A note on āṃreḍita-s in the Ṛgveda and issues of word distribution

sa darśataśrīr atithir gṛhe-gṛhe
vane-vane śiśriye takvavīr iva ।
janaṃ-janaṃ janyo nāti manyate
viśa ā kṣeti viśyo viśaṃ-viśam ॥ RV 10.91.2 by Aruṇa Vaitahavya

He, with visible auspiciousness, a guest in house after house,
in forest after forest lurking like a hunting falcon,
people after people, no people are overlooked by him,
The clan among the clans, he dwells in clan after clan.

The āmreḍita or the iterative is a duplicated compound like those seen in the above ṛk: gṛhe-gṛhe etc. While the variety of āmreḍita-s seen in the RV is no longer seen in modern Sanskrit, some forms have persisted from that time e.g. punaḥ-punaḥ or anyam-anyam. Their variants are also seen in other extant Indo-Āryan languages. This form of a compound is attested to my knowledge in the earliest branching lineage of Indo-European, Anatolian. It is also seen in Tocharian which probably branched off next. Forms equivalent to the “pra-pra” (forward and forward) found in the RV are also attested in Homeric Greek and are termed prepositional complements of verbs. However, the āmreḍita, with the involvement of all elements of speech, is most developed only in Indo-Iranian. Old Iranian in the form of Avestan displays forms like:
nmānē-nmānē = Skt: dhāmni-dhāmni; house after house (i.e. every household)
vīsi-vīsi = Skt: viśe-viśe; clan after clan (i.e. every clan)

Given their parallels in Sanskrit, it is clear that this expanded system of āṃreḍita-s was already present in the ancestral Indo-Iranian tradition. Here we briefly examine the āṃreḍita in the Ṛgveda. Given the temporal position of the RV, it likely presents a picture of the usage of such compounds close to the beginning of the Indo-Iranian tradition.

There can be anywhere between 1 to 4 āṃreḍita-s in a given ṛk. The number of ṛks-s with each number of āṃreḍita-s per ṛk is shown in Table 1.

AMreDita_tabTable 1

The maximum number of āṃreḍita-s in the RV is four, which is seen in a single ṛk, the one given in the opening lines of this article. By far the most common āṃreḍita is dive-dive (day after day), which occurs 47 times in the RV (the next most common, the ancient pra-pra, occurs only 12 times). A related usage ahar-ahaḥ meaning the same as dive-dive occurs 6 times. Thus, the āṃreḍita used in the sense of ‘daily occurrence’ seems to be a characteristic feature of the Vedic language. Another temporal āṃreḍita is yuge-yuge (6x), which famously occurs in later literature in the Bhagavadgītā. Whereas in that text it means ‘eon after eon’, in the RV it likely means some version of the pañcasaṃvatsara-yuga, which is explained by Lagadha in the Vedāñga yotiṣa. Some other temporal āmreḍita-s are also used on rare occasions in the RV: māsi-māsi (every month) and parvaṇā-parvaṇā (every fortnight).

Other āṃreḍita-s tell us about the old Āryan way of life and religion. Several of them indicate the sense ‘in every household’: dame-dame (6x), gṛhe-gṛhe/gṛhaṃ-gṛham (7x), vastor-vastoH (2x). Others denote the sense ‘in every clan’: viśe-viśe and variants (7x), jane-jane/janaṃ-janam/jāto-jātaḥ (5x). These meanings are shared with the most common Iranian āṃreḍita-s, suggesting that they were significant for the Indo-Iranians at large. Specifically, these terms indicate that the households and the clans were the primary organizational units of society. There is no mention of towns or even villages in any āṃreḍita-s although forests and trees are alluded to vane-vane/vṛkṣe-vṛkṣe. This is reflective of the semi-mobile life in the forest-steppe zone with its characteristic confederation of clans.

As we have noted before, the people of the RV were a warlike people. Thus, the āṃreḍita-s meaning something like ‘in every battle’ or ‘in every contest’ are common: bhare-bhare (6x), vāje-vāje (4x), raṇe-raṇe (2x). There is also a śatroḥ-śatroḥ, i.e. ‘of every enemy’. This indicates that frequent military encounters were a feature of the lives of the early Indo-Āryans.

We also have many terms relating to the Vedic religion as would befit a primarily religious text. First, we have devaṃ-devam/devasya-devasya/devo-devaḥ (10x) implying the worship of every god in the pantheon. We also here of groups or troops of gods: gaṇaṃ-gaṇam/śardham-śardham (3x). These are clear expression of the polycentric polytheism of the Ārya-s. Second, the soma- pressing and drinking sessions, which are part of the high Vedic rite, are alluded to in the terms made-made/sute-sute/some-some (7x). Third, rites and related terms are referred to multiply. The fire ritual: yajñe-yajñe and variants (8x); the ritual fires: agnim-agnim (3x); Ritual observances and actions: vrātaṃ-vrātaṃ (2x), karman-karman and variants (3x); ritual offerings: havir-haviḥ, samit-samit (2x); Incantations, recitations and invocations: dhiyaṃ-dhiyaṃ, girā-girā, have-have, brahma-brahma (5x)

To better understand the distribution of āṃreḍita-s in the RV we shall first take a detour to look at some basic statistics of the RV:

RV_stats_tabTable 2. Some basic numbers for RV including number of āṃreḍita-s occurring per maṇḍala

Figure 1 shows the number of words per hemistich (ardhark) with the alternate hemistichs colored in blue and red. The maṇḍala boundaries are marked with vertical dotted lines.

AMreDita_words_half_RkFigure 1

This plot shows that the RV has a generally consistent average behavior across most maNDala-s keeping with the relatively tight repertoire of Chandas (meters) used by most clans. However, the anomalies are also immediately apparent. The maṇḍala-8 dominated by Kāṇva-s clearly stands out as having a distinct structure. Moreover, this pattern is also seen in the initial part of maṇḍala-1 where multiple Kāṇva-s are represented. Further, in maṇḍala-1 we see an anomalous spike corresponding to the ultra-long meter, the atyaṣṭi used by prince Parucchepa Daivodāsi. In maṇḍāla-9, the soma-maṇḍala we again see an anomaly. Here there is a strong separation of the short meters (gāyatrī-s) in the first part and the long meters closer to the end. This reflects this distinct aggregation history of the maṇḍala-9, which unlike the family books tends to collect the sūkta-s used in the soma ritual from different clans around a Kāśyapa core. These anomalies again come out clearly in the first two panels of Figure 2, which show the average number of words per hemistich and the average number of words per sūkta across the 10 maṇḍala-s.

AMreDita_plotsFigure 2 shows some of the statistics for the RV with relation to the āṃreḍita distribution.

These features are related in part to peculiarities discussed by in our earlier analysis of related issues pertaining to the RV. We observe that the maṇḍala-s 8 and 9 have the lowest average number of words per hemistich, keeping with the dominance of the shorter meters in these maṇḍala-s (panel 1). However, we can see that Kāṇva-s tend to compose long sūkta-s; hence, they figure a higher number of words per sūkta on an average (Panel 2). However, in contrast, maṇḍala-9, which also has the lower average number of words per sūkta, has no such compensation and is dominated by short sūkta-s. These are peculiar to the soma ritual and the sāman-s composed on them. We also note that the Atri-s and Vasiṣṭha-s tend to compose more short sūkta-s than the others. Barring these anomalies, the RV is quite uniform, especially in terms of the average number of words per hemistich. This gives us the general background to investigate the distribution of āṃreḍita-s.

Panel 3 for Figure 2 shows the āṃreḍita-s in a sliding window of 2000 words through the length of the RV. The maṇḍala boundaries are marked by a vertical dotted line. We observe that the āṃreḍita-s are not uniformly distributed. There are whole regions with a low count and others with notable spikes. We find that the maṇḍala-s differ in their use of āṃreḍita-s (Panel 4 of Figure 2): maṇḍala-s 3 and 6 of the Vaiśvāmitra-s and Bhāradvāja-s are rich in āṃreḍita-s, whereas those of Vāmadeva, the Vāsiṣṭha-s and the soma maṇḍala are particularly poor. In the case of the soma maṇḍala, the metrical structure with a low average number of words per hemistich probably discriminates against āṃreḍita-s. However, in the case of the other maṇḍala the difference in āṃreḍita counts is in spite of their mostly average behavior in terms of word count per hemistich (compare panel 4 and panel 1).

To understand if this difference might have any significance, we simulated the distribution of āṃreḍita-s as a random process using the total number of āṃreḍita-s in the RV (Table 2). We created 10000 artificial sets corresponding to the size of each maṇḍala, checked the number āṃreḍita-s reached in each replicate and computed the Z-scores for the observed number of āṃreḍita-s and the probability of getting the observed number or more/less by chance alone (Figure 3).

AMreDita_simulationFigure 3

This experiment suggests that maṇḍala-s 1 and 10 have more or less the average number of āṃreḍita-s one would expect by chance alone. This probably reflects their composite nature rather than being the product of one dominant clan. However, maṇḍala-s 3 and 6 have greater than expected number āṃreḍita-s (p=0.016 and 0.014 respectively), whereas maṇḍala-s 7 and 9 have lower than expected number of āṃreḍita-s (p=0.035 and 0.037 respectively). This observation suggests there was possibly a conscious difference in the poetic styles of the Vaiśvāmitra-s and Bhāradvāja#-s on one hand and the Vāsiṣṭha-s on the other, with the former showing a predilection for the use of āṃreḍita-s. This makes one wonder if the reduced use of āṃreḍita-s by the Vasiṣṭha-s, who had some links to the Iranian side, represents a regional tendency also seen in the Avesta, which also uses a low number of āṃreḍita-s. As noted above the unique structure of the soma-maṇḍala probably accounts for its low āṃreḍita count.

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The amazonian banana republic: the strī-rājya in Hindu tradition

amazon_2

The śaiva tradition shows a dichotomy with respect to the role of the sex in ritual and purity. The earlier antimārga or pāśupata tradition focused on abstinence and the so-called “upward flow” or ūrdhvaretas. This indeed the underlying idea behind the ithyphallic depiction of Lakulīśa, the founder of one of the key atimārga traditions. However, within the śaiva tradition there was another ambivalent practice with earlier roots in the shared pool of ascetic practices, which were also inherited by the vaiṣṇava-s (e.g. Vaikhānasa-gṛhyasūtra). This was the asidhārā-vrata. Here the practitioner engages in kissing and coital contact with his wife or another beautiful and sexually active woman without spilling his seed. Successful practice of this for a certain fixed period is said to confer rewards on the practitioner. This practice continued within the śaiva-mantra-mārga both in the saiddhāntika (e.g. in the Mataṅga-pārameśvara tantra) and bhairava (e.g. Brahma-yāmala) streams. Thus, the practice was likened to walking on the sword-edge. Unlike this practice, which still emphasized the non-spilling of seed, among the practices within the bhairava-srotas of the mantramārga the full-fledged sexual ritual with actual ejaculation developed with many variations in doctrine and praxis. The founder of one of central traditions within this stream, namely the kaula tradition, was the siddha Matsyendranātha. A successor of his was another siddha Gorakṣa, who in certain late manifestations of the tradition is portrayed as superseding Matsyendra himself. This manifestation seemed to have been accompanied by a reversal to more abstinent practices and explicitly castigated the sexual activities of Matsyendra.

This is portrayed in famous story we narrated earlier, which is widespread in the eastern reflexes of the nātha tradition. Here, Matsyendra is described as going to a kingdom where only women existed, ruled by a female chief. It was termed the strirājya or Kadalirājya (the banana-kingdom). There Matsyendra engaged in sex with the queen and was about to die from total loss of vīrya, when his student Gorakṣa comes and saves him. This was the first time we learned of the strīrājya. A tale similar to this one of Matsyendra was also incorporated into the hagiography of the advaitācārya Śaṃkara presented in the Mādhavīya Śaṃkara-digvijaya. These accounts were consistent with our next encounter with the term strīrājya in the sūtra-s of Vātsyāyana. There, in his sexual ontology, he says that the women of strīrājya like violent actions in bed and also the use of kṛtrima-liṅga-s. Since then, we kept encountering the strīrājya in a number of Hindu sources and it struck us that this was a parallel to the amazons, who are frequently mentioned in Greek lore. We had to visit an art museum, where we saw a modern imitation of a Classical sculpture of an amazon, probably one of the famous amazons featured in Greek legend (Top). The person, whom were showing the museum, remarked that the amazon had an “Indian” touch to her – whether there was any truth to that or not – it prompted us to revisit the topic leading to the current discursion on the strīrājya.

While the amazons are frequently mentioned in the Greek epic and early literature, their counterpart, the strīrājya, finds only a rare mention in the Hindu epic, the Mahābharata. Yet a closer examination suggests that the inspiration for both probably stemmed from related steppe Iranic groups:
1) While there is no consensus it is most likely that the Greek word amazon does not have a Greek etymology. Rather, it is likely to have some kind of Iranic etymology such as ha-mazon, perhaps meaning a warrior band.
2) The Greek evidence from writers such as Herodotus associate them as mixing with the steppe Iranic groups like Scythians (śaka tigracūḍa) and spawning the Sarmatians (sairima). Consistent with this they are described as being experts of horse-borne archery.
3) The Roman leader Pompey records them as being in the army of Mithradata-VI the formidable Greco-Iranian king. The later Roman writer and general Ammianus mentions them as a neighboring tribe of the Iranic Alans (Aryans).
4) The archaeologist David Anthony notes that among the “Scythian-Sarmatian” warrior kurgans about 20% contain interred women in battle-suits like their male counterparts. Consistent with this, some Greek sources record the amazons being interred in large kurgans. This can also be placed in the context of Herodotus’ account of the death of Cyrus, where he marches against an Eastern Iranic steppe kingdom of the Massagetae which was led by a queen Tomyris.

This suggests that indeed these steppe Iranics with female participation in warfare might have inspired the yavana legends about the amazons. They may have been more familiar to the early Greek sources than the Indic ones because they launched a series invasions in the direction of the Greek sphere and are even credited to have built some temples in the Greek sphere, which were subsequently centers of Greek worship.

On the Indian side of the evidence we find a further mention from the great Gupta age naturalist Varāhamihira in his Bṛhatsaṃhitā:
diśi paścimottarasyāṃ māṇḍavya-tukhāra-tāla-hala-madrāḥ |
aśvaka-kulūta-halaḍāḥ strīrājya-nṛsiṃhavana-khasthāḥ || 14.22

He places the strīrājya in the northwest along with several other tribes including the aśvaka, madra-s and the Tocharians. This is consistent with strīrājya being associated with the steppe Iranics of the Northwest. In the second reference to strīrājya by Vātsyāyana it is situated along with Bāhlika (modern Balkh) again pointing to the northwest direction. This reference also mentions the strīrājya women sequestering youths in their antaḥpura-s comparable to the a Greek tale regarding how the amazons reproduce by sequestering males from other tribes. The Chinese bauddha traveler-scholar Xuanzang and the Tang-Shu record a country Lang-ka-lo with its capital as Su-t’u-li-ssu-fa-lo which has be rendered by some as strī-īśvara. It is explicitly stated as being under Iranian rule despite using Brāhmi script and having both bauddha-s and Hindus (hundreds of deva temples) on the way to the “Western woman country”. This would suggest that all these sources recognized the same Northwestern land, likely associated with one or more steppe Iranic groups.

This position is also in line with the mention by Kalhaṇa in the Rājataraṃgiṇi of the strīrājya. He describes strīrājya as being invaded in course of the expansive conquests of the greatest Kashmirian emperor Lalitāditya to the north of Kashmir. He mentions strīrājya as being conquered prior to the Lalitāditya conquering this Uttarakuru-s would again place it to the north and potentially in the steppes. Notably, he appears to attack it from the east crossing a desert which might have meant the southern reaches of the Takla Makan. In this regard we hear from the kavi who mixes the rasa-s of warlike and the erotic:

tad yoddhān vigalad dhairyān strīrājye strījano’karot |
tuñgau stanau puraskṛtya na tu kumbhau kavāṭinām || 4.173

Then the women-folk of strīrājya made the valor of [Lalitāditya’s] soldiers melt,
by placing to fore their high breasts and not the frontal lobes of their elephants.

strī-rājya-devyās tasyāgre vīkṣya kampādi-vikriyāṃ |
saṃtrāsam abhilāṣaṃ vā niścikāya na kaścana || 4.174

Seeing the emotions of trembling and the like exhibited by the queen of the strīrājya in front of him (Lalitāditya), no one could say for certain if it was due to to fear or eros.

ekam ūrdhvaṃ nayad ratnam adhaḥ karṣat tathāparam |
baddhvā vyadhān nirālambam strīrājye nṛhariṃ ca saḥ || 4.185

By placing one magnetic gem which pulled it upwards, and another one which pulled it downwards, he (Lalitāditya) installed an idol of Nṛsimha suspended in the air without support in the strīrājya.

Lalitāditya’s successor Jayāpīḍa is also mentioned as conquering the strīrājya:

citraṃ jitavatas tasya strīrājye maṇḍalaṃ mahat |
indriya-grāma-vijayaṃ bahv amanyanta bhūbhujaḥ || 4.587

After he conquered a large territory of the strīrājya it is a wonder that other kings considered his conquest of the field of his (Jayāpīḍa’s) senses [ever greater].

karṇa-śrīpaṭam ābadhya strīrājyān nirjitād-dhṛtam |
dharmādhikaraṇākhyaṃ ca karmasthānaṃ vinirmame || 4.588

He established the office of the court of justice and hoisted therein the auspicious silk of Karṇa, which he had seized from the conquered strīrājya.

Regarding his profligate successor Lalitāpīḍa we hear again from Kalahaṇa:

atṛptaḥ strībhir alpābhir ugrarāgaḥ sa parthivaḥ |
jaḍaṃ mene jayāpīḍaṃ strīrājyān nirgataṃ jitāt ||

The king (Lalitāpīḍa) with a raging passion and not satisfied with [just] a few women considered Jayāpīḍa impotent for having left the strīrājya after conquering it.

Thus, we see parallels of the Matsyendra story, where the king Jayāpīḍa is praised for having controlled his senses upon conquering strīrājya. Notably, the Karṇa-śrī-paṭa, while obscure in meaning, reminds one of the Greek legends of Herakles and Theseus taking away the girdle of the amazonian queen Hippolyta. A further account of the strīrājya in Hindu tradition is seen the Jaimini-aśvamedhaparvan, which presents itself as a fragment of the Mahābhārata of Jaimini. However, as it has come down to us it is much reworked text with a Vaiṣṇava focus. Here, the sacrificial horse reaches the strīrājya in course of its wanderings and is taken by Pramilā, the queen of strīrājya. Arjuna challenges her to battle and after a brief archery encounter a celestial voice advises Pramilā to give up and marry Arjuna. She releases the horse and accompanies Arjuna to Hastināpura, where she waits for him till his period of celibacy for the aśvamedha is complete. But in course of this account it informs us that there are no males in strīrājya. The females are apparently left male-less due a curse of Rudrāṇī. If the males go there they die in a month from the excessive and violent sex with the females of strīrājya in line with the Kāmasūtra-s comment in this regard (“māsamātraṃ striyaṃ prāpya paścāt prāpnoti vaiśasam | … tenaiva svena liṅgena praviśanti hutāśanam |”). This also reminds one of the Greek legend of the destruction of men by the sirens or Kirke during wanderings of Odysseus. There is indeed a reversal similar to that of Odysseus and Kirke in the Matsyendranatha tale where the women of strīrājya attack Gorakṣa when he leaves with the former, but by his yoga he turns them into birds.

Thus, we may infer both Greek and Hindu traditions had the memory of a land of females. In both traditions they were much embellished but it appears that the Greeks had much closer contact with the actual agents behind these legends. However, in both cases it seems they gave rise to a floating mass of legends, which were incorporated into various cycles in different ways. Given the relatively sparse occurrence of the strīrājya in Hindu tradition, one may ask if they really encountered them or if were merely stories borrowed from some other group like the yavana-s. In this regard we may note the following:
1) The pre-Mauryan bronze mirrors described Vassilkov of Indian origin suggest some kind of a contact between the Hindus and the steppe Iranics. Moreover the alternative account of the death of Cyrus given by Ctesias, where the Indians are said to form an alliance with a central Asia group the Dṛbika-s against the Achaemenids. These point to contacts between the Indians and the steppe long after the Indo-Aryans conquered and settled in India.
2) Stylistic similarities are seen between Northern Indian and horse-trappings (phalerae) and jewelry recovered from Sarmatian graves.
3) Recently Veeramah et al looked at ancient DNA extracted from individuals from Sarmatian and other graves from a wide swath of western Eurasia and studied their genetic affinities. Notably, one Sarmatian individual (labeled PR_10 in their study) from Russian Orenberg region (~400-200 BCE) and a Crimean individual (labeled Ker_1) with Hunnic-style deformed skull from around 200-400 CE show evidence for Indian admixture. A preliminary examination (needs more careful confirmation) does suggest that this reflects a relatively recent Indian admixture with SNPs private to greater India rather than some ancient Indo-Iranian relationship. This would imply that there was direct contact with individuals of Indian origin so as to result in gene-flow.

In conclusion, we hence believe that there was some real knowledge of the steppe Iranics with female warriors among the Hindus. They were a distant group with which the authors of the texts might not have had close familiarity. Nevertheless, the direct experience of those who had journeyed to those regions likely formed the historical core of the information presented by Hindu authors, which was then subject to poetic elaboration. It is known that among the steppe groups both Iranic and later Turko-Mongol there was some degree of participation of women in warfare (down to the Mongol times and even after their conversion in the west to Mohammedanism). This was probably the root of both the Greek amazons and the Hindu strīrājya.

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Sītā in the pyre

It was quiet, early Saturday evening. Vidrum, Jukuta, Sharvamanyu and Lootika were hanging out on the parkway of Somakhya’s house. They had assembled there for some curricular preparation on differentiation. Somakhya and Lootika had covered the chain rule and the differentiation of product functions. Buoyed up the lessons their classmate-students felt that they could sally forth for the impending tests with confidence. But Somakhya put the damper on them: “Differentiation is the easy part and one can rake up the marks on the exam like Gandulkar smashing sixes off the Rainbow Republic bowlers. The real challenges will come with the rising balls of integration.”

Lootika amplified the matter like an accompanying musician: “If you think you have mastered math, remember this tale in  Greek tradition: it was said that there was a mortal named Stentor who could yell with the voice of many men in unison. But once he attempted to contest with the god Hermes on hearing whose loud yell he died. Thus, when integration section opens you may come up against some Hermesian howlers that our classmate Hemaling spends all day integrating.”
Somakhya: “In any case we will cover that tomorrow along with some vectors.”

Vidrum flipping through the news and messages on his tablet chimed in: “I see some very disturbing news. A city man, Durjoy, has been arrested on the charges of trying to arrange aid to the socialist terrorists who have been trapped as part of the ongoing operation in Gondipura. Our classmate Manjukeshi who was supposedly taking preparatory lessons from him at that time has also been taken into custody!”
Sharvamanyu: “Good job by the security forces.”
Jukuta checking out the news herself: “That is really sad. Was this Durjoy the senior student from the St. Stephens institution? He would come to hangout outside our school and was quite good to me in those days. I do not know if was really involved in anti-national activities. But it is ridiculous on their part to arrest Manjukeshi. I believe she should be innocent.”
Lootika: “I think he was a lout. One day he accosted me outside school but I scampered away on my bike in fear, thinking he might be an acolyte of Shonit who had harassed me and my sisters in our former school. But Sharva got some interesting “stones”, from him, which he apparently found in Gondipura and gifted them to me. They turned out to be the fossil eggs and an ungual of the peculiar sauropod dinosaur Isisaurus ”
Sh: “Remember, he was always trying to seduce girls from our school. He left those fossils with me during an inter-school athletics event and never took them back. Then I gave them to you knowing you might find them interesting. Why do you think he was in Gondipura? I am sure it was not for dinosaur-hunting.”
Lootika: “Of course, now all that makes sense. I’d add Manjukeshi may not be innocent either.”
Vidrum: “Why do you think so? She is a nice girl. I know she was just taking classes from that Durjoy.”
Somakhya: “Vidrum, have you forgotten the day you first introduced her to us. She did strike us as a fighter for social justice. After all let’s not forget she used assemble for these lessons of ours whole of last semester. Until., well…I guess let it be.”
Jukuta: “I never really understood why she stopped coming. Somakhya don’t kill me for this, but she simply told me that Lootika was a horrible person. She added that despite being a girl, she was perpetuating gender stereotypes and horrible casteism. I tried to tell her that even though in school I had more than one tiff with Lootika, we are now grown up and in college and need to accept that she is truly brighter than all the girls in our class. She cut me off saying that I was either with her or with Lootika.”
Lootika: “But Jukuta you have not stopped on playing on both sides, right? Let me remind you we are under no obligation to have you here for these lessons.”
Vidrum: “Calm! Let us not get into one of those silly fights of you girls.”
Sharvamanyu: “I think it began with that article on the Rāmāyaṇa, which Manjukeshi authored for the college magazine, where she accused emperor Rāma of being a male chauvinist and a casteist. We castigated her for that and she got angry with Lootika for not supporting her silliness.”

Jukuta: “I did read that article. I don’t know much about the Ramāyaṇ, but if that incident of letting Sītā burn is in the book it does sound awful, like a dowry death by burning, you see. Lootika, I know you and your sisters are the only girls who know these things written by Brahmins in Sanskrit. Is it really there in the book?”
Lootika: “Jukuta, before anything else you must pronounce the Rāmāyaṇa correctly – repeat after me. When my parents first told me the tale, they left out this part. But then, when I read it myself, in order to narrate it to my sister Vrishchika, I encountered it right there in the original. I must confess I read that part with some tension, as to what was going to happen, and even surprise. It did not feel right to me then and I asked my mother about it. She gave me the explanation that Rāma was an avatāra of the god Viṣṇu and that the avātāra ended the moment he performed the superhuman task of killing the rakṣas-lord. Thus, beyond that point he behaved like an ordinary human and the act must be understood in that light. I then asked my father about it. He said that when Sītā was abducted by Rāvaṇa, the real Sītā was taken away by the gods and replaced by a māyā Sītā. As per his interpretation, at the time of the fire ordeal, the god Agni incinerated the māyā Sītā and returned the real Sītā to Rāma. So clearly, that incident was seen as bit unusual and rather than accept it without a question, people have come up with different explanations for it.”

Jukuta: “But Lootika at the bottom-line would that still not be misogynistic, as Manjukeshi had written?”
Lootika: “See, such words are neologisms, which make little sense in the context of old tradition. As a biologist I have learned not be fall prey to such emotionalistic constructs but coldly look at the hard realities on the ground. The sexes are always pitted in an inter-sexual conflict for maximizing their fitness. At same time, as we are eukaryotes, sex is institutionalized in us starting with the duplication into two paralogs of the ancient protein RecA and the HORMA protein acquired from bacteria more than 2.7 billion years ago. Like all other sexually reproducing organisms, we cannot make more copies of ourselves without the other sex. Hence, the games played by the players in this conflict should not be given any moralistic judgment. That said, if you read the original text, there is hardly anything of the kind the puerile Manjukeshi read in it. In fact, it can be read as not disallowing the option of her kind of unregulated sexuality: after all Rāma tells Sītā is that she can chose any other mate she wants. So she is offered unfettered agency in the sexual domain and it is she who chooses to remain in the union with Rāma by proving her chastity via undergoing the ordeal by fire. Moreover, you see her not caving in but giving a dignified and firm public defense of her character when it was attacked and this you can take from a brāhmaṇī who has actually read the text in the original language.”
Sharvamanyu: “Look at it from Rāma’s perspective too. There was a direct affront on his manliness as a warrior in the abduction of Sītā and he avenged it squarely. However, no man likes to be in doubt of his paternity or risk providing for his enemies’ offspring. Therefore, he had very natural reasons to be uneasy in accepting his wife who had been taken into his enemy’s custody for a while…”
Somakhya: “See, this discussion we are having here is one of the reasons for epic’s author to craft such a tale. Even though the Rāmāyaṇa, unlike our national epic the great Bhārata, is one of ideals, it has all these elements that bring out the grey shades of real life. If it were not for these, people would have had difficulty in having a discourse on dharma of the type as we are just having. When it comes to the knotted questions of dharma there are no black and white answers but only a difficult navigation though the shades in the grey zone. Incidents like this provide the framework for thinking about that path. But then the great bhārgava does not put things in the epic without a deeper metaphor, which only some grasp.”
Vidrum: “What is that deeper metaphor that you are alluding to? Does such exist in this particular incident?”
Somakhya: “It does. But I’d let this spidery girl expound it if she wishes as it concerns her ancestors.”

Lootika: “It was something I actually learned from Somakhya and that was when I came to a final understanding of the incident. Everything in the itihāsa of Valmīki is a reflection of deep elements of śruti. He has done so in order that people like you Vidrum and Jukuta can access the mysteries of religion that might be otherwise inaccessible. First, remember that the princess Sītā born of the furrow in earth is transparently the humanized form of the ancient goddess of agriculture, who was praised by my ancestor Vāmadeva Gautama in the mantra:

arvācī subhage bhava sīte vandāmahe tvā |
yathā naḥ subhagāsasi yathā naḥ suphalāsasi || RV 4.57.6

Be auspicious O Sītā, come come close to us. We worship you,
such that you confer us a good share [of things] confer the good fruits [of agriculture].

Second, my ancient clansmen, the Gotama-s, were the purohita-s of the Videgha-s, the clan of the Janaka-s to which Sītā of the Rāmāyaṇa belonged. Evidently, they transmitted the cult of Sītā to these kings. The brāhmaṇa on the Sāmidheni-s, the chants by which the fire is aroused in the śrauta rite, preserves an anachronistic ancient legend of one of the founders of my clan, Gotama Rāhūgaṇa, and his patron the founder of the Videha-s, Videgha Māthava. In that tale, it is said, alluding to the Eastward migration of the Indo-Aryans, that the land the Videgha-s eventually occupied was initially rich in water but difficult to cultivate and habit. But when Gotama uttered an incantation, the god Agni is said to have burst forth from Videgha Māthava’s mouth and burnt the land of the Sadānīra river making it habitable. It is this ancient legend that receives an epic reflection in the form of Sītā’s purification by the fire, representing that new land of the Videha-s and Kosala-s, along with the Gotama-s, to the east of the Kuru-Pāñchala, becoming fit for habitation and the furrow of agriculture. Rāma who is the earthly manifestation of the great Indra then receives that Sītā, even as my ancestor Vāmadeva says in his incantation:
indraḥ sītāṃ ni gṛhṇātu |
May Indra set his hold on Sītā (set down the furrow).”
Somakhya: “More generally, it might be seen as representing the agricultural practice of burning leftover plant material after winter (the grip of Vṛtra whose reflection in the epic is Rāvaņa) to get the fields ready for agriculture in spring under the fertilizing effect of Indra.”
Vidrum: “Interesting, but I guess such arcana might be beyond the lay user of the epics who might be satisfied with the social debate you brought up earlier.”
Lootika: “Of course this is mostly for the those who uphold the deeper language of tradition but we are just giving you a flavor of how overloaded the language of myth can be.”

Jukuta: “But what would you say about the killing of Jambukumār, the śūdra saint, by Śri Rām? Is that not very casteist?”
Sharvamanyu: “Who the hell is this Jambukumār? Never come across such in the Rāmāyaṇa.”
Lootika: “I believe she is referring to Śambuka.”
Vidrum: “That is a tough one. What would you guys say.?
Somakhya: “Again, not reading the original but going by the words of these social justice types can give you a wrong impression. Of course, it is a long story and we could sit here arguing both sides. for a while, like all these thorny points in the epic. Would you really like a story without such shades of grey, the tensions they create and the emotions they arouse? On the other hand there are at least three messages here. First, this Śambuka was a śūdra alright but he was not saint. He was aiming to ascend to and conquer the world of the gods, like the demon Rauhiṇa; hence, it was imperative that their representative on earth Rāmacandra Aikṣvākava nip such attempts in the bud. Remember that in the early days with the Ārya-s settled in the subcontinent of the Jambudvipa what was meant by śūdra was some kind of enemy of the Ārya-s. Overtime they were defeated and absorbed into Ārya society. This leads to the second message – it is actually one reflecting this social accommodation and change. The text clearly states: bhaviṣyac chūdra-yonyāṃ hi tapaścaryā kalau yuge || It presents a doctrine that successively over the four yuga-s each varṇa acquired the capacity to do tapas. Now this Śambuka was doing it out of turn in the previous yuga and hence he was punished for breaking the rules. However, in a positive message it indicates that in our yuga the śūdra has this option and the incident merely refers to the tensions of a bygone era. Thus, in the typical futuristic format of the purāṇa-s, the text is presenting this change in social reality vis-a-vis the śūdra against the background of the previous antagonism. Third, it delivers the message of conservatism for social change. Rapid social change by accommodation of those from without the fold presents dangers. Such elements are represented by Śambuka, who wish to emulate Ārya practices, not for good purposes, but for conquering the daiva realm. Hence, such revolutionary tendencies have to be suppressed in favor more gradual accommodation over the yuga.”

Jukuta: “Your uncompromisingly conservative outlook amazes me.”
Lootika: “You may appreciate it more as you age, though part of it is in the genes and we cannot do much about it. But whether you like our message or not remember that a love for facile, feel-good messages might turn you into a Durjoy, a Shonit or a Samikaran, who in the end might do more harm than good to society.”
Sharvamanyu: “And now Jukuta I hope you don’t bring up the issues of animal rights in the incident of the killing of Vālin.”
Jukuta: “Who is Vālin?”
Lootika: “The next time we assemble at my house I’ll gift you a bāla-Rāmāyaṇa. I really think you need to be reading it more than any of this curricular stuff.”

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

A sequence related to prime counting

The current note arose as an exploration branching off from the matter discussed in these earlier notes: this one and this one. As we saw before, Carl Gauss, while still in his teens, produced his first estimate of the prime number distribution in the form of the function:

\pi(n) \sim \dfrac{n}{\log(n)}

Here is \pi(n) is the prime counting function, which counts the number of prime numbers up to a given number n, and \log(n) is the natural logarithm of n. The \sim notation indicates that the prime counting function is asymptotic with \tfrac{n}{\log(n)}, i.e. as n\to \infty the ratio \pi(n)\big / \tfrac{n}{\log(n)} \to 1.

Subsequently, Gauss refined his fit for the prime counting function by using the famed logarithmic integral \textrm{Li}(x). We were curious if there was some arithmetic function, which was actually fitted by \tfrac{n}{\log(n)} rather it being merely a single term approximation of the \pi(n). In course of some arithmetic experiments, we stumbled upon a sequence, which we believe, without formal proof, is fitted by \tfrac{n}{\log(n)} in terms of average behavior.

This sequence f is defined thus: f[1]=1. Thereafter, add n-1 to all terms f[1:(n-1)]. Count how many of f[1:(n-1)]+(n-1) are primes. This count is f[n]. For example when n=2 we add 2-1=1 to 1 we get 2. Which is a single prime; hence, f[2]=1. Now for n=3 we add 3-1=2 to the first two terms and we get 3, 3. Thus, we have 2 primes; hence f[3]=2. For n=4, we add 4-1=3 to the prior terms and get 4, 4, 5, which yields a single prime, 5; hence, f[4]=1. Thus, the first few terms of the sequence goes: 1, 1, 2, 1, 3, 1, 4, 1, 1, 2, 7, 2, 7, 1, 1, 4, 11, 3, 9, 2, 4, 4, 11, 0, 2, 4, 4, 11, 11, 6. Figure 1 shows a plot of the first 20000 terms of the sequence.

prime_back_addition_fig1Figure 1

The blue line is the plot of this sequence and we notice right away that despite the fluctuations the average tendency is to grow with n. Via numerical experiments we were able to establish that this average growth is fitted best by the function \tfrac{n}{\log(n)} (red line in Figure 1). The green line in Figure 1 is the count of primes \pi(n). We observe that though some extreme values of f exceed \pi(n), the average behavior of f[n], i.e. \tfrac{n}{\log(n)} < \pi(n). This relates to a central development in the number theory: when Gauss conjectured the asymptotic relationship between \tfrac{n}{\log(n)} and \pi(n) the mathematical apparatus was not yet in place to prove it. This was finally developed by his last student Bernhard Riemann. Using those ideas, nearly century after Gauss’ conjecture, Hadamard and de la Vallée-Poussin proved it and it became known as the Prime Number Theorem. Further, de la Vallée-Poussin showed that \pi(n) was related to \tfrac{n}{\log(n)} thus:

\pi(n)=\dfrac{n}{\log(n)}+O\left(\dfrac{n}{\log^2(n)}\right)

Here, the second term is gives the error and is denoted using the big-O notation which was explained in an earlier note. This indicates that indeed \tfrac{n}{\log(n)} would be less than \pi(n). Thus, as can be seen in Figure 1 the average growth of f[n]<\pi(n).

We then used \tfrac{n}{\log(n)} to ‘rectify’ f[n] i.e. obtain:

f[n]-\dfrac{n}{\log(n)}

Prime_back_addition_fig2Figure 2

This rectified f[n] is plotted in Figure 2 and provides a clear picture of fluctuations in f[n] once we have removed the average growth trend. We observe right away that the amplitude of the fluctuations grows with n. To determine this growth trend of the rectified f[n], we first noticed from Figure 1 that \pi(n) tends to run close to the maxima of f[n]. Hence, we utilized the asymptotic expansion of \textrm{Li}(n), which is a better approximation of \pi(n) and captures the behavior beyond the basic \tfrac{n}{\log(n)} term:

\textrm{Li}(n) \sim \dfrac{n}{\log(n)} \displaystyle \sum_{k=0}^\infty \dfrac{k!}{(\log(n))^k}

\textrm{Li}(n) \sim \dfrac{n}{\log(n)}+\dfrac{n}{\log^2(n)}+\dfrac{2n}{\log^3(n)}+\dfrac{6n}{\log^4(n)}...

Using the first 4 terms to approximate the growth of the amplitude of rectified f[n] we get the red bounding curves shown in Figure 2. Thus, we conjecture that while f[n] grows on an average as \tfrac{n}{\log(n)}, the amplitude of its fluctuations is roughly approximated by \textrm{Li}(n)-\tfrac{n}{\log(n)} (Green bounding curves in Figure 2).

Posted in Scientific ramblings | Tagged , , ,

Convergence to a palindrome

This is a brief account of a sequence we constructed inspired by Dattatreya Ramachandra Kaprekar. It is not known to us if he had discovered it in one of his obscure publications from a small town in the Maharatta country. In any case we explored this sequence independently upon hearing of some procedures he used in his work. Consider a number like n=100. Its reverse is r(n)=1. Then, n+r(n)=101. We find that 101 is a palindrome. Consider another case n=155, then n+r(n)=155+551=706. This is not a palindrome so we continue the same process n+r(n)=706+607=1313 \rightarrow n+r(n)=1313+3131=4444. Thus, after 3 iterations of the process we have a palindrome. Thus, if we take any number and perform this operation of adding it to its digital reverse iteratively till we get a palindrome then our sequence f is defined as the palindrome to which each number n converges. Thus, f[100]=101; f[155]=4444. One question which arose was whether there are n that never converge to a palindrome. Between 1:99 all numbers converge to a palindrome even if a large one. Hence, we explored 100:999 in greater detail. Figure 1 shows a plot of f[100:999]

Palin_Fig1Figure 1. y-axis: \log_{10}(f[n])

Of these our experiments suggested that the following 13 numbers in the range 100:999 never converge to a palindrome (marked by red dots in Figure 1): 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986. One can see that barring 790 all of them come in pairs. The maximum value attained by f[n] in this range is the 13 digit number: 8813200023188. This value is attained when n is 187, 286, 385, 484, 583, 682, 781, 869, 880, 968. Comparable to the non-converging cases, here all ten cases come as pairs. There is also an interesting pattern for both the maxima and the non-converging cases: we observe that in each century it comes one number earlier but at some point a “new line of descent” emerges which then perpetuates the same pattern along with the older one. The same maximum value is attained for n in the range 1:99, with f[89], f[98]=16668488486661.

Omitting the single digit numbers, 11 is the primordial palindromic number and its multiples often tend to have a palindromic structure. Its multiples for k=1:9 are the only double digit palindromes. Multiples of 11 are also found among the triple digit palindromes: 121, 242, 363, 484, 616, 737, 858, 979. For n=1:99 all n>4 converge to a palindrome which is a multiple of 11. For n=100:999, f[n] have 64 unique prime factors ranging from 2 to 18209090957. Notably, the largest number of f[n], 641, are divisible by 11. Thus, the most frequent convergence even in this range is to a multiple of 11 (Figure 2).

Palin_Fig2Figure 2. The number of f[100:999], which are divisible by a given prime divisor from the set of all unique prime factors of f[100:999].

Figure 2 shows that other than than the first few primes (2, 3, 5, 7 etc) there are some anomalous standout values. Notable among these are 37 which with 3 reaches a palindrome 3 \times 37= 111 and other palindromic primes, most notably 101 and 131. The convergent might itself be a palindromic prime and for f[100:999] we have 101, 727, 929, 181, 383, 787. As can be seen in Figure 1 (Violet points) the n for which these palindromic prime convergents are reached have a distinctive pattern of distribution reminiscent of the maxima and non-converging values.

Finally, this sequence is notable for the very large values that are attained amidst otherwise pedestrian values (Figure 3).

Palin_fig3Figure 3. x-axis in \textrm{arcsinh}(f[n]) scale. Mean of f[n] is the red line while the median is the blue line

This makes for an interesting distribution where extreme events on the right end are rare but enormous in magnitude. This is reflected in the difference of several orders of magnitude between the mean and the median of f[n]. However, at least the actual occurrence of these extreme values is quite regular (Figure 1).

Posted in Scientific ramblings | Tagged , , , ,

A problem from 600 CE and some curiosities of Āryabhaṭa’s kuṭṭaka algorithm

Around 600 CE in the examinations of one of the Hindu schools of mathematics and astronomy one might have encountered a problem such as below (given by Bhāskara-I in his commentary on Āryabhaṭa’s Āryabhaṭīya):

dvayādyaiḥ ṣaṭ-paryantair ekāgraḥ yo ‘vaśiṣyate rāśiḥ |
saptabhir eva sa śuddho vada śīghraṃ ko bhaved gaṇaka ||

Quickly say, O mathematician, which number when divided by the numbers starting with 2 and ending in 6 (i.e 2:6) leaves 1 as the remainder, and is exactly divisible by 7?

This problem was given to illustrate the use of the kuṭṭaka algorithm first provided by Āryabhaṭa. Before we actually solve the above problem we will briefly examine the kuṭṭaka. The kuṭṭaka is a general algorithm deployed to obtain integer solutions for the indeterminate linear equations of the form ax-by=c, where a, b, c are positive integers. Thus, it is essentially the problem of finding the coordinates of the the integer lattice point through which the line ax-by=c passes. Of the three constants in the equation, a and b are given; c is not given but we have to find the smallest c for which the equation can be solved in integers. From that we can build other valid c For computational simplicity (with no loss of generality) we take a to be the bigger number and b to be the smaller number. The below presentation of it follows the matrix representation of Āryabhaṭa’s operation given by the mathematician Avinash Sathye:

Let us consider as an example the following equation 95x-25y=c. From a=95 and b=25 we can generate the below matrix which is the result of the kuṭṭaka procedure. We have written a function ‘kuṭṭaka’ in the R language that computes this matrix and few other details given a,b.

K=\begin{bmatrix} 19 & 4 & 95 & NA \\ 5 & 1 & 25 & 3 \\ 4 & 1 & 20 & 1 \\ 1 & 0 & 5 & 4 \\ 0 & 1 & 0 & NA \\ \end{bmatrix}

The process is initiated with column K[,3]. Write K[1,3]=a and K[2,3]=b. Kuṭṭaka in Sanskrit means ‘to powder’, common translated as ‘pulverizer’. We start ‘pulverizing’ a=95 with b=25, which means finding the K[3,3]=K[1,3] \mod K[2,3]. Then K[4,3]=K[2,3] \mod K[3,3]. We iterate this procedure until we get K[n,3]=0; that completes the column K[,3]. We call n as the number of iterations for convergence. Thus, the matrix will have n rows and 4 columns. The quotient of the division of K[1,3] \div K[2,3]=3 is written as K[2,4], that of K[2,3] \div K[3,3]=1 as K[3,4], so on till convergence. Thus, the cells K[4,1] and K[4,n] will always be empty (NA in the R language).

Then we fill in K[n,2]=1 and K[n-1,2]=0. There after we compute the remaining elements of this column working upwards from K[n-2,2] with the formula:

K[j,2]=K[j+1,2]\cdot K[j+1,4]+K[j+2,2]

We then fill in K[n,1]=0 and K[n-1,1]=1 and complete the column starting K[n-2,1] upwards with the formula:

K[j,1]=K[j+1,1]\cdot K[j+1,4]+K[j+2,1]

With that we have our kuṭṭaka matrix K and all the needfull stuff to solve the said indeterminate equation:
1) The greatest common divisor \textrm{GCD}(a,b)=K[n-1,3]. This is also the smallest positive c for which our indeterminate equation has integer solutions.
2) The least common multiple \textrm{LCM}(a,b)=K[1,1]\cdot K[2,1]\cdot K[n-1,3]
3) The integer lattice points through which the line passes are obtained from (K[2,2],K[1,2]) by assigning the appropriate signs. Thus, for the above equation we have the solution 95\times(-1)-25\times (-4) = 5. Thus, 95x-25y=5 will pass through the integer lattice at the point (-1,-4)
4) If we enforce the need for positive solutions then we can use (K[2,1]-K[2,2], K[1,1]-K[1,2]) to obtain the minimal integer solution: 95 \times 4 -25 \times 15 =5. Thus, 95x-25y=5 will pass through the integer lattice at the point (4,15) in the first quadrant.
5) We can write the following relationship, which helps us to more generally get the lattice points through which the line ax-by=c passes even if the values of a and b are interchanged or for c other than the minimal c:

b(K[1,1]\cdot p+K[1,2] \cdot q)-a(K[2,1]\cdot p +K[2,2]\cdot q)=K[n-1,3]\cdot q

Thus, if we set p=-1, q=1, we get (-4,-15) as further lattice points through which the line 95x-25y=5 passes.

Now we can tackle the original problem: Since it says that 7 divides the number r perfectly it can be written as r=7y where y will the y-coordinate of the lattice point. The numbers 2, 3, 4, 5, 6 leave a remainder of 1. Of them 4 is divisible by 2, and 6 by both 2 and 3. So all we need to consider are the numbers 4, 5, 6. Using the above kuṭṭaka or any other means we can show that \textrm{LCM}(4,5)=20 and \textrm{LCM}(20,6)=60. Thus, we can write 7y=60x+1, where x will the x coordinate of the integer lattice. Using kuṭṭaka on the equation 7y-60x=1 we get the matrix:

K=\begin{bmatrix} 60 & 17 & 60 & NA\\ 7 & 2 & 7 & 8\\ 4 & 1 & 4 & 1\\ 3 & 1 & 3 & 1\\ 1 & 0 & 1 & 3\\ 0 & 1 & 0 & NA\\ \end{bmatrix}

From this we can compose (60-17) \times 7 - (7-2) \times 60 = 1. Thus, our number is r= 43 \times 7 = 301, which is divisible by 7 but leaves a remainder of 1 for all integers from 2:6. More generally, if we say that r \mod 2:6 \equiv 1 then we can use K to compose the negative solutions r=-17 \times 7=-119 or r=-77 \times 7 =-539. Such triplets of solutions correspond to symmetric lattice points along the line.

In the final part of this note we shall consider the following operation: Take a number n and perform the kuṭṭaka operation with it (i.e. a=n) and all integers lesser than or equal to it (i.e. b=1:n). Then we count the number of iterations it takes with each of these integers to reach convergence. From above it is clear that the minimum number of iterations for convergence will always be 3. We term the result the kuṭṭaka spectrum of a number and plot this spectrum for the numbers 120, 123, 127 and 128.

kuTTaka_spectrumFigure 1

The kuṭṭaka spectrum displays several notable features:
1) It is pseudo-symmetric about the mid-point, i.e. either side of a/2 is an approximate mirror image of the other side but they differ in “height” by one iteration.

2) The number of times the kuṭṭaka spectrum hits a minimum (i.e. converges in 3 iterations) is equal to the number of divisors of a, D(n). Thus, for a highly composite number, as defined by Ramanujan, we get the record number of minima in the kuṭṭaka spectrum for any number less than it. Thus, in our example the highly composite number a=120 has 16 minima with the first six integers 1:6 giving a run of 6 successive minima. A minimally composite number like a=123=3 \times 41 in our figure we get 4 minima, namely 1 and the number itself and its two prime factors. The prime number in our figure, a=127, as expected has only 2 minima.

3) The more composite a number the lower its mean value of iterations (red line in Figure 1) than other integers in its immediate neighborhood. Thus, the highly composite number 120 has the lowest mean value in our set. In contrast, the primes have higher mean values than the integers in their immediate neighborhood. This is can be seen with a=127 in Figure 1.

4) A curious feature of the spectrum are the maxima, i.e. the value of b for which the maximum number of iterations are required for pulverizing a to convergence. For example the spectrum of a=128 shows 2 maxima: b=79 pulverizes it via the pathway: 128, 79, 49, 30, 19, 11, 8, 3, 2, 1, 0. The other one b=81 via 128, 81, 47, 34, 13, 8, 5, 3, 2, 1, 0. One immediately notices that the convergence in each of these two cases enters the Golden ratio convergent sequence. This feature can be investigated further by examining the distribution of the values of a/b for those b which result in maxima in the kuṭṭaka spectrum for a given a. In order to have have clear discrimination of these fractional values of a/b corresponding to the spectral maxima we chose a set of relatively large a, namely all integers from 500 to 1000. We then determined the kuṭṭaka spectrum for each of those numbers and extracted the maxima for each a and plotted a distribution of the a/b values (Figure 2).

kuTTaka_maximaFigure 2.

First, the maxima always occur in second half of the spectrum (Figure 2), i.e. b>\tfrac{a}{2}. This makes sense because smaller b would reach close to a in the first division itself and could pulverize it to a relative small number. However, b>\tfrac{a}{2} would fit only once in a and could leave a relatively large remainder that could need more steps for pulverizing. Second, strikingly, the dominant peak in this distribution is the Golden ratio \phi (Figure 2), suggesting the maxima tend to occur where \tfrac{a}{b} \approx \phi. Indeed in our above example \tfrac{128}{79}=1.620253. This can be intuitively understood as the b which generates a maximum may be seen as a Golden cut of a: if b>\tfrac{a}{2} is too big then the remainder generated will be small and might be pulverized quickly. If b>\tfrac{a}{2} is too small then it will leave a big remainder relative to b which might be quickly pulverized in the next step. Thus, the \phi could give you the cut that is just right. The next dominant peak is at 3-\phi \approx 1.381966. This is similar to \phi in its operation. These two are marked by a red dashed line in Figure 2.

There are further peaks in the distribution corresponding to other fractions on either side of \phi following a certain pattern of declining heights. Further they show the same symmetry as \phi and 3-\phi, with each peak m having a counterpart 3-m. We have thus far not been able to determine a more
general expression describing all these peaks or prove why they should be peaks but we were able to account for a subset of them as corresponding to other quadratic irrational numbers (marked by grey dashed lines in Figure 2). These include:

\sqrt{35/11} \approx 1.783765, 3-\sqrt{35/11} \approx 1.216235
\sqrt{3} \approx 1.732051, 3-\sqrt{3} \approx 1.267949
\sqrt{24/9} \approx 1.632993, 3-\sqrt{24/9} \approx 1.367007
\sqrt{2} \approx 1.414214, 3-\sqrt{2} \approx 1.585786

Of these the pair \sqrt{2}, 3-\sqrt{2}, especially the former is not the best fit to the peak but given the breadth of that peak it is possible that more than one attractor fraction is merged in that peak. It would be a good mathematical quest to discover the general expression for the peaks, their dominance and prove why they tend to be peaks. There might be a subtle fractal structure to them that might become apparent at large values of a.

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A brief note on some new developments regarding the genomics of Indians

When we wrote a previous article on this matter we had stated that new data will alter the details of our understanding of picture discussed therein. Indeed, two new manuscripts which were deposited in the past month by McColl et al and Narasimhan et al have done so. These are still deposited manuscripts and have not been formally published. Further other data might also come in the near future. Hence, we are not launching into any detailed presentation of the revised scenarios in this note. What we intend here is to simply provide a few illustrations of the authors’ results without much critical investigation.

EastAsians

A screen shot of McColl et al Figure 4

First, the study of McColl et al focuses on the far east bringing in new ancient DNA data. The main point of interest to the Indian scenario is that the Andaman Onge are part of a major push of hunter-gatherers into the far east and Pacific, which spawned several branches that in turn mixed among themselves in various combinations giving rise among others to the Austronesian groups and East Asians. Further, in deep Pacific there were admixtures of the basal branches of this radiation with the Denisovans, the signal of which is very clearly seen in Papuans and Australian aborigines. The basal-most branch of this group analyzed by McColl et al is the 40000 YBP Tianyuan man, suggesting that these populations were in the east by then. A basal branch of this radiation group also seems to have contributed to the ancestry of only a subset of native Americans (independently of the East Asian branch that also originated from this group). This suggests that they might have reached the New World independently in an earlier wave or mixed with one strand of the main East Asian line of Native American ancestry as they entered the New World. A deep sister group of the Onge and probably a basal member of this Eastern radiation was an ancient hunter-gatherer group that settled India, where they might have undergone admixtures with one or more preexisting non-sapiens species of Homo. This population is now defined as Ancient Ancestral South Indian (AASI) by Narasimhan et al, refining the earlier definition of “Ancestral South Indian” by Reich et al. We may term them the Indian hunter-gatherers.

The key point which Narasimhan et al make is that Neolithicization of the North-Western Indian Subcontinent proceeded via the entry of Iranian farmers from the west. Thus, this clarifies a previously uncertain situation based on archaeology alone. The entry of these Iranian farmers could have happened as early as the Mehrgarh Neolithic or in more than one wave of closely related western populations. In any case the authors posit that it had happened by 6700-5000 YBP. This Iranian farmer group mixed with the AASI in the NW of the Indian subcontinent and this admixture was likely the form of the population of the Harappan civilization that arose in this region. They term this population Indus periphery. Narasimhan et al also show that the Bactria-Margiana complex (BMAC) received some admixture from this population, likely of Harappan provenance, but did not contribute notably to the ancestry of the Indian subcontinent. Starting around 4100 YPB they start seeing Sintashta Steppe contributions appear for the first time in BMAC. This ancestry appears to have filtered south and reached the core Indian subcontinent thereafter. By 3700-3500 YBP they start seeing East Asian admixture on the Central Asian steppes, which continues down to the Scythian Iron Age. However, this East Asian ancestry is not visible in Indian populations. Hence, it appears that we are left with a window of 4100-3500 YBP when the Aryan invasion of the subcontinent happened. This is at the upper end of the mainstream invasion scenarios. Further, it is not inconsistent with the possibility that the invasion triggered the collapse of the Harappan urbanization around 3900 YBP. But it is also possible that the Aryans entered and occupied a landscape where the Harappan urban civilization had already collapsed or was in its last throes. It also provides support for the young age of the Veda, especially if one chose to place the Ṛgveda in the Panjab. Further, it lends some support to the scenario that the Soma cult was acquired by the Indo-Iranians and integrated with the older fire-cult as they reached the BMAC sites. It is notable in this context that one of the main proponents of the Soma cult in the Vaidika system, the Kaśyapa clan, was the default gotra for a brāhmaṇa who did not know his. There are issues with each of these points and interesting complications but we desist from discussing any of these now.

Interestingly, there was another recent publication by Vishnupriya et al applying the Bayesian phylogenetic methods to Dravidian languages. The results suggested a possible expansion of Dravidian happening around 4500 YBP. Narasimhan et al seem to mildly favor a Harappan origin of Dravidian. However, both the linguistic date estimate and several other linguistic arguments are against the Harappan civilization being that of Dravidian speakers. Rather, we suspect the Dravidians arose in the South as part of the Southern Neolithicization – this might have had genetic and memetic contributions from the Indus periphery but the Dravidian languages themselves were likely of Southern provenance, probably in the upper Godavari valley. In the aftermath of the Indo-Aryan reconfiguration of the north, it is likely that the Dravidians had their own expansions both South and North adopting various Indo-Aryan technologies and ideologies. This led to the Dravidianization of many AASI hunter-gathers, who might have earlier spoken other languages.

Narasimhan et al model extant Indian populations as a three-way mixture of the Indus-periphery, the Indian Hunter-gather (AASI) and the Steppe population related to the Sintashta complex. Below are some figures based on their model to illustrate the situation.

Indians_St_I_OFigure 1. A box plot showing the three modeled components of Indian Ancestry for the 140 populations studied by the authors. The gray line indicates the position of the genuine brāhmaṇa population with the lowest steppe ancestry (i.e. leaving out some groups which are not conventional brāhmaṇa, e.g. viśvakarman). It is clear that the brāhmaṇa-s show above average steppe ancestry and below average Indian hunter-gatherer ancestry.

Indians_st_I_O_histFigure 2. The same data is represented as a histogram. It is clear that whereas the Indus-periphery and Indian Hunter-gatherer ancestry is unimodal, the steppe ancestry is not with groups showing low and high steppe ancestry. This explains the authors’ earlier model of ASI and ANI.

We then sorted the populations into five categories: 1) braḥmaṇa-s (here we retained the viśvakarman); 2) Warrior caste (traditional kṣatriya-s) and their equivalents; 3) Middle castes: vaiśya-s, cattle-breeders and agriculturalists; 4) service castes: traditional service jati-s often included as other backward, backward and scheduled castes; 5) tribes. For this we had drop generic groups like Gujarati, Punjabi, Muslim and the like. This left us with 124 populations. These are plotted as a ternary diagram.

Indian_TernaryFigure 3. Ternary diagram of the 3 strands of Indian ancestry. The 5 caste-tribal groups defined above are colored: 1-red, 2-orange, 3-aquamarine; 4-blue; 5-violet. One can see the effect of the two admixtures with the steppe ancestry’s effect being predominant in the varṇa populations.

A closer examination of this is seen the next three figures:

Indian_steppeFigure 4. A box plot of the inferred steppe ancestry in the above-defined five groups. The steppe ancestry is arrayed in accordance with the caste ladder and tribals have the least of it on an average.

Indian_Indus_peripheryFigure 5. A box plot of the inferred Indus-periphery ancestry in the above-defined five groups. It is interesting to note that unlike the steppe ancestry’s the Indus-periphery ancestry is greater in the warrior and middle caste groups than in braḥmaṇa-s, who have a lower median value of this component. However, this difference is only mildly significant in the current data (p=.033) and sampling bias cannot be ruled out.

Indians_HGFigure 6. A box plot of the inferred Indian hunter-gatherer ancestry in the above-defined five groups. Here for the four groups from the warrior castes to the tribes we see a reverse of the scenario seen for the steppe ancestry. However, the braḥmaṇa-s show a slightly higher median value of this component. While again we should be clear that this could be due to sampling bias, taken together with the above plot, it might reflect some sociological reality. The braḥmaṇa-s probably to start with did not mix much with the preexisting populations of the subcontinent but as they expanded, especially while moving south, they mixed with directly with populations with lower Indus-periphery and higher hunter-gatherer components.

Together, these plots suggest a picture, which was long suspected from the physical appearance of Indians. The Indo-Aryans established themselves in the subcontinent entering via the NW, where they mixed with the older Indus-periphery populations that were likely part of or survivors of the old Harappan civilization. The groups with a wide-range of older Indian hunter-gatherer-Iranian farmer mixture were incorporated across the upper caste ladder but especially in warrior and middle castes where we see considerable dispersion (e.g. southern agnikula-kṣatriya with low steppe ancestry). The movement of braḥmaṇa-s into the south possibly also involved admixture with these groups.

Finally, a brief political note. The pro-Hindu pakṣa had acquired an aberration mainly in the past 3 decades known as OIT or the out of India theory for the origin of the Indo-Europeans. This never had a leg to stand on but is now dead and cremated. Unfortunately, the pro-Hindu side and mainstream H nationalism has invested so much in making Indo-Aryan autochthonism a centerpiece of their thought that it mostly ceded the writing of data-based Hindu prehistory to parties who are never going to be favorable to them. Even more tragically they do not even seem to recognize how wrong they were – there is a finite probability that most of the OIT proponents are going to continue that way. Further, there is an unsubstantiated rumor making rounds that the Indian side might have prevented the use of Indian aDNA in the current analysis fearing the inevitable end to OIT. If this were true then it would add to the scandal and only provide more fuel for the usual enemies of the Hindus. This intellectual failure of mainstream Hindu nationalism in framing its foundations is quite worrisome as it might reflect a deeper systematic failure in thought.

References:
The Genomic Formation of South and Central Asia, Narasimhan et al. https://www.biorxiv.org/content/early/2018/03/31/292581

Ancient Genomics Reveals Four Prehistoric Migration Waves into Southeast Asia, McColl et al. https://www.biorxiv.org/content/early/2018/03/08/278374

A Bayesian phylogenetic study of the Dravidian language family http://rsos.royalsocietypublishing.org/content/5/3/171504

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