The minimal triangle circumscribing a semicircle

Consider a fixed semicircle with center at O and radius r. Let \triangle ABC be the isosceles triangle which circumscribes it (Figure 1).

triangle_circumscribing_semicircleFigure 1

What will be the characteristics of the minimal form of the said triangle, i.e. triangle with minimum perimeter, which will circumscribe the said semicircle? Let the base angles of the isosceles triangle be \angle x. In order to determine the equation for the perimeter with respect to this base angle this we perform the following construction (Figure 2):
1) Join point O to one of the points of tangency of the circumscribing triangle ABC and the semi-circle (Point I_1, I_2).
2) This gives us two right triangles \triangle OI_2C and \triangle AI_2O.
3) Using their sides and \angle x we can determine the lengths of the segments \overline{OC}, \overline{I_2C}, \overline{I_2A} (Figure 2). Twice their sum will yield the equation of the perimeter of the triangle. It is:

P=2r\left (\tan(x)+ \dfrac{1}{\tan(x)}+\dfrac{1}{\sin(x)}\right)

triangle_circumscribing_semicircle2Figure 2

Thus, we will get the minimal triangle when the function y=\left (\tan(x)+ \tfrac{1}{\tan(x)}+\tfrac{1}{\sin(x)}\right) attains a minimum (Figure 3).

semicircle_triangle_minimFigure 3

The derivative of this function is (Figure 3):
y'=\left (\dfrac{1}{\cos^2(x)}- \dfrac{1}{\sin^2(x)}-\dfrac{\cos(x)}{\sin^2(x)}\right)

We get our minimum when y'=0. Thus at minimum,

\dfrac{1}{\cos^2(x)}=\dfrac{1}{\sin^2(x)}+\dfrac{\cos(x)}{\sin^2(x)}\\[7pt] \sin^2(x)=\cos^3(x)+\cos^2(x)\\[7pt] 1-\cos^2(x)=\cos^3(x)+\cos^2(x)\\[7pt] \cos^3(x)+2\cos^2(x)-1=0

Putting y=\cos(x) we get the cubic equation y^3+2y-1=0. It can be factorized as (y + 1) (y^2 + y - 1) = 0. From which we get the roots as (-\phi, -1, \phi'), where \phi=\tfrac{1+\sqrt{5}}{2} is the Golden ratio and \phi'=\tfrac{\sqrt{5}-1}{2}=\tfrac{1}{\phi}. x=\arccos(y). Since x is the base angle of an isosceles triangle, y=\phi' is the only valid value for which the \arccos(y) may be taken. Thus we get,

x=\arccos(\phi') \approx 0.9045569

Thus, when the base angles of the isosceles \triangle ABC are \arccos(\phi') then we have the minimal circumscribing triangle of a given semicircle (Figure 4).

triangle_circumscribing_semicircle_shrIFigure 4

The special features of this triangle can be seen in Figure 4, where the radius of the fixed semicircle is taken to be a unit: It is made up 2 right triangles whose sides specify the equation of \phi, i.e. \phi^2-\phi-1=0. The various powers of \phi are captured by it. The perimeter of the minimal circumscribing triangle is thus 2r \phi^{5/2}. The 2 points where the triangle touches the circular arc of the semicircle are specified by the angle \arcsin(\phi').

This triangle has a special place in the great geometric manifestation of the Bhairava and the Bhairavī, known as the Śrī-yantra, versions of which are worshiped in the Paścimāṃnāya (Kubjikā; Western lodge) and Dakṣiṇāṃnāya (Śrī-kula; Southern lodge) of the kaula tradition. This yantra is described in a key text of these traditions, the Saundaryalaharī, thusly:
caturbhiḥ śrīkaṇṭhaiḥ śivayuvatibhiḥ pañcabhir api
prabhinnābhiḥ śaṃbhor navabhir api mūlaprakṛtibhiḥ ।
tryaś-catvāriṃśad vasu-dala-kalāśra-trivalaya
tri-rekhābhiḥ sārdhaṃ tava śaraṇa-koṇāḥ pariṇatāḥ ॥

shrI_yantraFigure 5

The above verse from the Saundaryalaharī may be rendered as:
The (tri)angles in what constitute your dwelling, which is built of 9 mūla-prakṛti-s, i.e. the 4 Śrīkaṇṭha-s and the 5 Śiva-yuvati-s, all [stationed] apart from Śambhu, along with the lotuses with Vasu- (8) and lunar-digit- (16) petals, the 3 circles and the 3 lines, are 43 in number. (modified version of the translation of Subrahmanya Shastri and Shrinivasa Ayyangar)

This verse precisely captures the essence of the structure of the Śrī-yantra. The 9 mūla-prakṛti-s are the 9 basic triangles that constitute to the yantra. They represent the 9-fold form of the Bhairava of the Paścimāṃnāya, namely Navātman-bhairava. They are in turn divided into: 4 upward-facing or liṅga triangles, which are termed Śrīkaṇṭha-s. These are the manifest forms of the Bhairava. 5 downward-facing or yoni triangles, which are termed Śiva-yuvati-s. These are the manifest forms the Bhairavī. The central point, the bindu, lies apart from from them and represents the Bhairava in his state as the unmanifest singularity, i.e. Akulavīra. In contrast, the rest of the yantra with is manifold geometric structures can be seen as his śakti, Kaulinī, who manifests diversely as the kula-s of Yoginī-s. These Yoginī-s inhabit all elements of the structure: For example, the 8 mātṛkā-s (Caṇḍikā, Brāhmī, Rudrāṇī, Kaumārī, Vaiṣṇavī, Vārāhī, Indrāṇī, Cāmuṇḍā) inhabit the 8-petalled lotus (This is as per the mūla-prakrīyā of the Ānanda-bhairava tradition. Other paramparā-s place the Kāma-yoginī-s from Anaṅgapuṣpā to Anaṅgamālinī in these petals). The 16 Śrī-nityā-s inhabit the 16-petalled lotus.

The 43 triangles are reckoned as the 42 outward-pointing triangles and the inner-most triangle formed by the intersection of the 9 basic triangles. The outermost of these has 14 triangles known as the Saubhāgyadāyaka in which dwell the Sarva-saubhāgyadāyaka-yoginī-s. The second rung with 10 triangles is known as the Sarvārthasādhaka with eponymous yoginī-s. The third rung again with 10 triangles is the Sarva-rakṣākara with the 10 yoginī-s of that kula. The 4th rung with 8 triangles is the Sarva-rogahara with the 8 Vāgdevī-s as its resident yoginī-s. The 5th rung with the central triangle around the bindu has Kāmeśvarī as its yoginī.

So, where does our above minimal triangle figure in all this? It defines the Śrīkanṭha-1 which is the first triangle to be drawn while constructing the Śrī-yantra. Also, notably, the first Śiva-yuvati is an isosceles triangle formed by welding two 3-4-5 right triangles along the side of length 4. As we have seen before, this 3-4-5 triangle, the most primitive of the integral right triangles, emerges naturally as the maximal triangle defined by the characteristic ellipse of a right triangle. Thus, the two primary triangles of the Śrī-cakra natural feature as the ultima of certain problems of triangle geometry.

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Indo-Iranica: epic ruminations

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We first began our observations on the shared motifs in Indo-Iranian traditions when we were around 8 years of age after our mother introduced us to the Iranian epic, the Šahnāme. Over the years we built up these correspondences even as we expanded our studies on our ancestral Indo-European (IE) traditions. Several notable points have also been uncovered in the work by occidental Indo-Europeanists such as Dumezil, Watkins, and Puhvel. Yet our view of this is somewhat distinct — given that we are insiders of one of the last surviving Indo-European traditions, for us it is a reflection on our living tradition. We have recently learnt a lot about the history of the early Indo-Europeans due to archaeogenetics; this allows us to place certain things in a more correct historical framework. Hence, we thought it worthwhile to put down a few words on this matter. In particular, we felt that such an exposition might be of use to the Hindu students of their ancestral tradition — a kind of exploration of and introspection on tradition that is unfortunately rather uncommon among our people. Indeed, we would go as far as to state plainly that no amount of traditional mastery can substitute for the insights that comparative IE studies can provide for our tradition. Hence, ignoring it necessarily means an incomplete understanding of tradition. Accordingly, we present here a few comparative points on the IE epic traditions. Among the motifs we uncovered was the Garuḍa motif which is common to the Iranian Rostam and the Ikṣvāku heroes Rāma and Lakṣmaṇa on our side. We also noticed the motifs shared by the great pitāmahā Bhīṣma of the Hindu national epic and Rostam the hero of the Iranian national epic. In the late 1990s we learnt of the Iranicist Skjaervo’s work in this direction and found that several points coincided with our investigations. Nevertheless, we feel it might be useful to put down some of our thoughts in this this regard against the larger background of the I-Ir epic tradition. However, right at the outset, we should mention that this account is by no means aiming to be a comprehensive presentation of our thoughts on the matter and merely scratches the surface.

The Hindu epic traditions
The Hindu itihāsa tradition, which is a continuation of the ancient IE epic tradition that emerged on the Pontic–Caspian steppe, comes down to us in the form of the extensive Rāmāyaṇa of Vālmīki and the Mahābhārata of Kṛṣṇa Dvaipāyana. The first minutely captures the old “standard” religion, i.e. the ancestral Aindra religion of the Indo-Europeans [Footnote 1] as a humanized epic narrative focused on a historical hero Rāmacandra and his brothers, who at some point after the ārya conquest of Northern India occupied the dynastic seat of the ancient Ikṣvāku monarchs. While it was secondarily reinterpreted by the tradition which adopted the ancient IE deity Viṣṇu as its focus, it still bears the emphatic stamp of the old Aindra religion (wherein Lakṣmaṇa is the proper cognate of Viṣṇu) and also other ancient para-Aindra traditions like that of the great god Vāyu and Yakṣa Vaishravaṇa.

The Mahābhārata on the other hand might be termed the national epic because whatever kernel of history it contains is intimately linked with the emergence of a unified Indian state under the ārya-s in northern India sometime around 1300 BCE. However, it should be understood that the Mahābhārata is not history in the sense it is commonly understood today. It is again a classic IE epic narrative which “pours” diverse historical elements into the “bottle” of ancient mythic motifs associated with the IE religion and its I-Ir reflex. This is made explicit in the epic itself wherein its various characters are described as incarnations of asura-s, deva-s and other divine beings — e.g. the Bharadvāja hero Aśvatthāman is explicitly described as being the combined incarnation of the mighty god Rudra along with the god of death (Yama) and also the embodiment of lust and wrath (Mbh-“Pune” 1.61.66). These older religious elements are combined with the newly emergent strain of a Viṣṇu-focal system, the Sātvata tradition, emerging among eponymous clans, and to an extant the the Rudra-focal system. In historical terms, at least 4 distinct streams of “history” are melded in an anachronistic fashion into the story frame of the Mahābharata, despite the apparent chronological coherence in the epic: 1) The historical elements of the old Kuru-Pāñcāla confederation; 2) the emergence of Pāṇḍu power and their dynasty; 3) The Sātvata cycle with their heroes like Kṛṣṇa Devakīputra, Balarāma, Pradyumna, Aniruddha and Sātyaki (pañcavīra-s) among others; 4) The hero Bhīṣma.

The Iranic epic traditions
Sadly, the Iranic epic traditions have not come down to us in the pristine form of the Hindu epic traditions due the destruction of the Iranian empire and the satellite Iranic states by Islam and the cult of Jesus. There are two rather distinct streams of the Iranic epic tradition that have come down to us: 1) The Caucasian stream that is represented by the Nart epic material represents the traditions of the Northern Saka-s, Sarmatians (Sairima) and the Alans (Airya). The word Nart itself is a cognate of Sanskrit nṛ with a deep PIE etymology. Unfortunately, the remnants of these peoples were steam-rolled by the Christian Russians during their conquest of the region in the 1800s. Starting with Shora Begmurzin Nogma down to Colarusso’s good compendium the surviving material of these peoples has been collated and translated. 2) The Iranian stream which has come down to us in the form the Šah-nāme, the Garšasp-nāme, the Farāmarz-nāme and the Bānu Gošasb-nāme among several others. Of these the Šah-nāme appears to have been the model on which the other works were written. Importantly, these are very late works coming from a time well after the Mohammedan destruction of the Iranic religions. However, they appear to contain material from a much earlier Sassanian collation of the epic material in middle Persian the Khwadāy-nāmag. Of them, the Šah-nāme is well edited and studied. But recently the work of Iranicists like Gazerani and others has provide a lot of new material from the other epic cycles. Much of our discussion will be on material that can be gleaned from the Šah-nāme and the more recently presented texts. The Šah-nāme is an anachronistic medley of old I-Ir mythology, main-line (i.e. Zoroastrian-allied) Iranian epic tradition, the epic tradition of Sakastana (the region encompassing Eastern Iran, Balochistan and southern Afghanistan in modern parlance, where the Saka-s had established a kingdom starting around 2100 years ago), and the later history of the Achaemenids, Parthians and Sassanians.

Some examples of anachronism in the Šah-nāme
The lateness of the Šah-nāme is useful in one way: it allows us to understand how epic anachronism evolves because some of the details pertain to historical events that have independent records of their actual unfolding. The later Iranians had only a vague memory of certain periods of their history — in particular the Achaemenid period. In contrast, in non-Šah-nāme histories, including accounts of their rival, the Greeks, this period is remembered as one of their most glorious periods graced by great monarchs like Cyrus, Cambyses and Darius. Strikingly, the Iranians remember the dramatic Achaemenid architecture under names like Takht-i-Jamshid or Naqsh-i-Rostam, thus projecting their constructions to the ancient past to associate them with epic heroes like Yima Kshaeta (Jamshid) and Rostam. A parallel phenomenon is seen among the Hindus where temples built by historical kings are sometimes projected back to the time of the Pāṇḍava-s with complete loss of memory of the real builder among the masses. This chronological disconnect is most dramatically reproduced in the Šah-nāme in the transition between the ancient period — a blend of main-line Iranian and Sakastanian tradition — and the historical period beginning with Alexander of Macedon’s invasion and destruction of the Iranian empire.

The way the historical period is introduced is via Bahman, the last of the ancient kings, who puts an end to the lineage of Rostam. Bahman was smitten by the beauty of his own daughter Homāya and engaged in incestuous coitus with her. The result was the son named Dārāb (=Darius the Great). He made Homāya succeed him but this displeased his other son Sāsān through his original wife. This Sāsān is said to have hence left for India where he married a Hindu woman to give rise to a long line of Sāsān-s who eventually gave rise to the founder of the Sassanian dynasty (see below for a parallel account in the clan of Rostam which might have bearing on the origin of the Pallava-s of South India). Some other accounts instead place this account of Sāsān in Nishapur. In Iran itself Dārāb became the ruler and marching on the Greeks defeated them. He then took Nāhīd the daughter of the Macedonian king Filakus as his wife. But soon after impregnating her, Dārāb sent her back to Macedon repelled by her bad smell from poor hygiene. There she gave birth to Sekander (=Alexander of Macedon). In Iran his other wife gave him a son named Dārā (=Darius-III).

Sekander with Arsṭālīs (=Aristotle) as his minister refused to pay tribute to Dārā resulting in a war. After facing successive defeats Dārā fled to Kerman with Sekander in hot pursuit. Dārā sent a message to Porus (Paurava) the king of the Hindus to help. But before Porus could send assistance Dārā was stabbed by his own ministers. As he lay dying he gave his daughter Roxana in marriage to Sekander, asking him to preserve Avesta and the uphold Zoroastrian religion. Alexander killed Dārā’s assassins and made himself the emperor of Iran. Then Keyd (=Āmbhi), a king of the Hindus, sent him a Brāhmaṇa philosopher, a physician, a beautiful woman as a wife and an akṣaya-madirā-pātra. Pleased with these gifts Sekander spared him, attacked Porus and eventually defeated him.

It is immediately apparent that this tradition preserves a nucleus of history, i.e. the account of the destruction of the Achaemenid empire by Alexander followed by his going “native” by adopting Iranian customs to the distaste of his Macedonian and Greek entourage. However, we can also see a melding of mythic motifs and anachronistic conflation and telescoping of various historical elements: 1) the incest of Bahman is an old motif (c.f. the incest of Prajāpati from the Ṛgveda onwards). 2) the great historical rulers like Cyrus, Darius the Great etc are either forgotten conflated into the single character of Dārāb. 3) The Sassanian dynasty’s foundation is ahistorically coupled with the Achaemenid dynasty. Here, one may note that the supposed Hindu wife of Sāsān could be a motif that was recycled from the story of Farāmarz, the son of Rostam, who is said to have obtained his wife from an Indian island after rescuing her from a demonic Triśiras Tvāṣṭra-like daeva. This might also reflect a historical memory of a noted Sassanian emperor marrying a Gupta princess. 4) Alexander and Darius-III are made half-brothers — bringing in an old IE motif of the clash of clansman (e.g. Kaurava-s and Pāṇḍava-s).

With regard to Alexander it may be noted that one stream of Iranian tradition clearly remembers him as the “accursed, demonic one” who destroyed the Zoroastrian scriptures during his assault on Persepolis. In contrast, this tradition mentions Darius asking him to preserve the Avesta. It is conceivable that the two accounts preserve different aspects of the actual history — Alexander initially destroyed the Iranian religious center of Persepolis and brutally killed Spitamenes, the descendant of Zarathustra, who valiantly led the Iranian resistance against the Macedonian invaders. Indeed, the Iranian tradition that remembers the “accursed Alexander” specifically mentions his killing of Zoroastrian ritualists, i.e., Dasturs and Mobeds. On the other hand, Alexander adopting Iranian customs after the conquest of their empire might reflect the fact that he did eventual allow the restoration of Zoroastrian rituals in Iran. Likewise, in India too while certain Hindu sources close to the time (e.g. Yuga-purāṇa) remember the yavana-s as barbarous, we also know that they eventually extensively adopted and supported Hindu traditions in the so-called Indo-Greek kingdoms.

Thus, for a moment if we imagine a scenario where all alternative narratives of the period of history under consideration were lost and only the Šah-nāme survived then we would be left with a coherent narrative that despite its historical kernel does not capture the actual unfolding of events or relationships between the characters in it. Hence, this is something important to bear in mind while trying to read history in related epic narratives like the Mahābhārata — there is likely a historical pith to the narrative but it might not represent the actuality. This situation starkly comes to the fore as we move further back in time with respect to the Iranic and Hindu epic traditions. However, in both cases we are not entirely lost because we have the parallel records, often from the older scriptural texts like the Veda and the Avesta. We shall consider this by taking up the case of the clan of Rostam in the Iranian epic tradition.

The clan of Rostam and the intertwining of the Sakastana and main-line Iranian traditions
Reading the Šah-nāme one will be left with little doubt that Rostam was the greatest Iranian hero and that his clan consistently produced a whole line of such heroes (and heroines). This is quite like the Pāṇḍava-s in the Hindu national epic. Yet there is something striking about this family: except for its legendary founder it finds no mention at all in the Avesta, despite that text mentioning several Iranian heroes and ritualists who in the Šah-nāme are associated with the clan of Rostam. This strange situation closely parallels the conundrum we encounter in Hindu tradition where, the Veda mentions several Kuru-s, Pāñcāla-s and Kṛṣṇa Devakīputra, who appear in the Mahābhārata but almost completely fails to mention its main heroes, the Pāṇḍava-s.

Turning to the Avesta we see the following hero/ritualist figures who also figure in the Šah-nāme under their New Persian names given in a certain order that apparently reflects history: 1) Gaya Maretan (=SN: Keyumars) the primordial puruṣa of the Iranian world. 2) Haošyanha Paradāta (=SN: Hushang) the great ruler. 3) Takshma Urupi (=SN: Tahmuras) the great warrior. 4) Yima Vaivanant Kshaeta (=SN: Jamshid). 5) The interlude of the demon Azi Dahāka (=SN: Zohak). 6) Thraetaona Athwya (=SN: Feridun), who smites the Azi with his vazra (=SN: Gurz). 6) Keresāspa (=SN: Garšasp) the gadā-wielding warrior. 7) The kavi-s. These include among others Kavi Usan= (SN: Kay Us), Kavi Kavāta, Kavi Aipivohu and Kavi Haosrava (=SN: Kay Khosraw). 8) Zarathuštra. 9) Vīštāspa.

Now one can see that several of these early characters are shared in some form with the Hindus and mostly appear right from the RV onward: 1) Gaya Amartya. 2) Yama Vaivasvata Kṣatriya. 3) Ahi Dāsa. 4) Trita Āptya who, in the battles with the demons, assists Indra who wields the vajra. 5) Kṛśāśva, the “father” of the weapons. 6) Uśanas Kāvya. 7) Perhaps Iṣṭāśva. Some of these are gods while other are demons and yet others legendary men. These features along with their early appearance in both Indian and Iranian tradition suggests that they belong to the common Indo-Iranian past on the steppes.

What is notable is that neither this early Iranian nor the Hindu tradition remembers Rostam or most of his clan with the exception of Keresāspa, who is regarded as its founder in the Šah-nāme. In the early Iranian tradition of the Avesta he is remembered as an enormously strong man with his great gadā. He has a nearly semi-divine status (much like in Viśvāmitra’s mention of him in the Rāmāyaṇa) and his fravashi ( \sim pitṛ) is called upon by the Zoroastrians to protect them from rapacious attackers or thieves. He also slays a dreaded Ganderewa (Skt: Gandharva) who attacks the “good principle” in the world and several other adversaries. Notably, he is given the epithet naire-manah, i.e. of manly mind. This epithet becomes the name of his son Nariman in the Šah-nāme. In the surviving Pahlavi exegesis and the Dēnkard, he is termed Sāmān Kersāsp. Now, the epithet Sāmān is the name of his grandson as per Šah-nāme. Thus, it appears that the epithets of the ancient hero Keresāspa were used as names to provide a genealogy that connects Rostam to Keresāspa. Despite his heroic stature, he is also mentioned as sinning: as per the Pahlavi exegesis he worshiped images like the Hindus, while the Bundahišn describes this as worship of daeva-s by Sāmān. We suspect that behind these allusions lies a faint memory of Keresāspa belonging to the layer of the “un-divided” old daiva religion prior to Zarathuštra’s counter-religious modifications.

Thus, the Šah-nāme gives the lineage of Rostam as in Figure 1:

Rostam1.pngFigure 1. The figures are scaled according to their “importance” in the Iranic epic traditions. Some of the more “apocryphal” descendants of Rostam and his brother Zavāra are left out.

From Zāl onward we see considerable detail in the accounts of the heroes/heroines of the clan. However, many of these details are ancient mythic motifs. These include:
\bullet The romance of Zāl and Rūdāba. This encompasses classic motifs of the hero winning a beautiful wife from the enemy side found in many traditions. Specifically, it connects Zāl the Airya hero to Rūdāba who has Azi Dahāka as her ancestor, thereby bringing in the demonic element in the ancestry of the mighty hero Rostam. Her Hindu affinities are hinted by her father bearing the name Mahārāja in an Iranic transmutation. From an Iranian stand point the daeva-worshiper would be affiliated with the entities like Azi Dahāka. Such pairings between the “daiva” and the “dāsa” or “dānava” figures are seen across the IE world.

\bullet Zāl was born albino (c.f. Pāṇḍu) and was nursed by the Garuḍa-like divine bird meregha Saēna (=Skt śyena; SN: Simorgh), who gave him 3 of his feathers asking him to burn them whenever he needed the bird’s assistance. Rūdāba was about to die while giving birth to Rostam due his gigantic size. It was then that Zāl called for Simorgh who taught him the cesarean procedure to save his wife and son: The bird first asked Zāl to anesthetize his wife by giving her wine. Then it asked him to perform the cesarean section and take out Rostam. Post-surgery it revealed the herbs to heal her wound and the final healing was achieved by the stroking of Rūdāba with the feathers of Simorgh (see below). This Simorgh is mentioned as bearing the medicines on his tree in the incantation to the god Rashnu:
ýatcit ahi rashnvô ashâum upa avãm vanãm ýãm saênahe ýâ hishtaite maidhîm zrayanghô vourukashahe ýâ hubish eredhwô-bish ýâ vaoce vîspô-bish nãma ýãm upairi urvaranãm vîspanãm taoxma nidhayat
Whether you, O holy Rashnu! are on the tree of the eagle, that stands in the middle of the sea Vouru-Kasha, that is called the tree of good remedies, the tree of powerful remedies, the tree of all remedies, and on which rest the seeds of all plants; we invoke (Yašt to Rashnu-17).

This is consistent with the bird’s role as the teacher of the medical procedure to his protege Zāl and matches that of his cognate Garuḍa Tārkṣya Ariṣṭanemi who possesses cures. In the Rāmāyaṇa, he cures Rāma and Lakṣmaṇa when they were paralyzed by Meghanāda’s poison missile (see below) and Hindu tradition remembers the bird deity as the teacher of the medicine deity Dhanvantari:
nārāyaṇāṃśo bhagavān svayaṃ dhanvantari mahān ।
purā samudra-mathane samuttasthau mahodadheḥ॥
sarva-vedeṣu niṣṇāto mantra-tantra-viśāradaḥ ।
śiśyo hi vainateyasya śaṃkarasyośiśyakaḥ॥
The great Dhanvantari is a part of Nārāyaṇa himself. Formerly, he arose during the churning of the ocean. Learned in all the Veda-s and an adept in mantra-s and tantra-s he is the student of Vainateya and an adjunct student of Śankara.

\bullet The conflict of Sohrab and Rostam. This famous episode is replete with various motifs shared with the Mahābhārata. Rostam going to the foreign land to acquire a woman (Tahmina) is reminiscent of Arjuna going to the distant land to acquire Citrāṅgadā and Ulūpī. Tahmina approaching Rostam by herself for a son is again parallel to the approach of these Mahābhārata characters, especially Ulūpī. The consequence of their intercourse is a son who does not recognize the father but possesses similar martial ardor. They finally fight in an encounter that results in the death of one of them. In the case Rostam the son, Sohrab, is slain while Babhruvāhana slew Arjuna. In both cases there is the possibility of revival with a magical panacea. In the case of Arjuna that is obtained but Rostam fails to obtain it for Sohrab. Finally, one could also point out that the setting of the death of Sohrab at Rostam’s hand is in a great war and parallels the killing of Karṇa by Arjuna. In this regard, it may be observed that Rostam gives Tahmina an ornament to tie on their child before leaving her. This might be compared to the ornaments which Karṇa obtained at birth from Sūrya. In the case of Rostam, a similar incident again plays out with his other son Farāmarz and his Amazonian warrior daughter Bānu Gošasb who fight Rostam and his brother Zavāra. However, in this case they recognize each other before anyone is killed.

These are just a few examples of some ancient motifs being woven into the history of the clan of Rostam. The most striking part is that they play out during the reign of the Kavi-s as kings of Iran. While the Kavi-s are repeatedly mentioned in the Avesta not once are these Rostamian heroes mentioned alongside them. Thus, it almost appears as though they were superimposed on to the older history of the Kavi-s who are now reconfigured as kings of Iran. The end of the reign of Kavi Haosrava in the Šah-nāme bears a motif which is paralleled in the Mahābhārata. At the end of his long reign Kavi Haosrava renounced his kingdom for an ascetic existence and left for the mountains with 8 pahlavān-s. There their people beseech them not to leave the kingdom. Of the 8 pahlavān-s Zāl, Rostam and Gutarza (Gōdarz) [Footnote 2] return to the state. The remaining 5 pahlavān-s, Ṭus, Gēv, Faribarz, Bizhan and Gostaham continued with the Kavi until they reached a spring, where he took a bath, chanted the Avesta, bade farewell to his friends and vanished. The 5 pahlavān-s died shortly thereafter in the mountains in a snowstorm while searching for Kavi Haosrava. This motif to our knowledge is uniquely shared by the Indo-Iranians and is not found among other IEans. It parallels the last journey of the Pāṇḍava-s to the great mountain in Central Asia. 5 of them also die in the snowy mountains in the Mahābharata but the emperor Yudhiṣṭhira reaches the realm of the gods. On the Iranian side, Kavi Haosrava too does not die and awaits the coming of the Saošyant-s (Iranian Kalkin-like figures) at the end of time and assists them in those final days.

So where does the cycle of the Rostamians appear to enter Iranian tradition? Rostam is traditionally associated with Sakastana. In contrast, the Kavi-s being present in the Avesta itself can be associated with the mainline Zoroastrian tradition in Iran. Thus, it appears that the Iranian epic tradition of the Šah-nāme was formed by the amalgamation of this mainline tradition with the Rostamian cycle brought in by the Saka-s during their invasion of outer Asia and the conquest the region that came to be known as Sakastana.

Rostam, heroes in Hindu epics, and Bhīṣma
At first sight there is no reason to connect Rostam and Bhīṣma who superficially appear to play rather distinct roles in their respective epics. Indeed, some of the more overt features of Rostam can be identified with other heroes in the Hindu epics. For instance, above we mentioned the father-son killing motif shared with Arjuna. His inheritance of “divine weapons” like the the bow of Keresāspa and vazra Thraetaona Athwya remind one of Rāmacandra, the hero of the Rāmāyaṇa, who obtained the weapons of Kṛśāśva via Viśvāmitra. His proficiency with the vazra of Thraetaona Athwya, which is portrayed more like a gadā, his strength and frame which are explicitly compared to an elephant, and his elephant-killing capacity remind one of Bhīmasena. Indeed, his title in the Iranian epic would be rendered in old Iranian as Takhma-tanu: mighty bodied. Consistent with this, in the Šah-nāme Rostam is repeatedly presented as lifting giant rocks. Again Rostam’s battle with the powerful king Kok-e Kuhzād, who is said to rule the land between Sakastana and India from a powerful mountain stronghold, resembles the famous showdown between Bhīma and Jarāsaṃdha at his stronghold of Girivraja.

However, a closer look reveals many specific parallels between Rostam and Bhīṣma: In generic terms both of them are great warriors in the old IE mold who are basically unconquerable. Neither of them could be killed in a straight battle and certain tricks had to be used to finally kill them (see below). Both are also very long-lived: Rostam’s life encompasses the reigns of nearly all the Kavi-s. Similarly, the pitāmahā’s life spans the entire period from Śaṃtanu all the way to the time after Yudhiṣṭhira becomes king. Most interestingly, the name of Rostam’s mother Rūdāba means river water. Rostam’s own name in the old Iranian would have been *Rautas-takhma: river-strong. This is a rather strong homology to Bhīṣma who is the son of the goddess of the river Gaṅgā. Both of them are dynastic guardians who never attempt to become kings themselves though they might act as viceroys. Notably, in being dynastic guardians both side with the less-deserving or the less-legitimate successor to the throne. On the Hindu side, Bhīṣma despite professing his love for the successors of Pāṇḍu stands firmly by Duryodhana. On the Iranian side, Rostam with his circle of pahlavān-s favor Kavi Haosrava on the throne as opposed to the legitimate heir Faribarz.

Then there is their death: both had to be killed by a stratagem: In Bhīṣma’s case it involved the use of Śikaṇḍin as the front with Arjuna striking from behind. In the case of Rostam, the king of Kabul used his jealous half-brother Shaghād borne of a slave woman to Zāl to lead the former to his death. Zāl had sent Shaghād to live with his tributary the king of Kabul because his ritualists had foretold that he would bring destruction to his house. They hatched a plan to kill Rostam by feigning a fight in course of which he had also insulted Rostam. Then Shaghād went over to Rostam and asked him to attack the king of Kabul. When Rostam and his full brother Zavāra arrived to attack Kabul, the king claimed that he had accidentally misbehaved under the influence of alcohol. The king then agreed for peace and asked Rostam and Zavāra to join him in a hunt. In course of that Shaghād led Rostam to a trap which they had set up — a pit full of spears. Rostam and his long-lived horse fell into the pit and they were impaled through and through by the spears. Somehow Rostam climbed out due to his great strength and fooled Shaghād by saying that his end was at hand. Then he asked Shaghād to correct his ways and lead a proper life with Rostam’s powerful son Farāmarz. Thus, appearing to let him go, Rostam asked for his bow and two arrows in case a wild animal came to eat him as he lay dying. Shaghād gave them to him but still fearing him ran away and hid behind a tree. Rostam though dying from being pierced all over shot an arrow to transfix Shaghād to the tree; then he died. In the mean time his brother Zavāra had also also fallen in another spear-filled pit and died. Thereafter, Rostam’s son Farāmarz filled with wrath did an “Aśvatthāman” by slaying the king of Kabul with his entire family and wiping out the city.

Both Bhīṣma and Rostam are impaled all over by arrows or spears. Despite that they do not die immediately. Instead, they linger on only to die after they have either completed their final wishes. Thus, in both their birth and death these long-lived warriors of the Indian and Iranian worlds are united.

Rostam and Esfandiyār
Esfandiyār, old Iranic Spentdāta, is presented in the epic literature as the son of Vīštāspa, the patron of Zarathuštra. He was a holy warrior enforcing Zarathuštra’s counter-religion and his brother, the idol-breaking Pešōtanū, is said to be an immortal who would lead Zoroastrians against their enemies at the end of time [Footnote 3]. In the epic tradition, his father asked Esfandiyār to get Rostam bound in chains for his disregard towards the king. Moreover, Rostam was a follower of the older (ancestral I-Ir) religion and did not want his people to submit to the counter-religion Vīštāspa was pushing for. In this regard a parallel is again seen with Keresāspa the legendary ancestor of Rostam, who is presented as sinning by following the demonized old religion as opposed to the creed of Zarathuštra.

Esfandiyār told his father that a respectable and aged warrior like Rostam should not be disgraced thus, especially given that he had faithfully served the the kings of Iran. But at his father’s insistence Esfandiyār, Pešōtanū and the former’s sons went to arrest Rostam. Esfandiyār sent his son Bahman to ask Rostam to submit. He agreed to come and see the king but refused to be put in chains. He also offered to apologize to the king if he felt offended but firmly refused the idea of being bound. Pešōtanū suggested to his brother to accept this deal but Esfandiyār refused and challenged Rostam to combat. While the two were locked in an evenly poised combat, Rostam’s brother Zavāra and his son Farāmarz launched a counter-attack on Esfandiyār’s positions and killed two of his sons, one of whom was apparently an Athravan. After a prolonged fight Rostam’s mighty weapons failed to slay Esfandiyār but the latter’s special arrows breached Rostam’s armor and severely wounded him and his horse forcing him to retreat. It was at this moment he asked his father Zāl for a strategy. Zāl called the great bird Simorgh (Av: meregha Saēna) by burning one of the feathers it had given him [Footnote 4]. The bird removed the 8 arrow-heads embedded in Rostam’s flesh and soothed his wounds by stroking them with its feathers. It then instructed Rostam to bind the wounds and stroke them with its feather dipped in medicinal milk for a week for a complete a cure of Rostam and also his horse. The bird was also displeased with Esfandiyār for he had killed another eagle who was Simorgh’s associate [Footnote 5]. He then revealed to Rostam the secret of slaying Esfandiyār. It asked Rostam to make a double-headed arrow from the wood of the tamarisk tree and shoot at his eyes with it. The bird then took him to a distant land to obtain a branch of the tree and fashion the arrow. The next morning as the fight resumed with much fury; Rostam shot this arrow and struck down Esfandiyār.

However, Esfandiyār had a Zarathuštrian curse protecting him that whoever is responsible for this death would suffer in this world and the next. But Rostam told him that he should not be responsible because it was his father who sent Esfandiyār to fight him despite knowing that he would be slain by Rostam. Pešōtanū agreed that the blame was on Vīštāspa and Esfandiyār for not accepting Rostam’s deal. Rostam then took care of the upbringing and education of Esfandiyār’s surviving son Bahman. However, once Rostam had died, Bahman remembered how the Sakastanian herores had killed his father and brothers. Wanting revenge for this he suddenly attacked Rostam’s son Farāmarz and killed him (In some versions Zavāra survived the pit-trap and was also killed in this attack of Bahman). He also caged Zāl and devastated their realm.

This narrative is replete with several notable motifs:
\bullet We see ancient IE motif of the “Achilles heel” in the form of Esfandiyār’s eyes. More specifically, the vulnerability of Esfandiyār is similar to that of the Germanic god Baldr, who is killed by Loki’s plot by making the blind deity Höðr throw a spear or shoot an arrow made of mistletoe (to which he was vulnerable) at him. There is an apparent record of this motif among the Algonquin first Americans in the tale of their deity Glooskap and his rival Malsum (claimed by some to also bear the name Lox). Here Glooskap and Malsum had vulnerabilities to certain plants and owl feathers and eventually Glooskap slew Malsum with a fern root. While there has been a proposal that this motif might have been transmitted to the Northern Germanics during their contact with the New World, its presence among the Iranics suggests to a more ancient origin (Assuming the tale collected by Leland from the Algonquin is genuine).

\bullet The journey to a distant land to obtain the tamarisk tree missile can be compared in the Hindu epic to Arjuna and Kṛṣṇa going to the abode of Rudra to obtain the Pāśupata missile by means of which alone Jayadratha could be killed. The curse protecting Jayadratha from Vṛddhakṣatra can again be compared to that protecting Esfandiyār, which was likewise deflected on to his father Vīštāspa.

\bullet The feather motif: Zāl’s special connection to the great bird the Simorgh comes to aid Rostam on at least two occasions — once during his birth (see above) and again in this incident of his battle with Esfandiyār. Now the use of the feather is a clear connection with an ancient Indo-Iranian apotropaic ritual. On the Hindu side it is specified in the Yajurveda as part of the famous Sautrāmaṇi ritual in which the yajamāna is purified by the adhvaryu who strokes him with a śyena (hawk or falcon) feather twice above the navel and once below it:
yā vyāghram̐ viṣūcikā । iti śyena-pattreṇa yajamānaṃ pāvayati ॥ Mānava-śrauta-sūtra

The mantra used in the process from the Maitrāyaṇi-saṃhitā is:
yā vyāghraṃ viṣūcikobhau vṛkaṃ ca rakṣati ।
śyenaṃ patatriṇaṃ siṃhaṃ semaṃ pātv aṃhasaḥ ॥
My she, Viṣūcikā, who guards these two, the tiger and the wolf,
The lion and the winged hawk, may she guard this man (the yajamāna) from distress.
A comparable ritual is also found in the Taittirīya-brāhmaṇa

On the Iranian side we have comparable ritual in the Verethraghna Yašt (34-35):
verethraghnem ahuradhâtem ýazamaide
We offer sacrifice to Verethraghna, made by Ahura.

peresat zarathushtrô ahurem mazdãm:
ahura mazda mainyô spênishta dâtare gaêthanãm
astvaitinãm ashâum ýat bavâni aiwi-sastô aiwi-shmaretô
pouru-narãm tbishyañtãm cish ainghe asti baêshazô. 
Zarathuštra asked Ahura Mazda:
“O Ahura Mazda, most beneficent mind, creator of the material world,
you O holy one! If I have a curse upon me, a spell cast upon me
by the many men who hate me, what is the medicine for it?”

âat mraot ahurô mazdå: merekhahe peshô-parenahe vâreñjinahe parenem ayasaêsha spitama zarathushtra, ana parena tanûm aiwi-sifôish ana parena hamerethem paiti-sanghaêsha.
Ahura Mazda answered: “Take a feather of that bird with a wide wingspan, the Varenjina bird, O Spitama Zarathuštra! With that feather you should stroke your own body; with that feather you should back-hurl the spell to the enemies.”

\bullet Finally, we have specific point regarding the eagle-feather motif: the eagle helps Rostam in specie against the wounds of a difficult to repulse missile and does so by removing the arrow-heads and the stroking him with its feathers. This motif occurs in a rather similar form in the Rāmāyaṇa. There, the fierce rakṣas hero Meghanāda strikes the Ikṣvāku brothers with his nāga arrows. Riddled with Meghanāda’s arrows the two fall unconscious on the field. They were then healed by Garuḍa by stroking them with his wings and their power is doubled as a consequence:

śara-jāla-ācitau vīrāv ubhau daśaratha ātmajau ।
śara-talpe mahātmānau śayānāu rudhirokṣitau ॥ (R 6.50.3)
The two mighty valiant sons of Daśaratha enmeshed in a web of arrows are lying bleeding on a bed of arrows.

tato muhūrtad garuḍam vainateyam mahā balam ।
vānarā dadṛśuḥ sarve jvalantam iva pāvakam ॥
Then, all the monkeys saw in a moment Garuḍa, the son of Vinatā, of great strenght like a blazing fire.

tam āgatam abhiprekṣya nāgās te vipra-dudruvuḥ ।
yais tau sat puruṣau baddhau śara bhūtair mahā-balau ॥
Beholding him [the great eagle arrive], the snakes, which having become great arrows had bound those two strong men, fled.

tataḥ suparṇaḥ kākutsthau dṛṣṭvā pratyabhinandya ca ।
vimamarśa ca pāṇibhyām mukhe candra-sama-prabhe ॥
Thereupon, the eagle, having seen the two Kākutstha heroes offered them his good wishes, and with his wings stroked their faces that were radiant like the moon.

vainateyena saṃspṛṣṭās tayoh samruruhur vraṇāḥ ।
suvarṇe ca tanū snigdhe tayor āśu babhūvatuḥ ॥ 6.50.36-39
Their wounds, stroked by the son of Vinatā were healed. Their bodies quickly regained color and became smooth again.

Some concluding remarks
What is the significance of these mythic motifs for actual history of the Indo-Europeans? Archaeogenetics has finally offered us a framework for early IE history barring that of the Anatolian branch, which still remain somewhat mysterious. This is not the place to recount that in any great detail but briefly: It appears that the early phase of IE tradition corresponds to the Mikhaylovka culture starting around 3600 BCE in the Pontic–Caspian steppes north of the Black and the Caspian Seas. This culture had expanded considerably across the steppes by around 3300 BCE as the famous Yamnaya culture which likely spoke PIE (whether ancestor Anatolian was included remains unclear). Starting around 3000 BCE, PIE was starting to breakup into diverging dialects and the first to split was the eastern branch leading to the Afanasievo culture. This is probably the precursor of the Tocharian branch of IE. Subsequently, the Yamnaya culture spread westwards and acquired differing degrees genetic contribution from the earlier European residents who themselves were an admixture of western hunter-gatherers and west Asian/Anatolian farmers. This westward spread probably marks the split of the early western Kentum dialects that led to Germanic, and Italo-Celtic.

A population with the above European admixture likely moved backwards to give rise to the Satem branches which then split up into Indo-Iranian and Balto-Slavic by at least 2700 BCE. These were represented by the successor cultures of the Yamnaya like the Poltavka. Studies in the past 4 years have shown that a male individual dated from 2925-2536 BCE from the Poltavka culture bore the same Y-chromosome haplogroup as the Indo-Aryans of India (R1a-Z93). It is hence likely that the Poltvaka or similar cultures in the general middle Volga zone represent the ancestral populations from which the Indo-Iranians expanded. We hold that the early layer in the Vedic texts contain material from this period with clear memories of the even earlier PIE period (The exact scenario involves some complexities and will not be elaborated here). The proto-Greeks and probably the proto-Armenians (who were part of the kentum radiations) were also likely in the vicinity of this and interacted with these early Indo-Iranians. Subsequently, the Indo-Iranians expanded in two major stages eventually reaching their historical homelands. The earliest of these expansions is seen in the archaeological record as the Sintashta culture. The detection of the cognate word of Skt marya (a young man/warrior) in tablet from Tell Leilān in Syria dating 1800-1700 BCE suggests that their thrust into Outer Asia had begun by then. From the recently published genetic evidence, it appears that a massive military force of Indo-Aryans invaded the Indian subcontinent sometime around 1900-1700 BCE after quickly outflanking the Central Asian Bactria-Margiana complex and quickly conquered northern India to found the first Indo-Aryan states in the region. The so called Kassite records of Indo-Aryan deities suggest that smaller forces of Indo-Aryans were also active in West Asia where they even formed the elite of the Mitanni kingdom between 1600-1400 BC. The Iranian Aryans appear to have invaded Iran only later and are attested in Elamite records via words like Ašbayauda (horse-warrior).

Given this background, we suspect that the motif an apocalyptic war, which is common to several branches of Indo-European was something that emerged in the PIE tradition. It goes without saying that the early IEans were a warlike people. It is conceivable that their initial expansions marking the breakup of PIE was accompanied by internal wars among the early IE elite. Likewise, their expansions that followed brought them into intense conflict with other alien groups — traces of these are even seen in the archaeogenetic evidence. This probably gave rise to 2 epic motifs, namely that of the fight “within the family” and the battle with hostile aliens which might have involved abduction of women. Now, one may see these motifs were at the heart of the proto-itihāsa-s of the IE people. They were then reused in the narrations of various historical battles that the different branches fought far from their original homeland in their new homelands like India or Greece or Western Europe.

The weight of the evidence suggests that there was a fairly protracted period of an undivided Indo-Iranian period (probably \sim 500 years). In this period they probably continued to interact to a degree with the sister śatam branches (Balto-Slavic) and also the precursors of the Greco-Armenians resulting in some sharing of specific traditions. It is conceivable that the motifs such as the “river-born”, long-lived warrior emerged in this period and was woven into the later epics of both the Iranian and Hindu traditions. While these motifs were perhaps of early Indo-Iranian origin, we do see fainter echoes in the Greek tradition. For instance, Achilles is “made strong” by the River Styx, even as Rostam is “river-strong”. While motifs like the great eagle deity are probably of PIE or even earlier vintage [Footnote 6], it likely developed specific elements in this undivided Indo-Iranian period.

Admittedly, the Iranian epics are late and hardly pristine in their preservation. Yet, they illustrate an important process in epic-building, namely the weaving of narratives from different periods into a completely consistent yet anachronistic end-product. We posit that a similar process has been at work in the Mahābhārata, albeit at a much earlier time. This brought together the heroes of multiple Indo-Aryan traditions into a single whole, with the Pāṇḍava-s being superimposed onto the earlier Kuru-Pāñcāla epic. It is notable that even on the Iranian side we have at least three distinct elements being melded: the heroes of the undivided I-Ir period like Kṛśāśva, the heroes of the Zoroastrian counter-religious movement and the Sakastanian tradition around the clan of Rostam. It is notable that in some narratives there is clear memory of the religious strife between the Zoroastrian partisans and various adherents of the older religion, which include the clan of Rostam. However, overtime the latter were brought into the Zoroastrian fold and presented as champions of that counter-religious tradition. For example, in the Farāmarz-nāme we have Farāmarz debate with brāhmaṇa-s in India, convert them to Zoroastrianism and make forsake the worship of images of daeva-s.

Similarly, we propose that in India the Pāṇḍava-s who originally represented a secondary invasionary wave who overthrew the old Kuru-s, were subsequently presented as legitimate successors of the Kuru-s at Hastināpura. We also suspect that some of the heroic elements of the Sakastanian epic cycle and the Mahābhārata draw from a common tradition that simultaneously interacted with both the Iranian and Indo-Aryan mainlines. In this regard we may bring attention to the Pārataraja-s who founded a kingdom between \sim 125-300 CE in the region neighboring Sakastana in what’s today Balochistan. The names of their rulers include among other wise typically Iranic names (e.g. *Yodamithra, *Bagamithra, *Mithra-takhma) also names of the Pāṇḍava-s — Arjuna and Bhīmārjuna. This points to a tradition of honoring the Pāṇḍava-s even among these Iranic groups.

Finally, we may note that Iranicists like Gazerani and Pourshariati see the mark of much later history from the time of the rise of the Parthians in the epics of the Sakastanian heroes. They note that the powerful family of Surena, which played a central role in the great Parthian victory against the Romans led by Marcus Crassus, claimed descent from the clan of Rostam. This is indeed conceivable for certain narratives; however, we hold that the core narrative of the Sakastanian heroes is still of a much older period which was brought by the Sakas and Parthians (Pahlavān-s). Where we see this later narrative being superimposed onto to the earlier epic is in the likely Iranic memory of the foundation of the Pallava dynasty in southern India. We know from the Andhra inscriptions that their kings referred to their Parthian adversaries as Pallava-s. In the Iranic epic of Shahriyār we learn of the account of Sohrāb’s descendants. In course of his wanderings Rostam fought with Sohrāb’s son Barzu but learnt of his identity and brought him back to his land. Barzu had a son named Shahriyār, who eventually had a conflict with his cousins descending from Farāmarz. Furious over this he decided to leave Sakastana for India. He first went to Kashmir but the king of Kashmir was related to Farāmarz’s family by marriage and was his ally. Hence, he proceeded further to southern India towards the island of Sarandip (Shri Lanka) where he married the daughter of the Hindu king Arjang. He aided Arjang in his many battles, killed a demon called Sagsar, and later ascended the throne after his father-in-law. Whether a recent descendant of Rostam came to India in “real” history remains is hard to test. Nevertheless, it is entirely likely that that Iranics who founded the Pallava dynasty in south India were from Surena’s clan and thus claimed descent from Rostam. Hence, a memory of their settling in Southern India was likely superimposed back into the epic narratives of their old heroes.

Footnote 1: When we say Aindra we do not mean that the deity under the name Indra was present among the ancestral IEans. It merely means that Indra represents the Hindu reflex of an ancestral IE deity who was his cognate. In the rest of this account we shall use similar terms for other deities with proto-IE cognates, like Viṣṇu.

Footnote 2: Gutarza (Gōdarz) is believed by some to be the Parthian king Gutarza Gēvaputhra of history superimposed backwards into the age of the Kavi-s. We are not certain of this hypothesis. It is notable that Gēv is the husband of Rostam’s daughter Bānu Gošasb and the father of another Rostamian hero Bizhan. His name appears as the father of that of the historical Gōdarz and is at odds with the Šah-nāme.

Footnote 3: This is laid out in the Zoroastrian text known as the Zand i Wahman Yasn where in a Kalkin-like showdown he would be assisted by the deities Mithra, Rashnu, Sraosha and Verethraghna in an idol- and temple-smashing spree against the daeva worshipers.

Footnote 4: From the Farāmarz-nāme we learn that this ritual involved burning a piece of the feather with the agarwood, i.e. Skt Aguru, which is also used a fragrant material burnt in the ritual to the Ucchuṣma-Rudra-s specified in the Atharvaveda-pariśiṣṭha. It is also widely used in Hindu medicine.

Footnote 5: This attack on the holy bird is also an indication of the clash between the religion of Sakastanian heroes like Zāl, Rostam and Farāmarz, who are repeatedly helped by the great Saēna and the partisans of Zarathuštra.

Footnote 6: Probably, this developed from the observation of eagle taking back pharmacologically active plants to their nests.

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A modern glance at Nārāyaṇa-paṇḍita’s combinatorics-1

For improved reading experience one may use the PDF version.

Students of the history of Hindu mathematics are well-acquainted with Nārāyaṇa-paṇḍita’s sophisticated treatment of various aspects of combinatorics and integer sequences in his Gaṇita-kaumudī composed in 1356 CE. In that work he gives about 43 problems relating to combinatorics. Continuing with our study of various aspects of Nārāyaṇa’s work using a modern lens, in this note we shall look at some of his problems in combinatorics and what a modern (low level) student can learn from them.

The first problem we shall look at introduces a student to the discrete factorial function n! using a verse in the Vasantatilakā meter;
cāpeṣu khaḍga-ḍamarūka-kapāla-pāśaiḥ khaṭvāṅga-śūla-phaṇi-śakti-yutair bhavanti ।
anyonya-hasta-kalitaiḥ kati mūrtibhedāḥ śambho harer iva gadā’ri-saroja-śaṅkhaiḥ ॥
With a bow (1), an arrow (2), a sword (3), a double-drum (4), a skull (5), a lasso (6), a skull-topped rod (7), a trident (8), a snake (9) and a spear (10) — by changing them from one hand to another how many different images of Rudra come to be? Likewise, of Viṣṇu with a mace (1), a wheel (2), a lotus (3) and a conch (4).

For Rudra with 10 arms the answer is 10!=3628800, whereas for Viṣṇu it is 4!=24. This problem was well-known among earlier Hindu scientists and is not original to Nārāyaṇa. Here, he is merely reusing this verse without any change from Bhāskara-II’s Līlāvatī. In the case of Viṣṇu, each of these 24 permutations have a specific name starting from Keśava. There are correspondingly 24 forms of Lakṣmī. These forms are an important aspect of the Pañcarātra system where they are counted along with the 4 basic vyūha-s in the śuddha-mūrti-s (“Platonic” forms) in texts like the Nārada-pāñcarātra and are attested in iconography across the Indosphere. The general use of permutations in various endeavors is mentioned by Nārāyaṇa after he provides the procedure for writing out the permutations:
aṅka-prastāra-vidhiś caivaṃ mūrtiprabhedānām ।
sa-ri-ga-ma-pa-dha-nīty eṣāṃ vīṇāyā nikvaṇānāṃ ca ॥
This procedure generating the permutation of digits is also used in permutations of images [of deities], sa-ri-ga-ma-pa-dha-ni (the notes of Hindu music) and the notes produced by the vīṇā.

Problem-2 (A verse again in the Vasantatilakā meter):
dhātrī lavaṅga-dala-kesara nāga-railā vakraṃ kaṇāḥ samaricāḥ sasitā bhavanti ।
ekādibhiś ca militair gadināṃ katīha cūrṇāni bho vada gadāpanude gadajña ॥
O apothecary, how many different different disease-curing spice-powders come from mixing one etc (i.e. 1, 2, 3…) of gooseberry, clove, cinnamon, saffron, ginger, cardamom, Indian may apple (Sinopodophyllum hexandrum), cumin, pepper and sugar?

This type of problem is encountered widely in Hindu literature — we find a discussion of the combinations of tastes in the medical saṃhitā-s of Caraka and Suśruta. Subsequently, the great naturalist Varāhamihira in \sim 550 CE discussed the production of various perfumes by combinations of basic scents. Such combinations are also discussed by king Bhojadeva Paramāra in his chemical treatise in the 1000s of the CE. Related problems are also taken up in Bhāskara-II and by the polymath Śārṅgadeva in his musical treatise. This particular problem is rather typical of the combinations used in preparation of drugs in Āyurveda. As a pharmacological aside the Indian may apple if properly used can be quite effective treating tumors caused by certain papillomaviruses. Returning to the solution of the problem, we need to recall the formula for combinations:

{}^nC_k = \dfrac{n!}{k!(n-k)!}

Let N be the total number of powders that can be created via each set of combinations: by taking 1 at a time we get 10, by taking 2 at time we get {}^{10}C_2=45 and so on. Thus, we get:

N=\displaystyle \sum_{k=1}^{10} {}^{10}C_k=1023

nArAyaNa_combinations_Fig1Figure 1

In Hindu tradition, the study of combinations and their sum goes back to at least the Chandas-śāstra (the treatise on meters) of Piṅgala. This has been extensively discussed in the literature and we present it only briefly:

pare pūrṇam । pare pūrṇam iti । CS 8.34-35
Complete it by using the two distal ends. Repeat to complete using the distal flanking ends.

While these original sūtra-s of Piṅgala are difficult to directly understand, they have been explained in the glosses of several authors since (e.g. Kedāra-bhaṭṭa and Halāyudha). The two sūtra-s specify the construction of the Meru-prastāra or the combinatorial triangle. The first sūtra implies that you write out the flanking cells with 1 corresponding to row n:

1 \\ 1 \quad 1 \\ 1 \quad {. } \quad 1 \\ 1 \quad {. } \quad {. } \quad 1 \\ 1 \quad {. } \quad {. } \quad {. } \quad 1 \\ 1 \quad {. } \quad { .} \quad {. } \quad {. } \quad 1 \\

The second sūtra implies that you fill in the interior cell by repeating the procedure:

1 \\ 1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad 1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \\ 1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1 \\

While this is extensively discussed in the context of Chandas, one can also find a clear algorithm in Bhāskara-II’s Līlāvatī using the combination function to produce not just the combinatorial triangle (Meru) but also any row or cell of it. Thus, from the Meru we can write the formula for the expansion of a binomial as:

(x+y)^n=\displaystyle \sum_{k=0}^{n}{}^nC_k x^k y^{n-k}

This tells us that the null-product or 0!=1 (implicitly provided in Bhāskara-II’s algorithm: there only 1 way of not choosing anything). The magnitudes of the k^{th}-combination or the values assumed by the combination function for a given n as k changes determine the values of the individual terms of the above expansion. Hence, we use the problem-2 to introduce and illustrate to students the shape of the binomial distribution (Figure 1). Since the Meru itself can be seen as the triangle of (1+1)^n, we get the formula for the sum of combinations for a given n as,

N=\displaystyle \sum_{k=0}^{n} {}^{n}C_k=2^n

If we leave out the null combination k=0, we get N=2^n-1 as in the problem-2 where N=2^{10}-1.

As this note is part historical and part educational (for a low-level student), let us next consider another binomial expansion that played a cornerstone role in the origin of modern mathematics,

\displaystyle \lim_{n \to \infty} \left(1+\dfrac{1}{n}\right)^n

We can intuitively sense based on the easily-determined first 3 terms that it might converge to some number between 2 and 3 but what is that number? We can experimentally see that the above expression converges slowly: for n=10 we get 2.6; for n=100 we get 2.7; for n=200 we have 2.71. This is exactly where Jakob Bernoulli got to when he first encountered this problem and realized that it was converging to something around 2.71. However, we can do better by determining the limit:

\displaystyle \lim_{n \to \infty} \left(1+\dfrac{1}{n}\right)^n = \dfrac{1}{0! \cdot n^0}+ \dfrac{n}{1! \cdot n^1} + \dfrac{n\cdot (n-1)}{2! \cdot n^2}+ \dfrac{n\cdot (n-1) \cdot (n-2)}{3! \cdot n^3}...\\[10pt] = \dfrac{1}{n^0 \cdot0!}+ \dfrac{n}{n \cdot 1!} + \dfrac{n^2 \cdot (1-1/n)}{n^2 \cdot 2!}+ \dfrac{n^3\cdot (1-1/n) \cdot (1-2/n)}{n^3 \cdot 3! }...\\[10pt] = \dfrac{1}{0!}+ \dfrac{1}{1!} + \dfrac{(1-1/n)}{2!}+ \dfrac{(1-1/n) \cdot (1-2/n)}{3!}...

Taking the limit n \to \infty we get:

\displaystyle \lim n \to \infty \left(1+\dfrac{1}{n}\right)^n= \sum_{n=0}^{\infty} \dfrac{1}{n!}

Thus, our limit is the infinite sum of the reciprocal of the factorials. This is much faster-converging and with just 10 terms converges to 7 places after the decimal point to 2.7182818… The importance of this limit and the number it converges to comes to fore in another central result in the emergence of modern mathematics: What is the rate of change (derivative) of the logarithmic function? Let us start with a logarithm taken to some base b, i.e. y=\log_b(x). Hence,

\displaystyle \dfrac{dy}{dx}=\lim_{\delta x \to 0} \dfrac{\log_b(x+\delta x)-\log_b(x)}{\delta x} =\log_b\left( \dfrac{x+\delta x}{x}\right)^{1/\delta x} =\log_b\left( 1+\dfrac{\delta x}{x}\right)^{1/\delta x} \\[10pt] =\log_b\left( 1+\dfrac{\delta x}{x}\right)^{x/\delta x \times 1/x} =\dfrac{1}{x}\log_b\left( 1+\dfrac{\delta x}{x}\right)^{x/\delta x}

Now we can write \tfrac{\delta x}{x} as some \tfrac{1}{n}; \therefore \tfrac{x}{\delta x}=n. As \delta x \to 0, n \to \infty. Thus, we can rewrite our limit as:

\displaystyle \dfrac{d}{dx}y= \lim_{n \to \infty} \dfrac{1}{x}\log_b\left( 1+\dfrac{1}{n}\right)^{n}

We observe that this is the same limit we evaluated above. Now, if we define e as the sum of the reciprocal of the factorials, which is the limit, and set b=e then \tfrac{d}{dx}\log_e(x)=\tfrac{1}{x}. Thus, we get e to be the natural base of logarithmic function and the derivative of \log(x). Conversely, the area under a unit rectangular hyperbola, i.e y=\tfrac{1}{x} is the logarithmic function with base e.

Armed with e, we can next retrace certain developments in the history of early modern mathematics. What is the relationship of an arbitrary exponential curve y=a^x to e. For this we need to first determine the derivative of a^x. This is trivially done now that we have the derivative of \log(x):

y=a^x \; \therefore \log(y)=x \log(a)\\[10pt] \dfrac{d \log(y)}{dx}=\log(a)\\[10pt] \dfrac{d \log(y)}{dy}\cdot \dfrac{dy}{dx}=\log(a)\\[10pt] \dfrac{1}{y}\cdot \dfrac{dy}{dx}=\log(a)\\[10pt] \dfrac{dy}{dx}=y\log(x)=a^x \log(a)

exponentials_eFigure 2

With this in hand we can see the relationship of any exponential curve y=a^x to e (Figure 2):
\bullet Consider the family of exponential curves y=a^x (Figure 2; the red curve is y=e^x). From the above result we see that the tangent to a exponential curve will have the slope m=\log(a)a^x

\bullet Let x=\tfrac{1}{\log(a)}. Then: m=\log(a)a^{1/\log(a)}=\log(a)a^{\log(e)/\log(a)}=\log(a) a^{\log_a(e)}=e\log(a)

\bullet A line passing through origin has the equation y=mx. We set m=e\log(a); when x=\tfrac{1}{\log(a)} the equation of the line yields y=e. Similarly, the equation of the exponential curve yields y=a^{1/\log(a)}=e. Thus, the line y=e\log(a)x is the tangent to y=a^x from origin.

\bullet Thus, the tangent to an exponential curve from the origin will touch it at a height of e from the X-axis at x=\tfrac{1}{\log(e)}

Given the derivative of the exponential function, it is obvious that the derivative of e^x is e^x. This in turn allows one to establish the relationship of any power of e to the reciprocal of factorials. Consider the infinite series:

\displaystyle \textrm{f}(x)= \sum_{n=0}^{\infty} \dfrac{x^n}{n!}=1+x+\dfrac{x^2}{2}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+...\\[10pt]\\ \dfrac{d \textrm{f}(x)}{dx} = 1+x+\dfrac{x^2}{2}+\dfrac{x^3}{3!}+...\\[10pt] \therefore \dfrac{d \textrm{f}(x)}{dx} = \textrm{f}(x)

Now the function whose derivative is the same as the function itself is e^x; hence,

\displaystyle e^x= \sum_{n=0}^{\infty} \dfrac{x^n}{n!}

Thus, this gives the relationship of a power of e to the reciprocal of factorials. If we put x=1 in the above we get the same infinite series for e as we obtained from the above limit. With this in hand, we can arrive at one of the most remarkable functions discovered in the history of early modern mathematics that is key to our understanding of the universe.

nArAyaNa_normal_Figure3Figure 3

In problem-2 we saw the magnitudes assumed by the combination function. We see that they appear to define a bell-shaped curve (Figure 2, 3). What is the curve that best approximates the binomial coefficients as n \to \infty (Figure 3; shown for n=50). For this we can begin by noting the following. It is a symmetric curve around the central or the highest value binomial coefficients. It falls sub-exponentially and is asymptotic to the x-axis. Given this we can try to construct this basic shape with its maximum centered on (0,1) using an infinite series approach (Figure 4). Given that it is symmetric, we only need to consider even powers of x in such a series.

bell_curve_expansionFigure 4

We start with y=\tfrac{x^0}{0!}. This in the least captures the maximum but little else. So the next term corrects this by a subtraction to get a curve around the maximum; thus y=\tfrac{x^0}{0!}-\tfrac{x^2}{1!}. However, this correct falls straight down and we have to add a term to get closer to the asymptotic behavior with respect to the X-axis. Thus we get: y=\tfrac{x^0}{0!}-\tfrac{x^2}{1!}+\tfrac{x^4}{2!}. We continue this process (first 8 steps are shown in Figure 4) and get the infinite series:

y=\displaystyle \sum_{0}^{\infty} \dfrac{\left(-x^2\right)^n}{n!}

From the above series for a power of e we can immediately see that:


This is the famous equation of the shape of the normal distribution, which is a limit of the binomial distribution as n \to \infty. With this we can now provide the continuous approximation for the combination function normalized by the maximal combination (Figure 3):


Thus for the actual combination function we get:


vadāśu rupādi navāvasānaiḥ ।
bhedām̐ś ca labdhy aṅka-mukhāntya-bhedān
ūrdhvāṅka-yogaṃ sakalāṅka-yogam ।
aṅka-prapātaṃ ca sakhe pṛthak te
vadā ‘ṅkapāśe ‘sti pariśramaś cet ॥
Snakes (8), fires (3), deficit (9): (9,3,8); two (2), guṇa-s (3), limbs [of Veda] (6), moon (1): (1, 6, 3, 2); starting from form (1) to nine (9): (1,2,3…9); Quickly state: (i) the number of permutations; (ii) the number of permutations either beginning or ending in one of those digits; (iii) sum of digits in a particular place; (iv) sum of all numbers [formed by permutation of the digits]; (v) the total number of digits; O friend state these for each set separately if you have labored on combinatorics.

Let n be the number of the objects participating in the permutations without replacement and s be those objects, in this case digits. Given this, the problem systematically takes you through several interesting questions:
(i) The bheda-s, i.e. permutations: n!. For s=1..9 it is 362880.

(ii) The aṅka-mukha-s or aṅkāntya-s, i.e. number of permutations that either begin or those that end in a particular digit: (n-1)!= \Gamma(n). This is so because we keep one position constant and allow the remaining to vary freely; thus, n-1 positions are available for permutation. For s= 1..9 it is 40320.

(iii) The ūrdhvāṅka-yoga, i.e. the sum of the numbers in a particular column. \Gamma(n) \cdot \sum s. From the above we saw that the number of permutations starting with a particular digit is \Gamma(n). Thus, for a given column, we will have that many permutations with each digit. Thus, \sum s multiplied with \Gamma(n) will give us the sum for a given column. For s= 1..9 it is 1814400

(iv) The sakalāṅka-yoga, i.e. the sum of all the numbers formed by the digit permutations. \Gamma(n) \cdot \sum s \cdot (\displaystyle \sum_{k=0}^{n-1} 10^k). We have the expression for the sum of a column from above. Now, consider a small example of the given problem with 3 digits. We can rewrite the numbers formed by the permutations to keep the same total thus:

\begin{matrix} 1 \quad 2 \quad 3\\ 1 \quad 3 \quad 2\\ 2 \quad 1 \quad 3\\ 2 \quad 3 \quad 1\\ 3 \quad 1 \quad 2\\ 3 \quad 2 \quad 1\\ \end{matrix} \; \to \; \begin{matrix} 1 \quad 1 \quad 1\\ 1 \quad 1 \quad 1\\ 2 \quad 2 \quad 2\\ 2 \quad 2 \quad 2\\ 3 \quad 3 \quad 3\\ 3 \quad 3 \quad 3\\ \end{matrix}

As a result we can express the sum of all numbers formed by the permutation of the digits to be the sum of a column multiplied by \sum_{k=0}^{n-1} 10^k; 111 for the above example. Thus, for s=1..9 we get 1814400 \times 111111111= 201599999798400.

(v) Finally, the aṅka-prapāta, i.e. the total number of digits in all the permutations. n^2\cdot \Gamma(n). Since there will be n! permutations and n starting digits it is easy to see that the total number of digits across all permutations will be the above expression. For s=1..9 it is 3265920.

One would have noticed that we have used \Gamma(n) for (n-1)!. When Gauss studied the continuous form of the factorial function he merely took it as x!; however, the French mathematician Legendre defined it using \Gamma(n)=(x-1)!. We take the Legendre definition of the famous Gamma function as it naturally emerges in solutions of problems such as that of Nārāyaṇa. Indeed, this definition also naturally emerges from the famous integral of Euler for \Gamma(x) that behaves just like the (n-1)! function. Being an integral this also gives the continuous form of the \Gamma(x) function specifying the value of the function for non-integer x. Euler’s integral:

\Gamma(x) = \displaystyle \int_0^\infty t^{x-1}e^{-t}dt

This integral can be handled using the rule for integration by parts:
\int f(x) \cdot g(x)dx = f(x) \int g(x) dx - \int f'(x) (\int g(x) dx) dx
Using f(x)=t^{x-1} and g(x)=e^{-t} we get:

\Gamma(x) = t^{x-1} \int e^{-t}dt - \int (x-1) t^{x-2} (\int e^{-t}dt) dt \\[7pt] = -t^{x-1} e^{-t} + (x-1)\int t^{x-2}e^{-t} dt

Taking the limits we get:

\Gamma(x) = \displaystyle \left. -t^{x-1} e^{-t} \right\rvert_{0}^{\infty} + (x-1) \int_{0}^{\infty} t^{x-2}e^{-t} dt\\[10pt] \therefore \Gamma(x) = (x-1)\Gamma(x-1)

By putting x=n and doing the above repeatedly we get \Gamma(n)=(n-1)(n-2)... until we reach 1 at which point the integral becomes:

\Gamma(n)=\displaystyle (n-1)(n-2)..2 \cdot 1 \int_{0}^{\infty} t^0 e^{-t} dt =(n-1)!

In the final part of this note we shall consider the integer sequence defined by the aṅka-prapāta: n^2\cdot \Gamma(n). Now let us do this for the sets of n=1, 2, 3, 4... permutable symbols. We get the integer sequence f[n]:
1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920…

This sequence has a notable property. It defines the number of integers from 1..k! that are not divisible by k for k=2, 3, 4.... Why this is so is easy to apprehend: Since we start from 2, we have k=n+1. Now the numbers that will be divisible by k between 1..k! will amount to \tfrac{k!}{k}=(k-1)!=n!. Therefore, the numbers that will not be divisible by k will amount to (n+1)!-n!=n!(n+1)-n!=n (n-1)! (n+1-1)=n^2 \Gamma(n).

If we take the sum of the reciprocals of this sequence we see that it converges to a constant:

\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{n^2 \Gamma(n)} = 1.3179021514544...

Now, what is this number? We discovered that this number emerges from the solution of an interesting definite integral:

\displaystyle \int_0^1 \dfrac{e^x-1}{x} dx= 1.3179021514544...

The integral can be split up as:

\displaystyle \int \dfrac{e^x}{x} dx - \int \dfrac{1}{x} dx= \int \dfrac{e^x}{x} dx -\log(x)+C

Eix_etcFigure 5

It is immediately apparent that the first integral \int \dfrac{e^x}{x} dx is a tricky one: the function y= \dfrac{e^x}{x} diverges to \infty as x^+ \to 0 (from positive side) and to -\infty as x^- \to 0 (from negative side). Remarkably, these opposite divergences cancel each other and the integral converges to a fixed value. Thus we can evaluate it to a given x as:

\textrm{Ei}(x) = \displaystyle \int_{-\infty}^x \dfrac{e^t}{t} dt

This function \textrm{Ei}(x) is the exponential integral with deep connections with permutations. The two divergences of y= \dfrac{e^x}{x} exactly cancel each other when x=\log(\mu)=0.37250741..., i.e. \textrm{Ei}(x)=0. This \mu=1.451369234 is the Soldner-Ramanujan constant that was first discovered by Johann von Soldner and independently by Ramanujan who arrived at it when he discovered multiple series for the logarithmic integral \textrm{Li}(x)=\int_0^x \tfrac{dx}{\log(x)} (Figure 5), which Gauss had shown to provide the asymptotic description of the distribution of prime numbers. The famed \textrm{Li}(x) = \textrm{Ei}(\log(x)). Returning, to our original integral we can thus write its indefinite solution as:

\displaystyle \int \dfrac{e^x-1}{x} dx= \textrm{Ei}(x) -\log(x) +C

Now we observe that as x^+ \to 0,\; \textrm{Ei}(x) \to \infty,\; \log(x) \to -\infty (we only consider the approach to 0 from positive side for only there the real \log(x) is defined). The two remarkably balance each other such that as x^+ \to 0 the above integral converges to \gamma=0.577215664..., which is the famous Euler-Mascheroni constant with a deep connection to the Gamma function (See below). Thus, the definite integral (Figure 5):

\displaystyle \int_0^1 \dfrac{e^x-1}{x} dx= \textrm{Ei}(1)-\gamma=1.3179021514544...

This leads us to the formula:

\textrm{Ei}(1)=\displaystyle \gamma + \sum_{n=1}^{\infty} \dfrac{1}{n^2\Gamma(n)} = 1.89511781635...

From this and the above indefinite integral we can obtain the general formula for \textrm{Ei}(x) as:

\textrm{Ei}(x)=\displaystyle \gamma +\log(x) + \sum_{n=1}^{\infty} \dfrac{x^n}{n^2\Gamma(n)}

If we now substitute x by \log(x) we get the series for the logarithmic integral as:

\textrm{Li}(x)=\displaystyle \gamma +\log(\log(x)) + \sum_{n=1}^{\infty} \dfrac{\log^n(x)}{n^2\Gamma(n)}

This was the series for \textrm{Li}(x) that Ramanujan arrived at unaware of work of Gauss, Soldner and their successors in Europe. He then went on to discover other series that converged even faster to \textrm{Li}(x). With these relationships one can finally obtain a relationship between the mysterious Euler-Mascheroni constant \gamma that appears in various formulae pertaining to both the number world and the natural world and the Soldner-Ramanujan constant \mu of number theory. Since \textrm{Ei}(x)=0 when x=\log(\mu) by substituting this into the above series for \textrm{Ei}(x) we get:

\gamma = -\Gamma'(1) =\displaystyle \lim_{n \to \infty}\left( \sum_{k=1}^{n}\dfrac{1}{k} -\log(n) \right) = -\log(\log(\mu)) - \sum_{n=1}^{\infty} \dfrac{\log^n(\mu)}{n^2\Gamma(n)}

The first expression \Gamma'(x) is the derivative of the Gamma function. The second expression is Euler’s original definition of \gamma as a limit. The third is what we obtain from the above substitution, which gives it in relationship to \mu as derived from Ramanujan’s series.

Thus, in the works of the last great mathematicians of the Hindu tradition like Bhāskara-II, Nārāyaṇa and Mādhava we see the preamble to the developments of modern mathematics, which revealed the deep links between the number world and the natural world. Nārāyaṇa’s interest in combinatorics, sequences and sums may be compared with that of Euler. Armed with a photographic memory and an enormous capacity for numerical calculations, Euler was much like a paṇḍita of yore. Indeed, he dealt with infinite sums and definite integrals almost like a continuation of that old tradition. But among the Hindus it was Ramanujan, who close to 600 years after Nārāyaṇa and Mādhava nearly seemed as if he was channeling them to single-handedly take their tradition to a conclusion.

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

Kaiṭabha, poison and death: meanderings through tradition

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The medical Suśruta-saṃhitā, kalpasthāna, chapter 3 contains an unusual mythologem about the origin of poison. It clearly belongs to the Prājāpatya tradition and presents some unusual features. It presented in full below:

prajām imām ātma-yoner brahmaṇaḥ sṛjataḥ kila |
akarod asuro vighnaṃ kaiṭabho nāma darpitaḥ ॥
tasya kruddhasya vai vaktrād brahmaṇas tejaso nidheḥ ।
krodho vigrahavān bhūtvā nipapāta+atidāruṇaḥ ॥
sa taṃ dadāha garjantam antakābhaṃ mahābalam ।
tato ‘suraṃ ghātayitvā tat tejo ‘vardhata+adbhutam ॥
tato viṣādo devānām abhavat taṃ nirīkṣya vai ।
viṣāda-jananatvāc ca viṣam ity abhidhīyate ॥
tataḥ sṛṣṭvā prajāḥ śeṣaṃ tadā taṃ krodham īśvaraḥ ।
vinyastavān sa bhūteṣu sthāvareṣu careṣu ca ॥
yathā ‘vyakta-rasaṃ toyam antarikṣān mahīgatam ।
teṣu teṣu pradeśeṣu rasaṃ taṃ taṃ niyacchati ॥
evam eva viṣaṃ yad-yad dravyaṃ vyāpyāv avatiṣṭhate ।
svabhāvād eva taṃ tasya rasaṃ samanuvartate ॥

All these beings were emitted from his own womb by Brahman formerly. The arrogant demon Kaiṭabha created an impediment [for this task]. Indeed [on being obstructed], the fiery splendor of his wrath emerged from Brahman’s face. His anger having taken an embodied form rushed out with utmost ferocity. It burnt down that roaring, death-like mighty [demon]. After having slain the demon, that fiery splendor [of the wrath] grew most awfully. Seeing it, there arose consternation among the gods. Having arisen from consternation (viṣāda) it was known as poison (viṣa). Thus, having emitted his progeny in entirety, the lord (Brahman) thereafter placed the toxin in the organisms, mobile and immobile. Just as the atmospheric water with no taste on falling to earth acquires the flavors of whichever places [it falls at], so also the toxin by nature takes up the properties of the substances it associates with.

We shall use this myth as a foundation to discuss to the following points:
1) Here the demon Kaiṭabha is seen independently of his usual partner Madhu. Moreover, he is killed by the embodied wrath of Brahman rather than by Viṣṇu as in the paurāṇika records that have come down to us. Is this an unusual fossilization in a medical text of an old Prājāpatya tradition where Prajāpati killed Kaiṭabha by himself, which was then appropriated for Viṣṇu by his special votaries, or was it taken from Viṣṇu for Prajāpati by Brahma-sādhāka-s? It is tempting to propose the former due to the brāhmaṇa-like form of the narrative with a “folk-etymology” for viṣa, the ascendancy of Prajāpati in the late Vedic and epic period, which are coeval with the earliest layers of the two great medical saṃhitā-s, and a resemblance to some epic Prājāpatya narratives. For example, in the Mahābhārata Brahman creates seductive women to commit to lust men, who are on the path of becoming gods by observing dharma. However, the closest to this is the narrative of the origin of Mṛtyu seen in the Drōṇaparvan (Mbh 7.49; insert with respect to the Pune edition). It is said to have been narrated by Nārada to king Akampana in the Kṛtayuga when his son Hari, who is said to have had the splendor or Indra or Viṣṇu, died in battle:

Formerly, when Brahman emitted the creatures of the original creation they were immortal. As he worried about this, the wrath of Brahman manifested as a fire which filled the world and started consuming the organisms. Then Rudra, the lord the ghosts, who was born of Brahman, approached him and falling at his feet beseeched him not to destroy the diverse families of organisms he had created and sustained with such care. Hence, Rudra asked him to take back the fire. Brahman responded that he did not wish to destroy them but their growth and absence of death was weighing on the earth greatly and distressing it. But Rudra continued to press on Brahman, mentioning that the fiery substance he had emitted was emerging in a volcanic fashion blasting rocks, boiling rivers and destroying all vegetation and asked him the boon of controlling his destructive process. Brahman relented and did so. Then from his sensory apertures there emerged a dark woman wearing red clothes with a red tongue and eyes and ornamented with blazing earrings. She traversed the space and went towards the south. She was Mṛtyu. Brahman commanded her to be the death of all beings. On hearing that she began to cry with a melodious wail. Brahman caught her tears in his palms. She told Brahman that being female she did not want to join the god Yama and perform the cruel deed of destruction of life. She the went to various holy spots and performed tapasya with a sole focus on the god Brahman. Pleased with her tapasya Brahman asked her what she wanted. She told him that she had no desire to slay the creatures as they did not even inflict the slightest harm on each other. She wanted to boon from him of not having to perform the sinful acts of killing them.

Brahman then assured her that he along with all the other gods would confer on her the boon of being perfectly free of sin, eternally unsullied in her reputation. He told her that Yama the god of the South and the diseases would aid her in her task and commanded her to bring death to the four (why 4?) categories of organisms. She in return asked Brahman that the organisms be seized by tendencies like greed, anger, jealousy, malice, betray and delusion. Brahman granted that wish and said that the organisms would without restraint pierce the bodies of each other in manifold ways with utmost harshness (ahrīś cānyonyaparuṣā dehaṃ bhindyuḥ pṛthagvidhāḥ ।). He then bade Mṛtyu to kill the organisms without fear of sin. He assured her by saying that her tears which he had collected in his palms would become diseases that would arise from the bodies of the organisms themselves. Thus, when a man is killed by a disease sin would not stick to Mṛtyu. Nārada concluded his sermon by stating that it is not Mṛtyu with her rod who brought death to organisms but it was disease and conflict between themselves.

But before concluding, Nārada utters a remarkable verse which we shall take a detour to examine:

vāyur bhīmo bhīmanādo mahaujā 
bhettā dehān prāṇināṃ sarvago ‘sau ।
naivāvṛttiṃ nānuvṛttiṃ kadā cit
prāpnoty ugro ‘nantatejā viśiṣṭaḥ ॥
That terrible Vāyu, roaring dreadfully, of great energy, and going everywhere will shatter the bodies of living beings. The fierce [Vāyu] of endless energy neither attains restraint nor retreats in this matter.

At first sight, this verse is somewhat unusual given that the entire narrative does not mentioned Vāyu elsewhere. But it clearly appears to preserve an old concept regarding the bringer of death that goes back to the Indo-Iranian tradition. As we saw before, the gods Vāyu and Prajāpati (Brahman) have deep connection with roots in that tradition. Hence, the sudden introduction of Vāyu should not be seen as entire surprising. Importantly, this is the one clear Indo-Aryan reference to the death-bringing Vāyu, who also appears in the Iranian tradition as bone-breaking Asto-Astovidhātu and the “pitiless Vāyu (vayaosh anamarezhdikahe)” in the text known as the Aogemadaeca. The Aogemadaeca, termed by the Iranian ritualist Dastur Jamaspji as a treatise that “inculcates a sort of serene resignation to death”, covers themes relating to death quite like this narrative of Nārada from the Mbh. It has a series of incantations to Vāyu, which like in Nārada’s verse, states that there is no means to escape the way of Vāyu.

2) The legend of Madhu and Kaiṭabha also plays an important role in the origin mythology of the early Pañcarātra-vaiṣṇava tradition (the doctrine of the five nights). This is laid out in the Śānti-parvan of the Mahābhārata (12.335 in the Pune edition) in the context of the manifestation of Viṣṇu as Hayagrīva. As per the account the legend of this manifestation was narrated after that of the dual Nara-Nārāyaṇa and Varāha manifestations. Thus, these three are the primary vibhava-s of Viṣṇu beyond his four vyūha-s that are central to the Pañcarātra doctrine.

This narrative of Madhu and Kaiṭabha begins with an account of the end and re-emergence of the universe as per early Pāñcarātrika-sāṃkhya-yoga. Its is stated that all physical bodies are made up of the five primal substances born of the buddhi of the Īśvara who is none other than the great god Nārāyaṇa. At the end of universe, the solid substance merges into the liquid substance and the result is a vast liquid expanse known as the Ekārṇava. Then the liquid substance is absorbed into the heat and light substance. That in turn is absorbed into the gaseous substance and that is absorbed into vacuum. Vacuum is then absorbed into the universal mind, the manas. The manas is absorbed into vyakta and the vyakta in turn is absorbed into avyakta. The avyakta is absorbed into the puruṣa and the puruṣa is finally absorbed into the “final state”. Then there is only complete darkness or nothingness.

In that nothingness the brahma (neuter) manifests bearing in it the potential of natural law. That brahma then manifests as a puruṣa termed the Aniruddha (the generator vyūha of Pañcarātra) with the triple guṇa-s. He is equivalent to the pradhāna of Sāṃkhya and possessed of knowledge [of manifesting the universe] he is also known as the Viṣvaksena. The Aniruddha generating a realm of primal fluid rests upon it in yoga-nidrā. In that state he contemplates the creation of the wondrous universe with diverse properties and in the process reflects on his on great qualities. This self-consciousness (ahamkāra) results in a blazing 1000-petaled lotus emerging from the Aniruddha and from that lotus emerged the great 4-faced god Brahman also known as Hiraṇyagarbha or the golden egg. Brahman taking his Parameṣṭhin form he began planning the emission of the universe.

Just then two drops of fluid from the sea of primal fluid had fallen on the lotus of Brahman. When Nārāyaṇa looked at them, one of them full of pure Tamas became a demon Madhu. The other made of pure Rajas became Kaiṭabha. Holding maces in their hands they began roving in the pistil of the great lotus. Just then Brahman was commencing creation, starting with the Veda-s. Madhu and Kaiṭabha forcibly took the Veda-s away from him and dived into the primal ocean. Alarmed Brahman alerted Viṣṇu stating that the Veda-s were his eyes and without them he would not be able to create the universe. He then praised Viṣṇu with a hymn that recalled his 7 births from the different parts of the body of Viṣṇu.

Hearing this, Viṣṇu assumed the form of the macranthropic Hayaśiras with his body comprised of the entire universe. He began reciting the Veda loudly. Hearing this, the two daitya-s threw the Veda-s in the nether regions of the universal ocean and proceeded to see what was happening. As the did so, Hayaśiras dived into the great ocean and retrieved the Veda-s and returned them to Brahman. Not finding anyone the two daitya-s returned to the spot where they had cast the Veda-s only to find them missing. Wonder who had taken them away they rushed to the lotus from which they had first emerged. There they saw the white Aniruddha stretched in yoga-nidrā on the great snake, surrounded by a great circle of flames. They began to laugh loudly and, stating that he was the person who had retrieved the Veda-s, wondered aloud who that being was, desiring to battle him. This awakened the god, who taking on his Aśvaśiras form fought and slew both Madhu and Kaiṭabha. Thus, having restored the Veda to Brahman and having slain the obstructing demons Viṣṇu vanished letting Brahman to proceed with his act of creation. The krama of worship this god Hayaśiras was acquired by the Pāñcāla monarch from Rāma (the Bhārgava).

Thus, in this early Pāñcarātrika account the agent of Kaiṭabha’s destruction is Viṣṇu in his Hayagrīva form. However, here he is coupled with Madhu who only occurs in the Vaiṣṇava reflex of the myth. However, here too, obstructions to Brahman’s act of creation are central to the myth. This raises the possibility that there were two parallel versions of the myth: one among the Prājāpatya-s centered on Kaiṭabha as in the account of Suśruta and another among the Vaiṣṇava-s centered on Madhu, with the Vaiṣṇava-s subsequently acquiring Kaiṭabha for their focal deity. We see a further transformation at work in the later śākta tradition (e.g. Mārkaṇḍeya-purāṇa) where the role of arousing Viṣṇu to kill the two daitya-s is attributed to Mahākālī who herself is the Yoga-nidrā of Viṣṇu. Again, in later Vaiṣṇava tradition Hayagrīva gets decoupled from the Madhu-Kaiṭabha episode, though his role in retrieving the Veda is retained.

We may conclude this point by noting that the arthropod bites which form the backdrop of the section containing the above myth in Suśruta-s medical saṃhitā make subliminal return in later kāvya. For example, Māgha in his Śiśupāla-vadha states:

matkuṇāv iva purā pariplavau sindhunātha-śayane niṣeduṣaḥ ।
gacchataḥ sma madhu-kaiṭabhau vibhor yasya naidra-sukha-vighnatāṃ kṣaṇam ॥ 
Formerly, swimming like bugs Madhu and Kaiṭabha had proceeded to afflict the lord of the ocean in his bed. They interrupted the pleasure of the god’s sleep but just for a moment.

Bed bugs do not swim actively. But water-bugs are great swimmers and the bite of some like the water bug Notonecta glauca can inflict pain. This would then indicate that Māgha understood the water bugs to be of the same category as bed bugs.

3) Returning to the myth with which we started, we may note that beyond Brahman there is another god with a much more natural connection to poison. Right from the śruti we see that connection with the god Rudra. In the Ṛgveda we notice him drinking poison from a cup along with the Muni-s (RV 10.136.7). In the Atharvan, we have an incantation imploring Rudra not to strike down the ritualist with poxes or poison or the heavenly fire (AV-vulgate 11.2.26). In the Yajurveda, Rudra is termed Nīlagrīva alluding to his neck turning blue from drinking poison. This is elaborated in the paurāṇika tradition as happening during the churning of the world-ocean during which the poison known as halāhala or kālakūṭa arose distressing the deva-s and the asura-s. Rudra stepped forward to drink it. In the Pañcaviṃśa-brāhmaṇa of the Sāmaveda we have an account of Rudra smearing the plants with poison in particular years resulting in death of the livestock (PB 6.9):

viṣeṇa vai tāṃ samām oṣadhayo ‘ktā bhavanti yāṃ samāṃ mahādevaḥ paśūn hanti yac chaṃ rājann oṣadhībhya ity āhauṣadhīr evāsmai svadayaty ubhayyo ‘smai svaditāḥ pacyante ‘kṛṣṭapacyāś ca kṛṣṭapacyāś ca ॥
Indeed the plants becomes smeared with poison in that year in which Mahādeva kills the livestock. By uttering the incantation “Auspicious [be] the king of the plants [Rudra]” he makes the herbs fit for consumption. He makes consumable both types [of herbs] those which ripen without cultivation and those which are cultivated.

This is probably an allusion to the years when plants producing toxins in response to grazers grown in the pastures or when outbreaks of diseases like anthrax occur. Even more apposite to our legend, is a similar account from the Maitrāyanī-saṃhitā of the Kṛṣṇa-yajurveda with regard to the Prājāpatya-samidh (fire-stick) offered in the Agnihotra ritual. This is placed in the context of Prajāpati creating the organisms: after Prajāpati had emitted the organisms (beginning of MS 1.8.4) Rudra smeared the plants with poison (end of same section) resulting in them not being eaten by Rudra:

oṣadhīr vā imā rudrā viṣeṇāñjaṃs tāḥ paśavo nāliśanta te devāḥ prajāpatim evopādhāvan sa prajāpatir abravīd vāryaṃ vṛṇai bhāgo me ‘stv iti vṛṇīṣvety abruvant so ‘bravīn maddevatyaiva samidasad iti tasmāt prājāpatyā samid deveṣu hy asyaiṣā vāryavṛtā dve samidhau kārye dve hy āhutī ekaiva kāryaiko hi prajāpatir ekadhā khalu vai samiddha uta bahvīr āhutayo hūyante tā agninānvavākarot tā asvadayat tāḥ punarṇavā ajāyantaitarhi khalu vā agnihotriṇe darśapūrṇamāsine sarvā oṣadhayaḥ svadante yat samidham ādadhāti sarvā evāsmā oṣadhīḥ svadhayām akaḥ ॥
These plants being smeared with poison by Rudra (unusual MS saṃdhi) were not eaten by the animals. As the deva-s then ran to Prajāpati, he, Prajāpati, chose a boon: “May there be a [ritual] share for me”. [The gods said] choose. He [P] said: “May the fire-stick be [offered] with me as the deity.” Therefore the Prājāpatya fire-stick [is offered], for it was boon chosen from the gods by him. Should there be two fire-sticks in the ritual as there are two offerings? One indeed is the ritual action [offering the fire-stick] for Prajāpati is verily one. Thus, just one fold is the offering of the fire stick even though many oblations [of ghee] are offered. Those [plants] are successively strewn by Agni. They were made palatable. They were born anew again. Now indeed, when the ritualist who performs Agnihotra or the New- and Full-moon rituals offers [Prajāpati’s] fire-stick all plants become consumable. He has made all plants consumable for him.

A very similar brāhmaṇa passage for the single Prajāpati fire-stick in the Agnihotra is presented in the Kaṭha-saṃhitā and here too Prajāpati relieves the toxin of Rudra in the plants by letting loose Agni on them. This might be an allusion to the burning of pastures contaminated by disease or filled with toxin-producing herbage. (As an aside, the corresponding brāhmaṇa of the Taittirīyaka-s TB onward does not mention Rudra or Prajāpati or the plants. It simply mentions the single fire stick as being offered to Agni.)

Finally, we may note that several brāhmaṇa texts see Rudra as being born from Prajāpati even as in the above-stated legend regarding the origin of Mṛtyu from the Mbh. In the Kauśītaki brāhmaṇa (KB 6.1-9) he is born from Prajāpati with a 1000 eyes bearing a 1000 arrows with an 8-fold form (aṣṭamūrti). In the Śatapatha brāhmaṇa (ŚB he is born again from Prajāpati in an eight-fold form and then is joined with his 9th form Kumāra. In the Maitrāyaṇī-śruti (MS 4.2.12) he is born from sweat of Prajāpati combined with that of the three realms first generated by Prajāpati: Agni, Vāyu and Sūrya. Here, we are told that Bhava and Śarva are the dreadful names of the god; hence, one should not take them while making the offerings, instead once should take his pacified names Rudra and Paśupati. But most tellingly for the myth with which we started we can see a connection to an account of the birth of Rudra in the Śatapatha brāhmaṇa mentioned in the context of the great Śatarudrīya offerings to the 100-headed Rudra upon completion of the piling of the altar for the Soma ritual:

yadv evaitac catarudriyaṃ juhoti । prajāpater visrastād devatā udakrāmaṃs tam eka eva devo nājahān manyur eva so ‘tinn antar-vitato ‘tiṣṭhat so ‘rodīt tasya yāny aśrūṇi prāskandaṃs tāny asmin manyau pratyatiṣṭhant sa eva śataśīrṣā rudraḥ samabhavat sahasrākṣaḥ śateṣudhir atha yā anyā vipruṣo ‘pataṃs tā asaṃkhyātā sahasrāṇi+imāṃ lokān anuprāviśaṃs tad yad ruditāt samad rudrāḥ so ‘yaṃ śataśīrṣā rudraḥ sahasrākṣaḥ śateṣudhir adhijyadhanvā pratihitāyī bhīṣayamāṇo ‘tiṣṭhad annam icamānas tasmād devā abibhayuḥ ॥

As to why one makes this Śatarudrīya offering: When Prajāpati had decayed, the deities departed from him. Only one god did not leave him, Manyu (wrath): extended he remained stationed within. He (P) cried. [By] Whichever of his [P’s] tears that dropped down and settled on Manyu, he [M] became the hundred-headed Rudra, with a thousand-eyes and a hundred-quivers. Then the other drops that fell down, entered all these worlds in innumerable thousands; inasmuch as they originated from crying (rud), they were called Rudra-s. That hundred-headed Rudra with a thousand-eyes and a hundred-quivers, with his bow strung, and his arrow fitted to the string, was inspiring fear, being in quest of food. Therefore gods were afraid of him.

To appease the 100-headed Rudra and his thousands of followers, the other Rudra-s, the ritualist makes the 425 oblations to Rudra-s as specified in the Yajurveda. (As an aside, one may note that this dread of Rudra, even among the gods, is an old Indo-European sentiment: One may compare it to his Greek cognate Apollo entering the hall of the gods with his bow in the Homeric hymn to that god.)

Thus, in this narrative of the ŚB, Rudra is born of the wrath of Prajāpati. Hence, lurking behind the mysterious embodied wrath of Brahman, which became the poison of Suśruta, we suspect that there is an implicit allusion to none other than Rudra, whose wrath elsewhere in the medical saṃhitā-s is seen as the originator of diseases.

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Chaos, eruptions and root-convergence in one-dimensional maps based on metallic-sequence generating functions

Over the years we have observed or encountered certain natural phenomena that are characterized by rare, sudden eruptive behavior occurring against a background of very low amplitude fluctuations. We first encountered this in astronomy: most remarkably, in the constellation of Corona Borealis there are two stars that exhibit this kind of behavior albeit in opposite directions. There is T Coronae Borealis, which for most part remains fluctuating rather dimly in a narrow magnitude band between 9.9 and 10.6, well below naked eye visibility, and then once in a century or so (e.g. in 1866 and 1946) explosively blazes forth at magnitude of 2-3 ( \approx 1000 \times brighter) changing the shape of the visible constellation. On the other side, we have the equally charismatic R Coronae Borealis, which for most part very mildly fluctuates around the magnitude of 6 at barely naked eye visibility and then suddenly once in several years suddenly drops to the magnitude of 14 or lower ( \approx 1500 \times dimmer), beyond the reach of even a typical amateur telescope.

It has become increasingly clear that versions of such behaviors are observed across the domains of science. In biology, recent studies in foraging behavior have shown that diverse animals follow a pattern of foraging movements, which are characterized by routine small movements punctuated by the rare large movements. This kind of behavior allows the escaping of local resource limitations by episodic saltations to reach distant resource-rich regions. An unrelated phenomenon, earthquakes, also displays similar behavior, where small low-magnitude tremors are punctuated by rare episodes of major earthquakes with noticeable effects. This might also be seen in sociology/geopolitics where long periods of low intensity conflicts are interrupted by the rare cases of major warfare. The world wars could be seen as such manifestations against a background of low intensity conflict. This is relevant to the clash between the thinkers Taleb and Pinker regarding whether there is a real trend of the world having become more peaceful or not.

One of the great mathematicians of our age Benoît Mandelbrot provided a framework to understand these disparate phenomena under the rubric of random walks. He called a regular random walk, where the step sizes are normally distributed, as the Rayleigh flights. In contrast, if they instead show some distribution that has a tail with a slower than exponential decay then he defined them as Lévy flights after the mathematician Lévy. One example, of this is the so called Cauchy flight which results from the steps of the walk showing the famous Cauchy distribution (originally discovered by Poisson but attributed to Cauchy). Such random walks are characterized by typical steps that are smaller in magnitude more common than the typical steps under a normal distribution and the extreme steps are rarer than under a normal distribution but way more in magnitude than one would see under a normal distribution. Thus, they capture the eruptive behavior quite well.

We have been long interested in creating simple mathematical analogies for such behaviors observed in nature. Given that a random walk can be reduced to its simplest form, i.e. a one dimensional change in magnitude, we have been interested in one-dimensional maps that can display such behaviors. We describe below a class of one-dimensional maps, which we discovered, that show such a behavior. They are all related to quadratic roots known as metallic ratios.

The metallic ratios and generating functions of the metallic sequences
Metallic ratios can be defined as irrationals which are produced by the following formula:

m_n=\dfrac{\left(n+\sqrt{n^2+4}\right)}{2}; n =1, 2, 3...

They are called “metallic” after the smallest of them, the famous Golden ratio. As we have noted before the first few metallic ratios are “interesting” because they appear in various unrelated contexts but the larger ones do not seem to do so. m_n and its conjugate m_n' are root of a quadratic equation of the form:

x^2 \pm nx -1=0

The two roots are opposite in sign but correspond to m_n, m_n', which show the relationship:


Accordingly, they count among the the so-called Pisot-Vijayaraghavan numbers. We use m_n for the larger and m_n' for the smaller root in absolute magnitude. Below are the first few metallic ratios, which may assigned special symbols as below:


They are called ratios because they are the convergents of integer sequences which are specified by the rule:

f[n]=k \cdot f[n-1]+f[n-2]; k=1, 2, 3...; f[1]=1, f[2]=f[1]+k
With k=1 we get: 1, 2, 3, 5, 8, 13, 21…
With k=2 we get: 1, 3, 7, 17, 41, 99…
With k=3 we get: 1, 4, 13, 43, 142, 469…
With k=4 we get: 1, 5, 21, 89, 377, 1597…

Notably these sequences are related to the quadratics whose roots they are via certain generating functions. For a function y=f(x) the series expansion of f(x) at the value of x=a is given by:


Now, for instance, consider the function:


For x=0 its series expansion becomes:

y= -(x + x^2 + 2 x^3 + 3 x^4 + 5 x^5 + 8x^5...)

We notice that the coefficients of the expansion polynomial are 1, 1, 2, 3, 5, 8.... Thus, this expansion becomes the generating function of the Golden sequence. Similarly, if we take:


For x=0 we get the series expansion:

y=1 + x + 4x^2 + 13x^3 + 43x^4 + 142x^5...

Here, the coefficients of the expansion polynomial correspond to the Bronze sequence. The functions of the above type are tripartite curves with two parallel asymptotes at x=-m_n and x=m_n' (Figure 1). The central branch of the curve is bounded between these parallel asymptotes. The left and the right hyperbolic branches are bounded respectively on the right and left sides by the two asymptotes.

golden_bouncer_Fig1Figure 1

One dimensional maps based on the metallic sequence generating functions
The above type of functions specify our one-dimensional maps. The maps operate by using the well-known cobweb construction used in the study of chaotic maps (Figure 1). Figure 1 shows the process for y=f(x)=\tfrac{2x+1}{x^2+x-1}, which has the series expansion at x=0 as y=-(1 + 3x + 4x^2 + 7 x^3 + 11 x^4 +18x^5+29x^6...). The convergent of the coefficients of this expansion polynomial is the Golden ratio. We start with an initial point X_0. We then project the point on the the curve f(x) defined by one such metallic sequence generating function to obtain point Y_1. We then project Y_1 on the line y=x to obtain point X_1. We then repeat the above procedure with X_1. The resulting projection segments are plotted as a cobweb diagram (Figure 1). In algebraic terms the map is expressed as x_{n+1}=f(x_n). Our empirical examination revealed that not all maps of this type produce interesting behavior — they simply converge to a single fixed attractor value. Moreover, we did not observe interesting behavior with other small quadratic Pisot-Vijayaraghavan roots, e.g.: x^2-3x+1=0; x= 1+\phi, 1-\phi' or x^2-2x-2; x=1\pm \sqrt{3}.

In our exploration we found the following maps to show interesting behaviors.
1) Golden-ratio-based:



2) Silver-ratio-based:


3) Bronze-ratio-based:



3) Copper-ratio-based:


One point to note regarding these maps is the high degree of numerical instability of most of them. Hence, we have to use high precision numbers to study the evolution of a starting value under the map. We found that one needs a precision of 1500 bits in order to obtain a proper evolutionary trajectory for 20000 iterations of the map. The investigations discussed below are with this precision and number of iterations. Evolution under these maps can be classified into 4 distinct types and we shall discuss examples of each of the distinct types of behaviors in greater detail below.

Type-1: Chaotic and eruptive behavior
The archetypal member of this type is Golden-ratio-based map:

Golden_bouncer3_Fig2Figure 2.

Figure 2 shows the evolution of x_0=\tfrac{1}{4} under this map. It is immediately apparent that it is characterized by predominantly low amplitude movements interrupted by rare episodes of extreme eruptions that can be several orders of magnitude of the typical values. A feel for this can be obtained via the below table.

\begin{tabular}{|l|c|} \hline Quantile & Value \\ \hline Minimum & -6114.33 \\ Octile 0.125 & -2.406 \\ Octile 0.25 & -0.995 \\ Octile 0.375 & -0.413 \\ Median & 0.0036 \\ Mean & 1.75 \\ Octile 0.625 & -0.418 \\ Octile 0.75 & 1.003 \\ Octile 0.875 & 2.418 \\ Maximum & 35903.36 \\ \hline \end{tabular}

We observe that 75\% ot the values are in rather narrow band of \pm 2.41; however, the extremes are roughly 2500-15000 times greater than that band. Thus, we see what is plainly eruptive behavior. It is also rather obvious from these values that the distribution of the values attained under this map are dramatically different from a normal distribution with comparable dispersion. The rarity and the magnitude of the extreme values results in the mean being greatly different from the median.

Golden_bouncer3_Fig3Figure 3.

To get a better look at the nature of the movements as x_0 evolves under this map we plot the same on the \mathrm{arcsinh}(x) scale (Figure 3). The central blue line corresponds to 0. It is flanked on either side by two red lines which correspond to \pm 0.2. In this case \approx 12.5\% of the x_n lie in this band. The green lines correspond to the Golden ratios \phi', -\phi. The cases where x_n \ge 100, which is 0.64\% of the values, are marked with red points. We observe that:
1) The evolution of x_n follows a chaotic course.
2) The eruptions are triggered in x_{n+1} when x_n approaches \phi', -\phi (Blue points in figure 3). This is illustrated at greater magnification in Figure 4. The reason for this is rather obvious from the equation of the map — if x_n=\phi', -\phi then x_{n+1} will explode to \infty: thus, closer x_n gets to \phi',-\phi greater is the eruption in x_{n+1}. If the value of x_n>\phi', -\phi when approaching it then the eruption in x_{n+1} will positive, if x_n<\phi',\phi the eruption will be negative.

Golden_bouncer3_Fig4Figure 4.

3) Because the map tends to mostly produce small absolute values \approx 50\%\; x_n \in \pm 1 it is obvious that the smaller root \phi' triggers more eruptions than -\phi.
4) From Figure 4 one also observes a peculiar motif in the form of runs of relatively low amplitude fluctuations in the vicinity of \phi, \tfrac{-\phi'}{2} (marked as dark green horizontal lines).

Because of the rather dramatic dispersion of the values of x_n, we studied their distribution in the \mathrm{arcsinh(x)} scale (Figure 5).

Golden_bouncer3_Fig5Figure 5.

We observe that the distribution is symmetric about 0, as we would expect from the above quantile distributions of x_n. The blue curve shows an attempt to fit a Cauchy distribution to the observed frequencies. We see that it does not capture the distribution too well. The more general form of such a distribution, the t-distribution, also does not perfectly capture the observed frequencies. However, taking inspiration from that we were able to derive a curve that fits the histogram better than any of these distributions. It is defined by a shape function with 4 parameters a, b, r, s of the form (red curve in Figure 5):


Another map belonging to this type is similar one based on the Silver ratio (Figure 6):

Silver_bouncer2_Fig6Figure 6.

It displays the same type of chaotic movements as the above map with episodes of major eruptions.

Silver_bouncer2_Fig7Figure 7.

The zoom in Figure 7 shows that the eruptions in x_{n+1} are in this triggered by x_n approaching the Silver ratio and its conjugate (light green lines). Like the above map, here too there are motifs comprised of short runs of low amplitude oscillations around two values: 4-2\sqrt{2}, -1+\tfrac{1}{\sqrt{2}} (Figure 7).

Type-2: low amplitude oscillations punctuated with destabilization and eruptions
The maps exhibiting the second type of behavior are:

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

x_{n+1}=\dfrac{x_n}{x_n^2+3x_n-1}; x_{n+1}=\dfrac{2x_n+1}{x_n^2+3x_n-1}; x_{n+1}=\dfrac{x_n}{x_n^2+4x_n-1}

bouncers3_Fig8Figure 8.

Figure 8 shows examples of the evolution of the same x_0 under three of these maps based on the Golden, Silver and Bronze ratios. At first sight the eruptions are comparable to what we saw in the Type-1 maps — they are rare episodes and huge in magnitude relative to the rest of the movements under the map. However, these maps are distinct, in that other than the larger magnitude instability just before major eruptions the background movements are not even registered. To get a better look at what is happening we plot one of the examples (the map based on the Golden ratio) in the \mathrm{arcsinh(x)} scale (Figure 9).

Golden_bouncer2_Fig9Figure 9.

Here we can see that, as in the Type-1 maps, the eruptions are triggered when x_n passes close to -\phi', \phi (blue points in Figure 9 prior to the red points marking the eruptions greater than 100). But quite strikingly, even in the \mathrm{arcsinh(x)} scale, the background movements under the map are barely visible. Hence, we zoom in on a particular region of the \mathrm{arcsinh(x)} scale plot (Figure 10).

Golden_bouncer2_Fig10Figure 10.

Only here we observe that the map is for most part characterized by a very quiet behavior, which, unlike the Type-1 maps, takes the form of regular low-amplitude oscillations that asymptotically build up till they near one of the Golden ratio roots at which point they show unstable behavior leading to an eruption. After a major eruption the evolution settles in a very low amplitude oscillation as seen in the right part of Figure 10. Accordingly we get a paradoxical distribution of x_n values on the \mathrm{arcsinh(x)} scale with a very sharp peak and a heavy tail Fig 11. This distribution might also be described by some form the shape equation specified in the above type.

Golden_bouncer3_Fig11Figure 11.

These maps may also be used to describe a feature common to all these maps, i.e. extreme sensitivity to the initial conditions (Figure 12, 13). Even a difference of 10^{-8} in the starting x_0 results in completely different evolutionary trajectories. This is why we need to use extremely high precision numbers to get numerical stable evolution: e.g. 450-600 decimal digits.

bronze_bouncer_Fig12Figure 12.

An example based on the Bronze ratio.

silver_bouncer_Fig13Figure 13.

An example based on the Silver ratio. The second example illustrates how eruptive behavior can return after an extremely long phase of quiescence.

Type-3: Convergent root-seeking behavior after eruptions and instability
This is an interesting behavior prototyped by the Bronze ratio-based map:

Here, the map tends to show oscillatory behavior with occasional eruptions as in the above type. However, after 1 or a few eruptions associated with some chaotic instability the map settles to a fixed behavior where it cycles between a small set of values that gradually converge to roots of a certain polynomial equation (Figure 14, 15).

rootseeker_r1_Fig14Figure 14.

In this case after 2 large eruptions within the first 400 iterations it eventually converges to cycling between the three roots of the equation:

P_1: y=x^3-3x+1; \; \overbrace{x_1=2\sin\left(\dfrac{\pi}{18}\right) \rightarrow x_2=-2\cos\left(\dfrac{\pi}{9}\right) \rightarrow x_3=2\cos\left(\dfrac{2\pi}{9}\right)}

Here too, we notice that the eruption occur in x_{n+1} (red dots) when x_n \approx \beta', -\beta (blue dots close to blue lines), i.e. the bronze ratios. The map makes multiple attempts to settle into cycling between the roots of P_1 (brown horizontal lines) but each time slips away nears either -\beta or \beta' and then erupts. Between 600 and 700 iterations the map settles into what seems a permanent cycling between the roots of P_1 and by 800 iterations approaches within 10^-6 of those roots. Iteration of the map for an additional 100000 iterations showed no further eruptive behavior suggesting final convergence to the roots; however, we have not been able to prove this is indeed case.

rootseeker_r2_Fig15Figure 15.

In this case the map makes multiple attempts to cycle between the roots of P_1 and the roots of a second polynomial P_2 before finally converging to cycling between the roots of the latter (brown horizontal lines):

P_2: y= x^4+7x^3-6x^2-2x+1\\[8pt] \left.\begin{aligned} x_1 = \dfrac{-7 + 3\sqrt{5} + \sqrt{150 - 66\sqrt{5}}}{4}\\[8pt] x_2 = \dfrac{-7 - 3\sqrt{5} - \sqrt{150 + 66\sqrt{5}}}{4}\\[8pt] x_3 = \dfrac{-7 + 3\sqrt{5} - \sqrt{150 - 66\sqrt{5}}}{4}\\[8pt] x_4 = \dfrac{-7 - 3\sqrt{5} + \sqrt{150 + 66\sqrt{5}}}{4} \end{aligned}\right\}

There is one obvious static x_0 which remains unchanged under this map, namely the real root of the equation: x^3+3x^2-3x+1=0:

x_s = -1-2^{1/3}-2^{2/3}

If x_0 is exactly this x_s then it will return itself under the map. Even a point very close to x_s is unstable and will not remain there for too long, eventually converging to P_1 or P_2. For example, a x_0 that was identical to x_s till 600 places after the decimal point slid away from x_s by iteration 705 of map and within the next 20 iterations was on its way to converge to the roots of P_1. Thus, all other points under this map eventually converge to either the roots of P_1 or of P_2 (Figure 16).

bouncer_covergence._Figure16Figure 16.

The x_0 are sampled in the interval (-15,15) in steps of .025. The red x_0 converge to roots of P_1, while the green x_0 converge to roots of P_2. The roots of P_1 are marked by blue vertical lines and those of P_2 by orange lines. Notably, it seems that they converge to P_2 less often than P_1 such the ratio P_2:P_1 appears to converge to \beta' (is there a way to formally test this conjecture?). Interestingly, immediately below P_2: x_2 there is a region where every x_0 converges to P_1. There is a pseudo-symmetric region above P_2: -x_2 where they all converge to P_1. Overall, this convergence plot shows a strange pseudo-symmetry in terms of the iterations to convergence required by x_0 (Figure 16).

These roots show some interesting properties:
P_1: x_1+x_2+x_3=0; \; \dfrac{1}{x_1}+x_2=1; \; \dfrac{1}{x_2}+x_3=1;\; x_1 \cdot x_2 \cdot x_3=-1

Further, the roots of P_1 are connected to an interesting three-seeded Nārāyaṇa-like sequence with a subtraction rather than a sum:
f[n]=3f[n-1]-f[n-3];\; f[1]=0, f[2]=1, f[3]=3

This yields: 0, 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977…

The convergent of this sequence is 1+2\cos\left(\tfrac{\pi}{9}\right)=1-x_2 \approx 2.8793852. In principle, this could be used to construct approximations of a regular nonagon. The terms of this series are provided by a generating function which is reciprocal of P_1. Its series expansion at 0 is:

\dfrac{1}{x^3-3x+1}=1 + 3x + 9x^2 + 26x^3 + 75x^4 + 216x^5 + 622x^6 + 1791x^7 + 5157x^8 + 14849x^9 +...

For the second polynomial we have:
P_2: x_1+x_2+x_3+x_4=-7;\; \dfrac{1}{x_1}+\dfrac{1}{x_3}=1; \; \dfrac{1}{x_2}+\dfrac{1}{x_4}=1; \; x_1\cdot x_2\cdot x_3 \cdot x_4=1

These roots are similarly related to 2 sequences. The first is:
f[n]=7f[n-1]+6f[n-2]-2f[n-3]-f[n-4];\; f[1]=-1, f[2]=0, f[3]=1, f[4]=2

This yields: -1, 0, 1, 2, 21, 157, 1220, 9438, 73051, 565388, 4375926, 33868270, 262129619, 2028799713, 15702263239, 121530513443…

The convergent of this sequence is P_2: -x_2 \approx 7.739681318.

The second sequence is:
\displaystyle f[n]=7f[n-2]+f[]n-1]-\sum_{j=1}^{n-4} f[j]); \; f[1]=-1, f[2]=0, f[3]=1, f[4]=2

This yields: -1, 0, 1, 2, 10, 25, 95, 268, 921, 2760, 9075, 27995, 90199, 282083, 900320, 2833750, 9004640…

The convergent of this series is P_2: \dfrac{1}{x_1} \approx 3.165352

Notably, the terms of this sequence can be produced using the reciprocal of P_2 as the generating function. Its series expansion at 0 is:

\dfrac{1}{x^4+7x^3-6x^2-2x+1}=1 + 2 x + 10 x^2 + 25 x^3 + 95 x^4 + 268 x^5 + 921 x^6 + 2760 x^7 + 9075 x^8 + 27995 x^9 +...

Thus, this map function has a peculiar property with respect to the roots of the polynomials P_1 and P_2 in that applying it one root yields another in cyclic fashion. Given that for both these polynomials there is a root close to \beta', these can “capture” the x_n approaching it and drive them into a cycle. Thus, these might be seen as stable examples of the motifs encountered in the Type-1 behavior.

Type-4: Convergence to bounded bands after initial eruptions and instability
This type of behavior is exhibited by the Copper ratio-based map:

Evolution under this map has parallels to the Type-3 behavior. Like in Type-3, after initial instability, which might include some eruptions and chaotic fluctuations, the evolution under the map settles to cycling between specific bounded bands. Within those bands it wanders quasi-chaotically, i.e. with some discernible patterns, but never leaves those bands. Number iterations taken by different x_0 to converge to cycling within those bands can vary greatly (Figure 17).

bouncer_covergence_Fig17Figure 17.

Here, x_0 are sampled in the interval (-40,40) in steps of .01. The plot shows a pattern with zones of rapid convergence interspersed with clusters of much longer convergence times. When we zoom in on a region with a cluster of slowly converging x_0 we see that the number of iterations to converge to the band-cycle shows a fractal pattern (Figure 18).

bouncer_covergence_Fig18Figure 18.

Here, the x_0 are sampled in the interval (-4,4) at steps of .001. In both these cases a pseudo-symmetry similar to the iterations-to-convergence plot for the Type-3 behavior (Figure 16) is observed. Here too there is a static point x_s \approx -4.68577952..., which is the real root of x^3+4x^2-3x+1=0; it returns itself under the map. Unless one is exactly at x_s which has a very complicated closed form, any other x_0 in the vicinity continues to converge to the band-cycle.

band_convergence_Fig19Figure 19.

To examine this type of behavior more closely we consider the evolution of x_0=4.37 (Figure 19), which is particularly slowly converging in Figure 17. We find that there is a massive eruption little after n=150 (eruptions >20 are marked with red points), which is lodged in the midst of generally chaotic behavior which lasts till close to n=260 (see lower panel in \mathrm{arcsinh}(x) scale). Not surprisingly, within this region, we observe that, as in the previous types of behavior, the eruptions occur in x_{n+1} when x_n (marked with blue points) approaches \kappa', -\kappa (marked with blue horizontal lines). However, the striking feature of this type is that sometime after this region the map settles into a more regular cycling behavior and never leaves it to degree we have tested these maps (n=100000). We examine this cyclic behavior more closely in Figure 20.

band_convergence_points_Fig20Figure 20.

Here, we plot the evolution of x_0=4.37 for n=300..3000. We find that the x_n falls in very specific bands:
1) A very narrow middle band centered at \frac{2}{5+\sqrt{5}}\approx 0.276393... (Figure 20, red points).

2) A lower band bounded by \left(-2 -\frac{\sqrt{5}}{2}\approx -3.118033, -2 -\frac{1}{\sqrt{5}} \approx -2.447213 \right) (Figure 20, violet points).

3) An upper band bounded by \left(\sqrt{5}-1 \approx 1.236067, \frac{20+4\sqrt{5}}{15} \approx 1.929618\right) (Figure 20, blue points).

In the state of convergence the map cycles from the middle band to the upper to the lower band over 3 successive x_n. Beyond these bands, there are two lines:
1) in the lower band corresponding to \frac{2577-1221 \sqrt{5}}{58} \approx -2.6420517

2) in the upper band corresponding to \frac{5-\sqrt{5}}{2}

It is notable that the above band-bounds and lines are all related to \sqrt{5} which is the surd in \kappa, \kappa'. They are indicated by horizontal lines in the plot. From the two lines within the bands the points appear to alternately wander towards the upper and lower bounds of the upper and lower bands. As the map evolves, once they reach close the bounds the process repeats again starting from those lines (Figure 20). Each round of wandering is largely symmetric between the upper and lower bands but each round is different from the previous one. In this, type as in Type-3 it appears that x_n is captured into the band cycle as it approaches \kappa' which is close to the central band defined by \frac{2}{5+\sqrt{5}}.

There are several open questions (for us) regarding these investigations. Is there a formal way of deriving which roots a map may converge to if it shows a Type-3 behavior? Is there a way to formally establish the bands of convergence for Type-4 behavior? Is there way to say whether a map would go the way of Type-1 or Type-2 from its equation?

Whatever the case, the maps discussed here are most remarkable because they can produce complex behavior similar to that observed in natural systems with very simple underlying equations. They illustrate cases of long quiescence (Type-1 and Type-2) with eruptive behavior. These behavior suggests that prolonged low amplitude oscillations or secular changes do not guarantee the absence of sudden eruptions. This is important in realizing that situations like prolonged peace do not mean that there would no major catastrophic conflict. This suddenly emerges as the system moves towards a superficially unremarkable low magnitude state that in reality is something like the metallic ratio trigger seen in these maps. Then we have the reverse behavior where after an initial chaotic phase the map settles into a more regular behavior from which it never emerges due to being captured by certain similarly superficially unremarkable values (Type-3 and Type-4). In history we see that the dramatic movements that occur early in a civilization are never reproduced in the later stages from which it might be unable to break out.

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Big fruits and dead giants

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

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

elephant_fruits_smallFigure 1

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

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

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

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

elephants_pngFigure 2: The phylogeny of elephants according to Jeheskel Shoshani

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

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

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

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

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

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

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

sloths_smallFigure 4. Sloths by Jorge Blanco

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

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

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

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Pearl necklaces for Maheśvara

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

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

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

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

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

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

Table 1

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

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

Table 2

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

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

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

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

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

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

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

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

The families of Maheśvara’s necklaces

Figure 1

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

Figure 2 (click on figure to magnify)

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

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

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

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

Figure 3

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

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

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

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

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

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

Figure 4

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

Figure 5

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

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

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