization has to be more carefully considered in our context.

Vrishchika and Indrasena were quite satisfied for they had just published a paper where they found that an unusual gain of PAX6 binding sites in the delta-catenin regulatory region in family of brāhmaṇa-s which had consistently produced several significantly big-headed and intelligent individuals.

Posted in Heathen thought, Life, Scientific ramblings |

## A note on the asterisms forming the nakṣatra-s

In Hindu tradition the ecliptic is divided into 27 parts of $13\tfrac{1}{3}^o$ which correspond to 27 asterisms known as the nakṣatra-s. In the earliest extant layers of our tradition this number is 28 implying division into sectors of $12\tfrac{6}{7}^o$ or insertion of a nakṣatra with adjacent compressed sectors. The earliest complete nakṣatra lists are found in the Taittirīya-śruti and the Atharvaveda where they are recited as part of the nakṣatreṣṭi ritual which places the deities of pantheon in these celestial compartments. Indeed, ever since it has been common practice in Hindu astronomy to use the names of the deities corresponding to a given compartment alternatively for the nakṣatra itself. The old nakṣatra-lists in the TS, Laugākṣi’s sūtra-s and the Atharvaveda begin with Kṛttikā i.e., the Pleiades, suggesting that the system as described in these texts was put in place sometime in the interval of 4500-4000 years before present. However, we hold that the nakṣatra system did not begin with these texts. These texts are mostly predated by the ṛgveda, where we find stray references to specific nakṣatra-s and also the term itself in its generic sense. Thus, we believe a form of the system was already in place even in the days of the ṛgveda. Further, there are some homologies between nakṣatra-s and Iranian asterisms suggesting that some form of the system might have existed even in the Indo-Iranian past on the Eurasian steppes.

Now, one question which is often asked is whether there is a way of knowing precisely how nakṣatra-s were defined in the Vedic period. In classical Indian astronomy each nakṣatra is defined by giving the latitude and longitude of a star called the yogatārā, which was defined by the scientist Brahmagupta in 665 CE as the brightest star in the asterism. This makes the nakṣatra quite unambiguous but then there are nakṣatra-s, which are far away from the ecliptic in the Vedic reckoning raising the question if the later definitions differ from the Vedic ones. The best way to start answering this is by using the earliest surviving list in tradition that gives the number of stars in each nakṣatra from the Nakṣatra-kalpa of the Atharvan tradition:

ṣaṭ kṛttikā ekā rohiṇī tisro mṛgaśira ekārdrā dve punarvasū ekaḥ puṣyaḥ ṣaḍ āśleṣāḥ ṣaṇ maghāḥ catasraḥ phalgunyaḥ pañca hasta ekā citrā ekā svātir dve viśākhe catasro ‘nurādhā ekā jyeṣṭhā sapta mūlam aṣṭāv aṣāḍhā eko ‘bhijit tisraḥ śravaṇaḥ pañca śraviṣṭhā ekā śatabhiṣā catasraḥ proṣṭhapadau ekā revatī dve aśvayujau tisro bharaṇyaḥ | iti saṃkhyā parimitaṃ brahma || NK2

Kṛttikā -s: 6. This clearly coincides with the traditional understanding of the 6 Pleiades being this asterism. However, an older text the Taittirīya saṃhitā names seven of them explicitly:
ambā dulā nitatnir abhrayantī meghayantī varṣayantī cupuṇikā nāmāsi

Likewise in Greek tradition there was an older record of 7 with Aratus claiming that one of them had faded away. This is generally believed to be Ambā (Greek Pleione). This has led to the debate that originally the 7 were of comparable brightness with 28 Tauri fading away later. The parallel between the ārya and yavana records suggests that indeed such a fading might have happened.

Rohiṇī: 1. Hindu tradition has always identified Rohiṇī with $\alpha$ Tauri (Aldebaran); hence, the single star assigned to Rohiṇī should be taken as this one. The name is also indicative of the redness of he star.

Mṛgaśiras: 3. Some take this to be the 3 stars on the head of Orion ($\phi 1, \phi 2, \lambda$ Orionis), which is how they are denoted in classical astronomy. This remains a possibility even in the Vedic reckoning. In Vaidika reckoning the core of Mṛga was Orion with the arrow of Rudra shot through it (See below). The arrow is identified with the three stars of the belt of Orion ($\zeta , \epsilon , \delta$ Orionis), known in the Veda as the Invaka-s. Hence, we could entertain the possibility that originally the 3 could have meant the Invaka-s. In support of this proposal we note that the Taittirīya Brāhmaṇa 1.5.1.1 states:

somasyenvakā vitatāni ।
Soma’s are the Invaka-s [which are] drawn [from the bow to be fired].

This explicitly identifies Mṛgaśiras with the Invaka-s.

Ārdrā: 1. The Taittirīya, Kaṭha and Atharvaṇa-śruti-s are all consistent in identifying Ārdrā with the god Rudra. Going by classical astronomy the coordinates of the yogatārā of Ārdrā would indicate $\gamma$ Geminorum. There is another commonly held view that $\alpha$ Orionis is Ārdrā. The Vedic text says there is a single star associated with it and the evidence within the Veda suggests that it was not $\gamma$ Gem. First the Taittirīyas-śruti is unambiguous is stating:
ārdrayā rudraḥ prathamānam eti ।
With Ārdrā, Rudra goes forth luminescent.

This means that he original Ārdrā was likely seen as a bright star. Now, while both $\alpha$ Ori and $\alpha$ Canis Majoris are bright stars, $\alpha$ Ori is too close to Mṛgaśiras making $\alpha$ Can Ma more likely, and is also closer to the position of the later yogatāra when projected on to the ecliptic. In the brāhmaṇa on the nakṣatra ritual in TB 1.5.1.1 the name Ārdrā is replaced by Mṛgayu which is always understood as Sirius:

rudrasya bāhū mṛgayavaḥ
The two arms of Rudra are the Mṛgayu-s (Stars in Can Ma).

Further, this is supported by the evidence of the Aitareya Brāhmaṇa on the famed āgnimāruta-śastra recitation, which we provide in full:

Prajāpati desired his own daughter.

divam ity anya āhur uṣasam ity anye |
The sky some say and others the Uṣas.

tām ṛśyo bhūtvā rohitam bhūtām abuyait |
Having become a stag he approached her as a red deer.

taṃ devā apaśyann: ākṛtaṃ vai Prajāpatiḥ karotīti |
The gods saw him: “Prajāpati is doing something that is not done”.

te tam aichan ya enam āriṣyaty etam anyonyasmin nāvindaṃs
They wished to punish him. They did not find find him among one another.

teṣāṃ yā eva ghoratamās tanva āsaṃs, tā ekadhā samabharaṃs
Whatever most terrible forms exist they brought together in one place.

Brought together they became this god; hence, his name has the word “bhūta” (Bhūtapati)

bhavati vai sa yo ‘syaitad evaṃ nāma veda ||
He who knows his name thus prospers.

taṃ devā abrūvann: ayaṃ vai Prajāpatir ākṛtam akar, imaṃ vidhyeti |
The gods told him: “this Prajāpati has verily done a deed that is not done; pierce him.”

sa tathety abravīt, sa vai vo varaṃ vṛṇā iti | vṛṇīṣveti |
He said: “So be it” He also said: “let me choose a boon from you.” They said: “Choose”.

sa etam eva varam avṛṇīta: paśūnām ādhipatyaṃ |
He chose this boon: “The overlordship of animals”.

tad asyaitat paśuman nāma paśumān bhavati yo ‘syaitad evaṃ nāma veda |
Hence, his name contains the word animal (Paśupati). He who knows this name thus becomes rich in cattle.

tam abhyāyatyāvidhyat, sa viddha ūrdhva udapravata |
He took aim and pierced him [Prajāpati]. Pierce thus he flew upwards.

tam etam Mṛga ity ācakṣate |
They know him as the [constellation of the] deer.

ya u eva mṛgavyādhaḥ sa u eva sa, yā rohit sā Rohiṇī,
He who is the piercer of the deer [is the asterism] known as that, she who is red is [is the star] Rohiṇī.

yo eveṣus trikāṇḍā so eveṣus trikāṇḍā |
That which is the 3-pointed arrow is the [asterism] of the three-pointed arrow.

The semen of Prajāpati spilled out and ran; it became a lake [the Milky Way].

te devā abruvan: medam Prajāpate reto duṣad iti |
The gods said let this semen of Prajāpati not get ruined.

As they said: “let the semen of Prajāpati not be spoiled” it became “not spoiled”

tan māduṣasya māduṣatvam |
The state of not being spoiled is of not spoiled.

māduṣaṃ ha vai nāmaitad yan mānuṣaṃ |
From “not spoiled” is the name which is “man”.

tan māduṣaṃ san mānuṣam ity ācakṣate parokṣeṇa,
That which is “not spoiled” they know by metaphorical meaning to be linked to man.

parokṣa-priyā iva hi devāḥ
For it is as if the gods like the mysterious.

This narrative clearly identifies Rudra with the killer of Prajāpati. Prajāpati is unambiguously identified with the constellation of Orion and positioned with respect to Rohiṇī. Further, he is described as “flying above” his hunter when pierced. Together these identify the constellation of Rudra his hunter with Can Ma. Hence, we may conclude that originally $\alpha$ Can Ma was Ārdrā. Further, the name Ārdrā means moist indicating a link with the wet season. The Iranian equivalent of Sirius, Tishtrya is also associated with rain suggesting that Ārdrā inherits this ancestral association.

Punarvasū: 2. There is a fairly uniform agreement that the two stars of Punarvasū are $\alpha , \beta$ Geminorum (Castor and Pollux). The simile of these two stars is frequently encountered. In the Rāmāyaṇa (1.29.25; Gita Press edition):

śaśīva gata-nīhāraḥ punarvasu-samanvitaḥ ॥
The refulgent great sage (Viśvāmitra) entered the path of the āśrama, even as the moon free from fog accompanied by the two stars of Punarvasū (i.e. Rāma and Lakṣmaṇa.

Vedic tradition associates Punarvasū with the goddess Aditi. Under this association the simile of the moon in Punarvasū is again seen in the Rāmāyaṇa (6.67.161):

kumbhakarṇa-śiro bhāti kuṇḍalālaṅkṛtaṃ mahat ।
āditye’bhyudite rātrau madhyastha iva candramāḥ ॥
The giant head of Kumbhakarṇa adorned with earrings shone forth even as the moon having arisen at night in the midst of the constellation of Aditi (i.e. between Castor and Pollux).

The Taittirīya Saṃhitā has an incantation in the Soma procurement ritual (in 1.2.4): aditir asy ubhyataḥ śīrṣṇī sā naḥ suprācī supratīcī saṃ bhava ।
You are Aditi, she is two-headed, be good to us together eastward and westward.

Śatapatha Brāhmaṇa (3.2.4.16) states:

You are Aditi, the two-headed. Because he speaks the right in an inverted manner and makes what comes comes first last, and what comes last first by her, therefore she is double-headed. That is why he says: “You are Aditi, the double-headed”.

These allusions indicate that the two-headed nature of the constellation of Gemini was transposed on to the presiding deity Aditi and the inversion associated with the two heads along with the eastward and westward paths might indicate an old memory of the start of the ecliptic at Aditi in prehistoric times (>7000 years BP).

Puṣya: 1. This single star is generally taken to be $\delta$ Cancri which was close to the ecliptic. However, we have evidence from the Ṛgveda that originally it meant the Praesepe open cluster (M44). The great ṛṣi Śyāvāśva ātreya:

yuṣmādattasya maruto vicetaso rāyaḥ
syāma rathyo vayasvataḥ |
na yo yuccati tiṣyo yathā
divo ‘sme rāranta marutaḥ sahasriṇam ||
5.054.13c

May we bear as charioteers of the great wealth
given by you, discriminating Maruts.
That which does not depart, even as Tiṣya does not from the sky,
to us in thousands, Marut-s rejoice.

The comparison of great riches, in thousands, is indicative of the great mass of stars in the open cluster supporting the identification of the old Tiṣya with M44.

Āśleṣā: 6. This constellation is associated with the snakes in Vedic tradition. It corresponds to the head of the Greek constellation of Hydra, suggesting that the link to a snake goes back to early Indo-European times. Āśleṣā is specifically associated with the head of the snake. Hence, the 6 stars should correspond to $\theta , \zeta , \epsilon , \delta , \sigma , \eta$ Hydrae. The Vedāñga Jyotiṣa states that the summer solstice began in the middle of this constellation suggesting that it was composed around ~3350 YBP.

Maghāḥ: 6. While today Magha is associated with $\alpha$ Leonis, the Vedic tradition indicates 6 stars for this asterism. This would mean it included the entire sickle of Leo: $\epsilon , \mu , \zeta , \gamma 1 , \eta , \alpha$ Leonis. The Atharvaveda Nakṣatra sūkta states that the summer solstice happened in this asterism pointing to a period of ~4400 YBP.

Phalgunyaḥ: 4. There are 2 Phalguni-s pūrva and uttara together with 4 stars. These can be identified with $\theta , \delta$ Leonis (pūrva) and $\beta$, 93 Leonis (uttara).

Hasta: 5. Tradition unequivocally identifies Hasta with Corvus. Hence the 5 principal stars of Corvus are the 5 listed for Hasta: $\alpha , \beta , \gamma , \delta , \epsilon$ Corvi.

Citrā: 1. Spica in Virgo. The star itself is one the nakṣatra-s mentioned in the RV (according to us contra white indological opinion). In the TB 1.5.1.3 Citrā is described as an additional star of the god Indra.

Svāti: 1. Arcturus in Bootes. Also known as Niṣṭyā in the Yajurveda.

Viśākha: 2. $\alpha , \beta$ Librae. The constellation of the Ikṣvāku-s according to the Rāmāyaṇa.

Anurādhā: 4. $\beta , \delta , \eta , \rho$ Scorpii

Jyeṣṭhā: 1. Antares. The TB describes this star as a second Rohiṇī keeping with the red color of the star.

Mūla: 7. These seven stars are in the tail of Scorpio. Which stars exactly were identified with the asterism is hard to say but most likely were: $\zeta , \eta , \theta , \iota , \kappa , \upsilon , \lambda$. This is the constellation of the goddess of the nether regions Nirṛtti in the Veda. In the Rāmāyaṇa it is associated with Rākṣasa-s who are supposed to have emanated from Nirṛtti.

Aṣāḍhā-s: 8. These eight stars are in the two Aṣāḍhā-s: The pūrva group may be identified with the 4 stars associated with the spout of the teapot of Sagittarius: $\delta , \gamma , \epsilon , \eta$ Sagittarii. The uttara group may be identified with the handle of the teapot: $\phi , \sigma , \tau , \zeta$ Sagittarii. The Taittirīya Brāhmaṇa’s nakṣatra-sūkta identifies the pūrva group with divine waters (yā divyā āpaḥ payasā sambabhūvuḥ ।) and all other waters as emerging from it. We take this identification as an allusion to the bright center of the Milky Way just next to the pūrva group.

Abhijit: 1, Vega. This star is way off the ecliptic and is omitted in latter lists. However, its name meaning the all conquering is equivalent to the Iranian name for the same star Vanant. This suggests that it might have been an ancient association. The Aitareya brāhmaṇa indicates that it was used to mark the day just before the svarasāman days during the annual sattra. Tilak we believe rightly realized this was the reason why Abhijit was important in the early period to mark this ritual day which in turn is critical for marking the days leading up to the Viśuvān day.

The Mahābharata preserves a curious tale regarding Abhijit’s fall and also involves other asterisms pointing to a precessional legend. The great god Indra tells the god Skanda:

abhijit spardhamānā tu rohiṇyā kanyasī svasā |
icchantī jyeṣṭhatāṃ devī tapas taptuṃ vanaṃ gatā ||

Abhijit, the younger sister of Rohiṇi, contested with her desiring seniority. She went woods to perform austerities.

tatra mūḍho ‘smi bhadraṃ te nakṣatraṃ gaganāc cyutam |
kālaṃ tv imaṃ paraṃ skanda brahmaṇā saha cintaya ||

I am dumbstruck by the fall of that auspicious star from the sky.

dhaniṣṭhādis tadā kālo brahmaṇā parinirmitaḥ |
rohiṇyādyo ‘bhavat pūrvam evaṃ saṃkhyā samābhavat ||
Time was specified by Brahmā starting with Dhaniṣṭhā around [the ecliptic]. Formerly, they started from Rohiṇi and thus their number was complete around [the ecliptic].

evam ukte tu śakreṇa tridivaṃ kṛttikā gatāḥ |
nakṣatraṃ śakaṭākāraṃ bhāti tad vahni-daivatam || 3-219.8-12 (“Critical edition”)
Thus told by Indra, Kṛttikā-s went to the third heavenly realm.
There they shone forth in the shape of a cart presided by the god Agni.

While several authors have attempted to decode this legend it remains rather obscure. The only clear parts are the memory of a transition from the Rohiṇi period to the Kṛttikā period and an allusion to the loss of Abhijit from the nakṣatra reckoning. This might relate to Abhijit having lost its utility as a marker of important rituals close to the solstices due to precession.

Śravaṇa/Śroṇa: 3. These are quite unambiguously identified as $\alpha , \beta , \gamma$ Aquilae. It is possible that it was associated with the celestial footprint of Viṣṇu in his three strides.

Śraviṣṭhā/Dhaniṣṭhā: 5. While the Nakṣatra-kalpa gives 5 stars for this asterism the older Taittirīya-śruti seems to indicate that there were 4. In any case this group is unambiguously identified with Delphinus. The older reckoning likely took 4 of the brightest stars, $\alpha , \beta , \gamma , \delta$. The NK included one further the star.

Śatabhiṣā: 1 This is today take to be $\lambda$ Aquarii. But was it the star meant in the Vedic texts is unclear. There is an asterism of Iranians known as Satavaēsa, which we hold to be the equivalent of the Vedic one. The Iranian asterism was associated with the sea while the Vedic one with Varuṇa. The possibility of Fomalhaut ($\alpha$ Pisces Austrinisis) being this star is not implausible.

Proṣṭhapada-s: 4. The two Proṣṭhapada-s are given 2 each. Identifying each pair with the two vertically adjacent stars of the 4 stars comprising the square of Pegasus seems the most likely for these.

Revatī: 1. Classical astronomy identifies it with $\zeta$ Piscium. This is a really undistinguished star. So we cannot be sure if that is what was originally meant or a higher up star like $\beta$ Andromedae was used. Narahari Achar holds that the goddess Pathyā Revatī mentioned in the Svastisūkta of the Atri-s implied this asterism. While this is not impossible we are not entirely sure of that especially given the undistinguished nature of the star identified with it.

Aśvayujau: 2. $\alpha , \beta$ Arietis. Concerning this asterism there is a problematic issues concerning the the Yajurvaidika incantation known as the Uttaranārāyaṇa. This text describing the cosmic Viṣṇu now bearing a special name Nārāyaṇa states:

aho rātre pārṣve । nakśatrāṇi rūpam । aśvinau vyāttam ।
His sides are the day and the night. His form [is comprised of] the asterisms. The two Aśvin-s his jaws.

Here Nārāyaṇa is identified with the constellations even as Prajāpati was earlier identified with them in the ritual of the Nakṣatra-rūpin Prajāpati specified in the Yajurveda (Taittirīya Brāhmaṇa 1.5.2.2). This identification continues through later Vaiṣṇava tradition. The ritual itself has further continuity going down to the Gupta age where it is described by the naturalist Varāhamihira who states that by performing it a man becomes attractive to women and women attain beauty. Now the question is whether the account of the Aśvayujau at the mouth of Nārāyaṇa have some significance of the date of this text. The text is clearly a late one clinging to the edge of the Vaidika productions but when exactly was it composed. If one takes Aśvayujau to imply the start of the nakṣatra cycle having shifted to this asterism it would yield a date of around 2300 YBP. This date resonates with the white Indologists who ascribe late dates to all Vedic production. However, we do not think the mouth should be taken as the beginning of the nakṣatra cycle. Rather, that position is usually reserved for the top of the head. Hence, the mouth likely implies the nakṣatra after it which might imply the equinoctial coelure passing before Aśvayujau suggesting a date of around 3300-3000 YBP. One also wonders if the tale of Prajāpati being fitted with a goat’s head after his beheading by Rudra’s agent Vīrabhadra alludes to this period, with the goat’s had representing Aries.

Bharaṇī: 3. This triad is uniformly understood to be the compact triangle formed by 41, 39, 35 Arietis.

Brahmagupta says in Khaṇḍakhādhyaka (1.9.1-2):
tvāṣṭra-guru-vāruṇa+ārdra+anila-pauṣṇāny eka tārāṇi ॥

brahma+indra-yama-hari+indu-tritayam ṣaḍ-vahni-bhujaga-pitryāṇi ।
maitrāṣāḍa-catuṣkam vasu-ravi-rohiṇya iti pañca ॥

Clearly by the early medieval period Hindu asterism-reckoning had changed to a degree from the Vedic period. Now the number of stars in each asterism was specified as:
Kṛttikā: 6; Rohiṇī: 5 (likely whole Hyades+Aldebaran); Mṛgaśiras: 3; Ārdrā: 1; Punarvasū: 2; Puṣya: 1; Āśleṣā: 6; Maghā: 5; Phalguni-s: 2 each; Hasta: 5; Citrā: 1; Svāti: 1; Viśākha: 2; Anurādhā: 4; Jyeṣṭhā: 3; Mūla: 1; Aṣāḍhā-s: 4 each; Abhijit: 3; Śravaṇa: 3; Śraviṣṭhā: 5; Śatabhiṣā: 1; Proṣṭhapada-s: 2 each; Revatī: 1; Aśvayujau: 2; Bharaṇi: 3.

Posted in Heathen thought, Scientific ramblings |

## Journeying through the fractal slopes of mount Meru with two-seeded recursive sequences

The Hindus have been fascinated by sequences and series from the beginning of their civilizational memory recorded in the Veda. This continues down to the medieval mathematician Nārāyaṇa paṇḍita, who discovered a general formula (sāmāsika paṅkti) that can be to obtain the ‘Meru-średhī’ (known in the west as Fibonacci’s sequence). He uses it as a model to explain the population dynamics of cows. In modern terms the formula goes thus:
Let $f[1]=f[2]=1$. These are two seeds of the sequence. Then,
$f[n]=f[n-1]+f[n-2]$, where $n=3,4,5...$.
One sees that this yields the famous sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… i.e. Nārāyaṇa’s Dhenu-saṃkhya or cow-numbers. We may denote this special sequence as $M[n]$. It is well-known that,
$\displaystyle \lim_{n \to \infty} \dfrac{M[n+1]}{M[n]}=\phi$, where $\phi$ is the Golden ratio.
Thus, as we have illustrated before, the Dhenu-saṃkhya can be obtained as integer values of a function based on $\phi$.

Closer to our times, following the discovery of a everywhere continuous but nowhere differentiable function by Bernhard Riemann, the Japanese mathematician Teiji Takagi discovered another remarkable curve of this type. Let function $s(x)$ be defined as,
$s\left(x\right)=\min \left(\left(x-\lfloor x \rfloor \right),\textrm{abs}\left(x-\lceil x \rceil \right)\right)$

Then the Takagi function is defined as,
$f\left(x\right)=\displaystyle \sum _{n=0}^\infty w^n \cdot s\left(2^n x \right)$ Parameter $w=0.5$ gives an aesthetically pleasing curve. As can be seen from the figure the curve as has fractal form keeping with its undifferentiable nature

Figure 1. The Takagi curve (red)

A lay person may wonder why we are mentioning these two seemingly disparate pieces of mathematics together. The connection between them becomes apparent via the remarkable sequences discovered by the scientist-philosopher Douglas Hofstadter. He first presented these ideas in his curious book ‘Gödel, Escher, Bach: An Eternal Golden Braid’.

Our own journey through these began in the days of our youth when we chanced upon Hofstadter’s book in a book-store. Not having the cash to procure it we spent sometime taking in its braided ideas right there. While that encounter was not enough to take in all the sequences he discussed in the book we got enough of the basic idea to experiment by ourselves. The basic idea behind the Hofstadter class of sequences is a generalization of the procedure behind the Meru-średhī. This can be illustrated with the sequence with which we first experimented:
$f[n]=n-f[f[n-1]]$, where $f[1]=f[2]=1$ and $n=3,4...$
Like $M[n]$ it is also initiated with two seed values but the definition of the subsequent elements involves a nested definition. The definition itself looks simple and yields the following sequence:
1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14…

One notices that unlike the Dhenu-saṃkhya-s this sequence covers all the positive integers. However, some are repeated multiple times. Interestingly, we observe that $f[3]=2, f[5]=3, f[8]=5, f[13]=8, f[21]=13$. Thus more generally $f\left [M[n] \right ]=M[n-1]$. Moreover, once we have this sequence we can also write out pairs of the form $(f[n],n)$. The set of such pairs can be represented as edges of a graph (i.e. $f[n] \rightarrow n$), which turns out to be a tree.

Figure 2. Graph of $(f[n],n=1:50)$

The tree displays an interesting structure which resembles an evolutionary process of descent through modification. In this we can recognize a “primitive” lineage which is the line of descent comprised of $M[n]$. The remain lines of descent branching off from it also display a Meru-średhī process albeit each at a different level. For e.g. we have the line of descent: 3, 4, 7, 11, 18, 29, 47…

Our next advance in understanding these sequences came from reading a paper by Clifford Pickover, which presented a means of graphically visualizing the structure of these sequences. Inspired by this we used a slightly different graphical representation to study the structure specified by these sequences. One of the notable Hofstadter sequences is,
$f[n]=f[f[n-1]]+f[n-f[n-1]]$

The technique we used to visualize the sequence is to subtract a certain factor proportional to $n$ from the $f[n]$ in order to render the values along the x-axis rather than as along the curve which the increasing $f[n]$ follows. We call this rectification. Thus we rectify the above sequence by plotting $(n, f[n]-\tfrac{n}{2})$

Figure 3. $(n, f[n]-\tfrac{n}{2})$ for $f[n]=f[f[n-1]]+f[n-f[n-1]]$

Remarkably we get ever larger copies of the Takagi curve, with each loop progressing in width as 2, 4, 8, 16, … i.e. the powers of 2 (Figure 3). Thus, the curve captures the binary base representation of integers up to a give integer and the fractal has a predictable form. The successive loops of the structure also represent a journey into the fractal structure of the Takagi curve.

Figure 4. $f[n]=f[f[n-2]]+f[n-f[n-2]]$ rectified as $f[n]-.68023n$

Another such sequence was discovered by Mallows (Figure 4), which has an overall similar behavior as the above sequence of Hofstadter. However, rather than the regular symmetric Takagi curve generated by the sequence it has a more jagged fractal curve with a distinct pattern of two peaks followed by three peaks.

Returning to the original sequence we studied, $f[n]=n-f[f[n-1]]$, we can rectify it using $f[n]-\tfrac{n}{\phi}$.

Figure 5. $f[n]=n-f[f[n-1]]$

It has a very different structure from the above sequences. Instead of ever-increasing loops this is characterized by rapid but fixed bandwidth of oscillation with basic reoccurring units incorporated into higher order reoccurring units, reflective of its tree structure discussed above.

One would notice that the generative formulae for these sequences, unlike that of the Meru-średhī, involve doubly nested specifications. Such specifications mostly lead to sterile sequences because the index $n$ for a given $f[n]$ can drop below 1 or above the current value thus killing the sequence. Nevertheless, there are several additional productive formulae beyond the above that lead to a range of interesting behavior. These are illustrated before.

Figure 6. $f[n]=f[n-f[n-1]]+f[n-f[n-2]-1]$ rectified using $f[n]-\tfrac{n}{2}$.

Somewhat similar theme to the above case with similar looking motifs nested at multiple levels giving rise to a pseudo-repetitive appearance.

Figure 7. $f[n]=f[n-f[n-1]]+f[n-f[n-2]]$ rectified using $f[n]-\tfrac{n}{2}$

This is one of Hofstadter’s discoveries which has ever larger pulses of high intensity fluctuations of similar form separated by regions of low intensity fluctuations. The repetition of the same basic form albeit with variation at larger and larger scales resembles the formula generating the Takagi curve.

Figure 8. $f[n]=f[f[n-1]]+f[n-f[n-2]-1]$ rectified using $f[n]-\tfrac{n}{2}$

Another Hofstadter sequence similar to the above but the pulses themselves are arranged on waves of ever-increasing wave-length giving it the form of ever-larger sigmoids.

Figure 9. $f[n]=f[f[n-1]]+f[n-2\cdot f[n-1]+1]$.

This is non-linear in terms of its central growth curve and cannot be perfectly rectified. Hence, the factor $f[n]-.42n^{.818}$ is used. This was discovered by Stephen Wolfram in an excellent introduction to these sequences for a lay reader although he does little to acknowledge his predecessors’ work. It shows a higher order wavy pattern of increasing wavelength but within it the pulsations show much more irregularity.

Figure 10. $f[n]=f[n-f[n-1]-1]+f[n-f[n-2]-1]$ rectified using $f[n]-\tfrac{n}{2}$

Unlike the above this shows neither regions of reduced intensity pulsation nor a higher order wavy pattern. Instead is simply shows a chaotic pattern of fluctuations which grow in magnitude with $n$.

If the above sequences were generated using doubly nested specifications we also find complex behavior emerging from triply nested formulations. We discovered for ourselves some such sequences which are illustrated below.

Figure 11. $f[n]=n-f[f[f[n-2]]]$ rectified using $f[n]-0.6823n$

This is a triply nested relative of the generative formula shown in Figure 5. Like it is shows a fixed bandwidth along with some basic motifs reoccurring at various scales.

Figure 12. $f[n]=f[f[f[n-1]]]+f[n-f[f[n-2]]-1]$ rectified using $f[n]-0.459n$

This formula resembles the doubly nested version seen in Figure 8 and like it shows a higher order wave-like structure with increasing wavelength. Each wave module is a “sea-horse”-like structure, which develops an increasingly complex pattern of fluctuations as it grows larger in size.

Figure 13. $f[n]=f[f[f[n-1]]]+f[n-f[f[n-1]]]$ rectified using $f[n]-\tfrac{n}{\phi}$

This triply nested formula shows a link to the Meru-średhī and $\phi$ similar to the one explored in Figure 5. The rectified form shows cycles of increasing size in the sequence 2, 5, 13, 34 … i.e. $M[2k+1],\; k=1,2,3,4...$. These cycles are a journey up and down a fractal mountain, which we term the journey through the slopes mount Meru. The mountain has some fractal faces and others which are sheer cliffs. Further, the set of edges $f[n] \rightarrow n$ constitute a tree graph as seen with the above-described doubly nested version. Remarkably, there is one primitive lineage from which all other lineages branch off which is the Meru-średhī with the Dhenu-saṃkhya-s (Figure 14). Now the branches also show a curious relationship to the Dhenu-saṃkhya-s: at any section through a given level in the tree we have that many branches as the numbers that would count up to the corresponding Dhenu-saṃkhya in that level. Thus at the level of first branch the numbers are 4, 5 (2 branches); at the next level the numbers on the branches are 6,7,8 (3 branches);
at the next level 9,10,11,12,13 (5 branches) and so on. Thus on each branch there is an inherent relationship to the underlying Dhenu-saṃkhya, which can be seen in the figure. Whereas the graph in Figure 2 shows a strict bifurcation this graph shows a pattern for n-furcation related to the Dhenu-saṃkhya: if a non-primitive branch emerges from a number the $M[n]$ then it will display $n-3$-furcation (Figure 14). Thus, the branch from $M[4]$=3 will show 1-furcation; the branch from $M[5]=5$ will show 2-furcation; the branch from $M[6]=8$ with show 3-furcation and so on.

Figure 14. The branching graph for $f[n]=f[f[f[n-1]]]+f[n-f[f[n-1]]]$

One could explore other interesting features of these sequences but we stop here with the philosophical insight the imparted to us. Simple two-seeded sequences like the Meru-średhī represent a process that is directly dependent on the state of the system at two former time points. Thus they evolve directly as a function of time like the unconstrained growth of an organism under ideal conditions, which Nārāyaṇa tried to model. However, the doubly and triply nested generative formulae, while starting from the same seeds, do not develop directly dependent on time but rather as a second or third order consequence of states at former time points. Thus, while the simple case might be seen as a direct reaction of two former reactants leading to a current product, the nested specifications can be seen as reactants at former time leading to further reactants which in turn specify the current product. What we observer is that by the simple act of including this kind of higher order specification we get several different kinds of complexity: Some forms of complexity like the Takagi curve while intricate have a regular pattern to them. Other forms show different degrees of higher order pattern but are much more irregular in their immediate behavior. Finally there are those that are very irregular and not obviously predictable beyond some general statistical features. Actions on simple strings can generate complex forms — hence, these sequences could serve as an analogy for how higher order dependencies naturally generates complex structure in nature. Such processes could also be behind the patterns of other systems like human history. A question that leads to a deep philosophical puzzle is whether such processes might be active in biological systems.

Posted in art, Scientific ramblings |

## Some pictures relating to incidence of tuberculosis and AIDS

This another note of the type mentioned earlier. The earlier mentioned caveats apply here too. These are simplistic and superficial examinations of the issues being considered.

Tuberculosis is caused by Mycobacterium tuberculosis an actinobacterium and is predominantly transmitted by means of aerosol expelled from the respiratory tract of patients with pulmonary infection. Very few live bacteria are sufficient to establish infection in a new host. AIDS is caused by HIV-1 and HIV-2 which are related retroviruses of lentivirus clade. It is predominantly sexually transmitted although other kinds of transmission via blood products and mother to child are also possible. TB is a disease known from the earliest human record. In Hindu tradition we find its earliest mention in the Atharvan collection (the yakṣma-sūkta-s; e.g. AV-vulgate 2.33) and might have been discovered by the great brāhmaṇa Kaśyapa. The disease balāsa described by the Atharvan-s also seems likely to have been osseous TB, which has also been noted in approximately coeval Egyptians. AIDS has a more murky medical history. Its origins can be squarely placed in Africa where both forms appear to have emerged from related retroviruses infecting chimpanzees (HIV-1) and the mangabey monkey (HIV-2). Its world-wide spread is something which has happened very recently in the aftermath of the European penetration of sub-Saharan Africa. Yet these two diseases are believed to have gotten entangled because HIV suppresses immune system of the host by targeting the CD4+ T cells, macrophages and dendritic cells which are cells central to the immune response itself. This is an interesting evolutionary phenomenon with some deep ramifications. The weakening of the immune system by AIDS is said to facilitate opportunistic infection by M.tuberculosis.

This supposed connection between the two diseases made us check out the actual data:
-The incidence of TB is from WHO for year 2012
-The incidence of HIV is from WHO for year 2012
-The other data is from UN for the latest available year.
-The incidence of HIV is based on those being recorded as going for antiretroviral treatment drugs, so it is an underestimate of the actual number( e.g. Bangladesh in this data).

Figure 1

The number of TB and HIV incidences are positively correlated across 175 countries in a log-log plot of this data (Figure 1). The correlation has $r^2=0.456$ and slope is 0.662 (indicating scaling as roughly power 2/3). This is consistent with the pathological entanglement of the two diseases but the correlation is not very high suggesting that they have their own independent spheres of action. Indeed, TB was already a widespread disease with large pool of infections ages before AIDS became a global issue and retained that network even after the somewhat effective vaccination and debilitating antibiotic treatments emerged for it.

Figure 2

We next looked at how the incidences of TB scale with population of a country (Figure 2) in a log-log plot. One notices that the two are strongly positively correlated ($r^2=0.774$) and slope 1.14722 indicating a nearly linear relationship between the two. This suggests that irrespective of the population size and continent the country comes from there an approximately fixed incidence of TB for a unit of human population (median value of ratio of TB incidences to population $\approx 4.5 \times 10^{-4}$).

Figure 3

When we do the same for AIDS we seen an interesting difference (Figure 3). The two are again positively correlated in the log-log plot with a slope of 0.93 suggesting an approximately linear correlation of the number of incidences of AIDS with population size. However, the correlation is much weaker than what is seen for TB: ($r^2=0.454$). What could be the reasons for this? We chose to take a closer look at these two diseases because currently they can infect people pretty much anywhere via “regular” human activities such as coughing, spitting or sex. They are not dependent on a special predisposing factors like malaria which needs a vector with a geographically localized distribution. Hence, we would say that the weaker correlation for AIDS reflects a fundamental differences in the “regular” human activities like sex. Right away one can see that African countries have pretty much distinctly higher incidences of AIDS that Asian countries with comparable populations. A major factor in this could be the greater tendency for risky sexual behavior arising from the promiscuous mating systems in Africa as compared to Asia. This contention is supported by the two exceptions in Asia, Thailand and Cambodia, which are known to be centers of risky sexual activities.

Thus these plots are a good example of how a simple illustration can lead one to key factors in the differential epidemiology of diseases if one keeps ones eyes open.

Figure 4

Figure 5

One would expect that the incidences of these diseases are negatively correlated with life-expectancy. The above two figures (Figure 4 and 5) show this correlation on a log-log plot with life-expectancy. Yes, there is negative correlation, but it is weak based data which we used. When we look at the world at large it is slightly more pronounced for TB than for AIDS. However, when we consider the continent of Africa alone the correlation jumps up for AIDS ($r^2=0.323\; vs \; r^2=0.153$)and falls for TB ($r^2=0.165\; vs \; r^2=0.224$) relative to the global correlation than suggesting that specifically HIV infection is a notable factor in reducing life-expectancy in the African continent.

Posted in Scientific ramblings | Tagged , , , ,

## Cobwebs on the golden hyperbola and parabola

The material presented here is rather trivial to those who have spent even a small time looking at chaotic systems. Nevertheless, we found it instructive when we first discovered it for ourselves while studying conics. Hence, as part of recording such little tidbits of trivia that over the years have caught our eyes we are putting it down here.

Consider the iterative map,
$x_{n+1}=1+\dfrac{1}{x_n}$
Notably, irrespective of what starting number $x_n$ you take (with one exception) it converges to the Golden ratio $\phi=1.61803...$ (Figure 1).

Figure 1

The one exception is when $x_0=-\tfrac{1}{\phi}$. In this case it is rather obvious why it remains fixed at $-\tfrac{1}{\phi}$. However, notably, say you are very close to $x_0=-\tfrac{1}{\phi}$, e.g. -.61804, then you hover close to $-\tfrac{1}{\phi}$ for around 7 iterations and then drift away rapidly to converge to $\phi$ quite rapidly. The rate of convergence measured as the number of iterations you take to converge within a certain range of $\phi$ (say $\pm 10^{-7}$ in below figure) shows an interesting pattern: for values closer and closer to $-\tfrac{1}{\phi}$ you take longer and longer to converge to $\phi$ whereas values closer to $\phi$, not surprisingly, converge faster. Thus, while $\phi$ serves as the near universal attractor and $-\tfrac{1}{\phi}$ as the near universal repellor for this map, the repellor is actually weaker at repelling values that lie closer to it (Figure 2).

Figure 2

Rather dramatic jumps but still convergent behavior can be obtained at certain points: When $x_0=0, -\tfrac{1}{2}, -\tfrac{3}{5}, -1$ we jump between 0 and $\infty$ before moving towards convergence. This behavior can be studied in terms of the maximum value reached by $x_n$ before convergence towards $\phi$. The distribution of these values for various starting $x_0$ shows a peculiar fractal structure with peaks of decreasing heights as one moves towards the repellor $-\tfrac{1}{\phi}$ from either direction (Figure 3).

Figure 3

So what it is the connection to conics alluded to above? One can see right away that the above map corresponds to the hyperbola,
$y=1+\dfrac{1}{x}$

The iterations of the map can be rendered geometrically as the famous cobweb diagram used by students of chaotic dynamics: Start with a curve and a point representing $x_0$ on the x-y plane. Move along the vertical direction from that point till you hit the nearest point on the curve. Mark that segment. Then proceed in the horizontal direction till you hit a point on the line $y=x$. Mark that segment. Then move vertically again to hit the curve and so on. Repeat this procedure till you converge. When you do this procedure for the above hyperbola (Figure 4) then every point except one with x-coordinate $-\tfrac{1}{\phi}$ converges to the point $(\phi, \phi)$, which is the attractor. The repellor is $(-\tfrac{1}{\phi},-\tfrac{1}{\phi})$. One can see that these fixed points are intersects of the said hyperbola and the line $y=x$

Figure 4

Now there exists a parabola with the same fixed points as the above hyperbola:
$y=x^2-1$

For this parabola, if a starting point for the cobweb diagram lies in the band delimited by the lines $y=\pm \phi$ then it moves towards $-\tfrac{1}{\phi}$. But it does not converge to that point. Instead it is eventually trapped in a four-point orbit: $(0,0); (-1,0); (-1,-1); (0,-1)$ (Figure 5). If the starting point lies outside the above band then it races away to $\infty$. But there are a some points where it actually converges to one or the fixed points. Any point whose x-coordinate is $\pm \phi$ converges to $(\phi, \phi)$. Any point whose x-coordinate is $\pm \sqrt{\phi}$ or $\pm \tfrac{1}{\phi}$ converges to $(-\tfrac{1}{\phi},-\tfrac{1}{\phi})$. The closer point’s x-coordinate is to one of the above values the greater the number of iterations it requires to be placed in the final four-point orbit. Thus, the parabola with the same fixed points as the said hyperbola displays a very different behavior — the outcomes are convergence to a fixed point, divergence to $\infty$ or eventually settling into a fixed orbit. However, for certain parabolas such as the logistic parabola, $y=k x(1-x)$ we see the famous chaotic behavior for certain values of $k$.

Figure 5

Posted in Scientific ramblings |