## The Rāmāyaṇa and a para-rāmāyaṇa in numbers-I: epic as religion

This note may be read as part of our studies on the Rāmāyaṇa and para-Rāmāyaṇa-s of which an earlier part is presented here.

A study of the epic in Indo-European tradition suggests that there were two registers of the old Indo-European religion. While today both of them survive together with any vigor only among the Hindus, until not too long ago these registers showed some survival even among their Iranian cousins. From these it is apparent the first register is the “high religion” which manifests as śrauta and smārta performance. Among the ārya-s this further evolved into other manifestations as seen in the tantra-s of the sectarian traditions. Nevertheless, the Vedic base remained the model for most of these later developments. On the other hand the lay manifestation of religion was by the medium of the epic or itihāsa-s in India. Their religious value elsewhere in the Indo-European world was apparent in Greece. Indeed, in the classical Greco-Roman confluence the last attempt of reviving the religion by emperor Julian, which was being swept away by the “Typhonic” evil of the preta-moha, involved a focus on the religious facet of the Homeric epics.

In both India and Greece there are two epics, which have numerous parallels in their motifs, and resonate even in their overall themes. However, in India each has a distinct character. The Rāmāyaṇa is what might be termed “the universal epic of ideals.” The Mahābhārata is on the other hand our national epic, the epic of the first ārya nation in India, the foundation on which the modern Hindu nation rests. The Iranians have a comparable national epic in the form of the Kśathāya-nāmag and its precursors but apparently lack the universal epic. Among the Greeks to an extent the Iliad probably played a national role but tended towards the universal in the later phase. It was the universal epic, the Rāmāyaṇa, which was the vehicle of the ārya-dharma beyond boundaries of Jambudvīpa. In its role as the foundation of the “lay religion” it was remarkably tenacious and withstood the assault of the other Abrahamistic evil in the form the marūnmāda in Indonesia. It also served as a means of preserving the ārya-dharma in both India and in the east against the assault of the Aryan counter-religions promulgated by the naked-one and the ground-toucher. Indeed, in India the powerful force of the itihāsa-s was realized by successors of both these heterodox promulgators, who either attacked the itihāsa-s or tried to have people not attend their exposition.

The remainder of this note we shall look at the Rāmāyaṇa via numbers, which was part of my self-discovery of its key religious facet. Most importantly, it reveals something about the deep layers of the ārya-dharma and its evolution over time. Before we get started, a few caveats should be stated upfront: The texts I am using are the so-called “critical editions” of the Rāmāyaṇa and Mahābhārata for the first para-Rāmāyaṇa, the Rāmopākhyana of Mārkaṇḍeya. These critical editions have their faults but are available in electronic form and are thus amenable to semi-automatic text analysis by regular expression searches. Almost all of these analysis were performed by means of such. The Heidelberg system has a very sophisticated text-parsing mechanism for several Sanskrit works but I did not use it except for one word search (inspired by an interlocutor on Twitter), which will be discussed as part of another note, as it was not quite compatible with my command line pipeline. So my system could have some deficiencies but manual checking of the results shows that it is largely correct and the magnitudes should be taken as genuinely representative. In general for this activity you need to have a good knowledge of the various names of the gods, characters and weapons used in the text. Although not a paṇḍita, being a brāhmaṇa, I believe that I have a level of command of this as a reasonable representative of my varṇa should, so the results might be taken as generally reliable. Finally, I am aware that in white indological circles some work in this direction has been done by the likes of Brockington. However, I did not consult his papers as I wanted to have my own unbiased experience of the data and conclusions from it. More generally, wherever there is tractable data I believe that an educated man should analyze it himself rather than wholly relying on hearsay of others.

First we shall look at the gross features of the Rāmāyaṇa (Figure 1):

Figure 1

-The text has seven kāṇḍa-s, which are composed of multiple sarga-s, which in turn are composed of śloka-s. The Ayodhyā, Yuddha and Uttara have much more than median number of sarga-s and śloka-s.
-However, it is notable that except for Yuddha the other kāṇḍa-s have a nearly constant median śloka count per sarga (~24-27). This was the likely count maintained by Pracetas and his son Vālmīki the original composers of the Rāmāyaṇa for a typical kāṇḍa, probably aiming to be around 25 śloka-s. The Yuddha in contrast is longer both in terms of number of sarga-s and also the number of śloka-s per sarga. Clearly, this is a distinct composition suggesting that a different style was adopted on purpose for the military narratives typical of Indo-European epics. Unlike the median, the mean śloka count per sarga is higher with anomalies for both the Yuddha and Sundara. We shall take a closer look at this in Figure 2

Figure 2.

-Here we see the actual frequency distribution of the sarga length across the Rāmāyaṇa and per kāṇḍa in śloka-s: Here the differences are more apparent.
-The first three kāṇḍa-s are “tighter” in distribution with modal sarga length close to the median length. The Kiṣkindhā shows some divergence in the form of a fat tail with several sarga-s in of great length (40-70 śloka-s).
-The Sundara is most unusual in having a bimodal distribution with short sarga-s peaking less than 20 in length and longer ones peaking around 35. This pattern suggests a deliberate compositional shift perhaps reflecting the peculiar nature of the Sundara as an avenue for display of poetic beauty.
-The Yuddha is clearly distinct with the general peak and median length being shifted to being between 30 and 40. There is also a sizable fraction of very long sarga-s above 40 going all the way to well over 80. This again emphasizes the distinctness of the battle narratives where the long recitations perhaps appealed to the war-like ancient ārya audience who might have been in similar battles in their own lives.
-Finally, Uttara shows a typical median distribution of sarga length with a major fraction of sarga-s distributed around this value. However, it is distinct in showing a bimodality with two peaks one with length between 10-20 śloka-s and another with length between 40-45 śloka-s. This suggests a certain composite character with the shorter sarga-s probably representing the several short narratives included in it and the long ones relating to battle-sequences comparable to the Yuddha.

Now coming to the core issue of religion we shall look at the frequency of occurrence of the gods in the Rāmāyaṇa (Figure 3)

Figure 3

-It is apparent that Indra is literally the leader of the gods. He occurs nearly twice as frequently as the next contender Prajāpati or Brahmā. He is the standard for all comparisons and the hero of the Rāmāyaṇa is frequently likened to him. Indeed, there is a the tacit understanding that Indra used his māyā to take the form of a man in order to slay Rāvaṇa. This is suggested by Mandodarī’s lament upon her husband’s death:

atha vā rāma-rūpeṇa vāsavaḥ svayam āgataḥ |
māyāṃ tava vināśāya vidhāyāpratitarkitām || R 6.99.10

Or indeed Indra himself appeared in the form of Rāma,
for ruining and slaying you using impenetrable illusion.

Thus, it is hinted that Indra, who right in the Ṛgveda is famous for his māyā, uses it to kill the rakṣas.

Now again, though the core kāṇḍa narrative itself mentions Rāma taking the weapons of Viṣṇu from Agastya, in the preamble it is mentioned that they were the weapons of Indra himself.

agastyavacanāc caiva jagrāhaindraṃ śarāsanam |
khaḍgaṃ ca paramaprītas tūṇī cākṣaya-sāyakau || R 1.1.34c

At Agastya’s words Rāma verily took up Indra’s bow,
sword and the excellent inexhaustible quiver.

Of course the grand finale of the Yuddhakāṇḍa has Rāma ride the chariot of Indra steered by Mātali himself and using Indra-s weapons:

sahasrākṣeṇa kākutstha ratho ‘yaṃ vijayāya te |
dattas tava mahāsattva śrīmāñ śatrunibarhaṇaḥ ||

O descendant of Kakutstha, the slayer of foes, one of great strength and opulence, the thousand-eyed Indra has given for your victory this chariot.

idam aindraṃ mahaccāpaṃ kavacaṃ cāgni-saṃnibham |
śarāś cādityasaṃkāśāḥ śaktiś ca vimalā śitāḥ ||R 6.90.9-6.90.10

[He has also given] this great bow of Indra and his armor which glow like fire,
as also these arrows blazing like the sun and this bright sharp spear.

Finally, to slay Rāvaṇa he is said to use the missile made by Brahmā. But even here it is a mighty missile made by Brahmā in the manner of Tvaṣṭṛ in the Veda for Indra to conquer the three worlds:

brahmaṇā nirmitaṃ pūrvam indrārtham amitaujasā |
dattaṃ surapateḥ pūrvaṃ triloka-jayakāṅkṣiṇaḥ || R 6.97.5c

[The missile] was formerly made by the god Brahmā of immeasurable might for the sake of Indra. It was given to the lord of the gods [Indra] when he formerly sought to conquer the three worlds.

The missile itself has characteristics that are clearly suggestive of the vajra of Indra:
ratha-nāgāśva-vṛndānāṃ bhedanaṃ kṣiprakāriṇam |R 6.97.8c”
The swift acting [missile] was the smasher of [entire] troops of chariots, elephants and horses.
dvārāṇāṃ parighāṇāṃ ca girīṇām api bhedanam |R 6.97.9”
It was capable of smashing its way through through bar-reinforced doors and also mountains”

Tellingly it is described as “vajrasāram” (imbued with the essence of the vajra), and “yama-rūpam” (of the form of Yama). The latter epithet directly recalls the the first person statement of Indra in the 10th maṇḍala of the Ṛgveda where he says that he wields a missile that is like Yama himself.

The deployment of this missile by Rāma on Rāvaṇa is again thus described thus:

sa vajra iva durdharṣo vajrabāhu-visarjitaḥ |
kṛtānta iva cāvāryo nyapatad rāvaṇorasi ||

The missile, difficult to defend against like the vajra hurled by the arm of Indra, unstoppable like the causer of death (Yama), hit Rāvaṇa on his chest.

Thus struck Rāvaṇa fell:

gatāsur bhīmavegas tu nairṛtendro mahādyutiḥ |
papāta syandanād bhūmau vṛtro vajrahato yathā || R 6.97.021

His life-breath having departed the lord of the nairṛta-s of fierce speed and great luster fell from his battle-car to the ground like Vṛtra struck by the vajra.

Thus, to the ancient ārya audience this recitation would have immediately evoked the imagery of the Ṛgveda, where Indra’s heroic deeds in battle are praised in the ritual.
In conclusion, this makes it is clear that the original Rāmāyaṇa was composed in a setting where the aindra flavor of the ārya-dharma was the still the main expression of the religion. It is indeed likely that that it was tacitly implied that Rāma was a manifestation of Indra in human form to kill Rāvaṇa.

Now what about the rest of the Vaidika pantheon. Was it like the late Vedic age or the saṃhitā-s themselves?

-We see considerable prominence for Sūrya, Vāyu, Viṣṇu, Yama, Rudra in addition the Prajāpati/Brahmmā. However, the Aśvin-s, the Marut-s, the distinct āditya-s are not prominent. Agni has a moderate presence although primarily in the sense of poetic similes. This suggests that period of composition while still marked by Aindra dominance was one which was probably positionally distinct and temporally much later than the saṃhitā period. Of the prominent deities the indistinct solar deity suggests the rise of the new Indic solar cult with links to the older Āditya system but certainly very distinct in its manifestation with parallels to those seen in the Iranian world.

The prominence of Vāyu is related to his association with Indra in battle against the dānava-s, a feature which was prominent in both the Veda and the para-Vedic tradition. The latter is partly reflected in the Rāmāyaṇa and also relates to the importance of his son, Hanumat in the epic. We should mention here that in counting Vāyu we have almost entirely avoided including the incidental occurrence of his name as a epithet of Hanumat. A similar situation accounts in part of the prominence of Viṣṇu; however, his story has more which will be further discussed below. If Indra is identified with Rāma, and the role of Vāyu is taken by Hanumat, then it is rather obvious that the place of Viṣṇu is taken by Lakṣmaṇa. The Rāmāyaṇa makes this obvious in the statement:

vikramiṣyati rakṣaḥsu bhartā te saha-lakṣmaṇaḥ|
yathā śatruṣu śatrughno viṣṇunā saha vāsavaḥ || 6.024.029c

Your husband [Rāma] with invade the rakṣas with his brother Lakṣmaṇa even as the foe-killing Indra against his foes along with Viṣṇu.
or:

sa dadarśa tato rāmaṃ tiṣṭhantam aparājitam |
lakṣmaṇena saha bhrātrā viṣṇunā vāsavaṃ yathā || R 6.87.9

He then saw the undefeated Rāma standing with his brother Lakṣmaṇa like Indra with Viṣṇu.

Like in the Veda the most frequently referred act of Viṣṇu are three world-conquering strides suggesting that this old motif was still of great importance in the age of the Rāmāyaṇa rather than later elements like his incarnations or battles with certain demons. His weapon, the cakra is frequently mentioned, unlike in the Veda, where other gods are described as wielding it but not Viṣṇu. This suggests that the Rāmāyaṇa marks a stage after the saṃhitā period where the cakra became established as the favored weapon of Viṣṇu. However, it does preserve the memory of Indra’s cakra mentioned in the śruti in R 1.26.5. Notably, Viṣṇu is mentioned as killing the demon Naraka in a conflict which was perhaps coupled with Indra’s battle with Śambara:
śambaro devarājena narako viṣṇunā yathā | R 6.57.7

Thus is appears possible that this exploit of Viṣṇu was transferred to his avatāra Kṛṣṇa in a later retelling of the legend. Indeed, the whole Kārṣṇī retelling has Viṣṇu only thinly veiled by the Yadu hero.

-Of the other gods, Garuḍa and Kubera despite having a presence in the Veda are not prominent there beyond specific rituals. Nevertheless, even there, there is an under-current that they had a role of some note in household rituals. Their importance clearly comes out in the Rāmāyaṇa. In particular it is clear that the whole epic has a frame that tries to highlight the might of Rāvaṇa as the expense of Kubera, implying that he was an important deity of the time. He is named as one of the great regal gods along with kings Varuṇa and Yama and his greatness is repeatedly mentioned. This importance of Kubera, as we have seen before has a strong para-Rāmāyaṇa tradition too as laid out in the Rāmopākhyāna. Notably, in that relatively short text he is 3rd most frequently mentioned deity (Figure 4) suggesting that his importance was visible throughout the whole early phase of the Rāmāyaṇa tradition.

Figure 4

His importance is also implied by his airplane the Puṣpaka playing a notable role in the epic. His son Nalakūbara is also seen as cursing Rāvaṇa resulting in the protection of Sitā’s chastity upon her abduction. Kubera is also described as providing a secret missile to Lakṣmaṇa in his dream that allowed him to counter the Yama weapon of Meghanāda in their final encounter.

lakṣmaṇo ‘py ādade bāṇam anyaṃ bhīma-parākramaḥ |
kubereṇa svayaṃ svapne yad dattam amitātmanā ||

Lakṣmaṇa of fierce valor also deployed another missile, which given [to him] by the incomparable Kubera himself in a dream.

When the two missiles collided a great explosion is said to have taken place with a fire breaking out as they neutralized each other – in a sense implying that Kubera is no less than the god of death in his might.

-Yama in the Ṛgveda is strictly associated with the context of the funerary and ancestor rituals. However, there is again the under-current in the other saṃhitā-s that he was an important deity in regular existence as the god of death. This role of his in the Rāmāyaṇa is rather prominent and both in terms of numbers and the way he is referred to as a great king suggests that he was an important god in the ārya-dharma of the time. The death-dealing rod of Yama and entering his abode are common similes.

-Prajāpati: This deity is hardly present in the core clan-specific works of Ṛgveda – he is mentioned only twice outside of maṇḍala-10. But in maṇḍala-10 he has already risen to being the supreme deity in certain sūkta-s. He is conceived as both the overlord deity as well as the protogonic “golden-egg”. Now this would suggest that he was a late-emerging deity, probably specifically in the Indic setting after the ārya-s had left their ancestral steppe regions. However, we do not think this is the case. Comparisons with protogonic deities in the Greek realm suggest that such a deity predated the Greco-Aryan split. Rather we posit that he was not a key protogonic deity of the normative Indo-European pantheonic system but was the focus of one of several Indo-European cults outside the standard polytheism. Some deities who were part of the standard polytheism were also foci of such extra-normative cults but others like Prajāpati were solely cultic to start with. In both India and Greece the proponents of such protogonic deities started acquiring great prestige and religious centrality. In India this is reflected in the late Ṛgveda of the maṇḍala-10 and the brāhmaṇa-s where we witness the meteoric rise of Prajāpati. In the process of his rise he began to eat into the dominance of Indra, the head deity of the standard IE model.

In the itihāsa-s his ectype Brahman is likewise prominent as the head of the pantheon, though he is already beginning to face competition from the radiations from the cultic foci around Skanda, Rudra and Viṣṇu. What we see in the Rāmāyaṇa is that he is without any close competitor the second most frequently mentioned deity (Figure 3). His prominence in this itihāsa seems to be similar to what we see in the brāhmaṇa-s: As a deity at the head of the pantheon Brahman shares the position with Indra, but his prominence is clearly eating into that of Indra. This suggests two possible scenarios: 1) He was already a prominent figure from the very beginning of the Rāmāyaṇa tradition and his “power-sharing” with Indra is reflective of the parallel scenario in the brāhmaṇa-s were he had already risen to the highest rank. Thus this would imply that both aindra and prājāpatya memes were already active as the epic was being composed. 2) The Rāmāyaṇa as proposed above was primarily an aindra epic and Brahman secondarily encroached on Indra’s share in an independent replay of what happened in the brāhmaṇa-s.

On historical grounds we favor the second scenario. A comparison of the nāstika productions of the ground-toucher and the naked-one’s cults clearly indicate that at their time the prājāpatya strand of the religion was primarily among brāhmaṇa “intellectuals”. This intellectual link continued to later times when we see mathematical and scientific authors like Āryabhaṭa and Brahmagupta invoke Brahman as their deity (contrast with older scientific tradition in the Caraka-saṃhitā where Indra is dominant). The rest of the people in large part seem to have still followed the aindra religion until pretty late in Indian history with some competition from the other cultic foci mentioned above. This is indicated by the fact that the two nāstika teachers accepted this aindra mainstream as their background and mention the prājāpatya tradition primarily in the context of their brāhmaṇa rivals. Notably, in the first of the many Rāmāyaṇa of the jaina-s, the Paumacariyaṃ, Vimalasūri explicit calls out the stupidity of the āstika versions on grounds of their denigration of the great god Indra. This historical background would imply that the prājāpatya-s first rose as a dominant force inside the Vedic intellectual circles. The mark of this rise was first seen in the brāhmaṇa texts. Then as the prājāpatya-s “conquered” the intellectual landscape they extended their influence to more “secular” intellectual activities such as the itihāsa-s and mathematics/science. This was when Brahman came to prominence in the Rāmāyaṇa tradition. However, by the time the purāṇa-s started taking shape in their extant form, the other cultic sectarian foci had radiated enough to catch up and supersede Brahman. Of the old cultic foci, Skanda after an initial rise faded away. In contrast, Viṣṇu and Rudra came up to Brahman and soon overtook him to the point that despite the three of them being acknowledge as a trinity Brahman sunk to the “junior” position of the trinity. In part the tale of him having no temples might reflect the inability of the intellectual-centered Prājāpatya system to capitalize on the rising āgama-dharma, despite an early attempt hinted by the Atharvaveda pariśiṣṭa-s.

So what do the numbers from the text tell us? First looking at the Rāmopākhyāna we find that Brahman/Prajāpati has gone ahead of Indra (Figure 4). It was created by an author(s) who were clearly Prājāpatya and did not see any need to emphasize or maintain the position of Indra beyond what was absolutely unavoidable. What this tells us is that the Rāmāyaṇa tradition passed through a distinct phase after its original composition where Prajāpati had become important in it and it was in this phase that the fork leading to the Rāmopākhyāna was created. More tellingly, this proposal is supported when we look at the by kāṇḍa counts of key deities (Figure 5: shown as percentage of verses featuring particular deva). Here we see that Brahman has a peculiar distribution that is distinct from that of Indra and Vāyu. While the latter two show clear kāṇḍa-specific differences, they are more uniformly distributed across the kāṇḍa-s. In contrast the occurrences of Brahman show a significantly higher occurrence in the Bāla and Uttara kāṇḍa-s while being greatly under-represented in the rest. We know that both these kāṇḍa-s were clearly subject to reshaping after the core epic was composed because they try to explain things which were not clear elsewhere in the epic (e.g. the origin of the heroes and villains of the text). This together with the above observation clinches the case for the second of the above proposals: after the original epic in an aindra form was composed the Prājāpatya-s refashioned it by primarily redacting the first and last kāṇḍa-s.

Figure 5

-Viṣṇu again: Two other major deities show a similar of kāṇḍa-wise pattern of distribution as Brahman: Viṣṇu and Rudra. Importantly, they are minor in their presence in the Rāmopākhyana (Figure 4). This suggests that the vaiṣṇava and śaiva redaction occurred later than the forking of the Rāmopākhyāna and acted in manner very similar to the prājāpatya action before them. That they were also directly in conflict with each other is suggested by the fight between Rudra and Viṣṇu which is inserted into the bāla-kāṇḍa. Another key point is that the vaiṣṇava material show no strong hints of the avatāra doctrine nor the early pāṅcarātrika tradition which is strong in the Mahābhārata. This suggests that the vaiṣṇava redaction comes from an early stream of the sect that underwent further evolution by time of the redaction of the Bhārata.

-Rudra: In the Rāmāyaṇa has his characteristic features of being dark-throated, three-eyed, with braided locks (Kapardin), having a bull as his banner/vehicle, holding a great bow, having Umā for this wife and displaying great ferocity. His destruction of the Tripura-s is frequently mentioned. Additionally, his slaying of Andhaka gets multiple references. These references frequently come in kāṇḍa-s 2-6 suggesting that they are indeed the ancient similes involving the deeds of Rudra. E.g.

sa papāta kharo bhūmau dahyamānaḥ śarāgninā |
rudreṇaiva vinirdagdhaḥ śvetāraṇye yathāndhakaḥ || R 3.29.27

He, Khara, fell to the ground being burnt by the fire of the missile even as Andhaka [fell] burnt down by Rudra in the White Forest.

Most of these features have direct or indirect reference in the Veda, often going back to the oldest layers. However, we do not hear of his exploits made prominent in the purāṇa-s like the killing of Jalandhara or Śaṅkhacūda. Thus Rudra in the Rāmāyaṇa has not changed in any notable way from his Vedic form.

-Finally one may note that in this Kāṇḍa-wise distribution Kubera is mostly uniform across kāṇḍa except for the uttara – paralleling Vāyu to an extent. This we believe suggests his ancient and intrinsic importance to the text with the Uttara merely serving as a receptacle for lore relating to him and Vāyu.

In conclusion, we can say with some confidence that the great Rāmāyaṇa of sage Vālmīki was originally an epic encapsulating the popular register of the Indo-European religion as manifest among the Indians – the ārya-dharma. Its heroes were set in the mold of the great deities Indra (Rāma and Vālin), Vāyu (Hanumat), Viṣṇu (Lakṣmaṇa), Kubera (perhaps some of the Kuberian element transferred to Vibhīṣaṇa), the opaque popular Āditya (Sugrīva), with simile-linkages to Rudra and the Maruts (encompassed in Hanumat). Despite the later sectarian redactions starting from the prājāpatya-s casting it in different light, it retained this ancient religious spirit of the ārya-dharma. It was this that erupted forth like the great ape Hanumat to animate the Hindus in their life and death struggle against the unadulterated evil of Mohammedanism when they seemed all but lost. That is why a memorial to the epic should be built at Ayodhyā after destruction of all marūnmatta elements in the holy city.

## The great faceless man

In the late Yajurvaidika upaniṣat, the Śvetāśvatara, which is the foundational text of the śaiva-śāsana, the god Rudra is described thus:

na tasya pratimā asti yasya nāma mahad yaśaḥ।
There is no one who his equal, whose name [itself] is great fame.

This sentence has also been taken to organically imply something else among Hindus too: Statues are not made of the great people – their name itself is great fame. Keeping with this we mostly do not have statues of many of the great figures of Hindu tradition. For instance, we do not know how Bodhāyana or Āpastamba or Āśvalāyana or Paippalāda looked, though we take their names on a daily basis. So also with great men even closer to our times, like say Vācaspati Miśrā. Now, speaking of other heroes, like Rāma Aikṣvākava, whose name might be almost taken daily in some of our households, we have some kind of a description in the opening of the Rāmāyaṇa:

ikṣvāku-vaṃśa-prabhavo rāmo nāma janaiḥ śrutaḥ |
niyatātmā mahāvīryo dyutimān dhṛtimān vaśī ||

Born in the Ikṣvāku clan, he is known among men by the name of Rāma. He is self-controlled, of great manliness, radiant, resolute, and has his senses under control.

buddhimān nītimān vāgmī śrīmāñ śatru-nibarhaṇaḥ |
vipulāṃso mahābāhuḥ kambugrīvo mahāhanuḥ ||

He is intelligent, politically astute, eloquent, opulent, and an extirpator of foes. He is broad-shouldered, of mighty arms, with a conch-like neck, and strong-jawed.

ājānubāhuḥ suśirāḥ sulalāṭaḥ suvikramaḥ ||

His chest is broad, he is a great archer, his collar-bones are well-concealed, and is a suppressor of foes. With arms reaching up to his knees, with a good head, shapely forehead and good gait.

samaḥ sama-vibhaktāṅgaḥ snigdha-varṇaḥ pratāpavān |
pīnavakṣā viśālākṣo lakṣmīvāñ śubha-lakṣaṇaḥ || 1.1.8-11

His body is well-proportioned, he is of smooth complexion and mighty. His chest is rounded, his eyes large, he is prosperous and with auspicious marks.

Similar accounts might be found elsewhere in the Rāmāyaṇa too. One thing which comes out of this account is that it is fairly generic for a mighty kṣatriya except for one specific, unusual feature namely “ājānubāhuḥ” – i.e. that his arms reached down to his knees, which might have been a peculiar characteristic of the man himself. Thus, while one can build a generic image of emperor Rāma as a mighty kṣatriya, we can still say we do not know how he *exactly* looked. Now, this is in part keeping with a the broader issue we have discussed earlier, namely the iconic depictions of deities among the early Hindus. As we argued before such existed but were not prominent and were perhaps “primitive” keeping with the archaeological evidence from several early Indian sites. In this sense the Indian iconography mirrored the primitivism of the early Greek iconography.

This is in stark contrast to Egypt where their great Pharaohs are known more from their portraits rather than epic narratives. When we see the images of the lordly Pharaohs, while stylized, there is clearly an element of individuality behind them. Over the ages of its heathen existence, in addition to statues, Egypt developed an even more realistic portraiture in other media. It is conceivable that this Egyptian tradition of portraiture spread through West Asia and then Europe influencing other cultures, first Semitic and then Indo-European. Thus, we see it emerge first among the Hittites to some extent and then eventually among the yavana-s (here collectively Greek and Macedonian) and romāka-s. Thus, by the time of the Macedonian invasion of India we we see a vigorous tradition of realistic royal portraiture on their coins, medallions and mosaic work that has moved far away from their ancestral primitivism. Early Indian coinage was abundant in symbolism and even primitive iconography of Hindu deities but not royal portraits. However, in the years following the Macedonian attack we see an emergent tradition of such portraiture both on coins and in the form of larger icons as seen in the case of Aśoka the mighty Maurya. This trend would suggest that the impulse for portraiture eventually reached the Indian world only via the Macedonians. Likewise, there have also been plausible suggestions that the movement of yavana-s eastwards along with contacts with the now fully portrait-using Hindus sparked the emergence realistic portraiture among the cīna-s along side the unification by the Chin. This manifest burst upon the scene on a truly cīna scale in the funerary statuary of the Chin conquerors.

However, it does remain a fact that the explosive spread of *extant* portraiture mostly post-dates the Macedonian invasion, which could imply a causal link between the two. Of this most traces have been erased outside the peninsular tip by the evil hand of the Army of Islam. In Nepal however some early specimens survive such as the image of king Jayavarma-deva from śaka 105= 185 CE. In a 3 century time window from that point we also see traces portraiture among the Śuṅga-s, Andhra-s, and Iranic rulers (Kuṣāṇa, Pahālava, Śaka) in India. In discovering and describing the image of king Jayavarman of Nepal, Tara-ananda Mishra has given an excellent account of the evolution of portraiture including several points we have independently arrived at. Other than royal portraiture there appears to have been a vigorous tradition of the imagery of persons endowing religious images and constructions. Moreover, śaiva teachers of both the mantra-mārga and atimārga, extraordinary śaiva and vaiṣṇava devotees, and siddha-s were also prominent objects of whole-body portraits. Statues of such were once common throughout India but have been mostly damaged or demolished in the north. Thus, in south India we can still see statues of emperor Kṛśṇadeva or Govinda Dīkṣita but we do not have the original image of the great Bhoja-deva or Lalitāditya.

Yet, despite all of this, on the whole the the majority of our great figures have not survived in portraiture, perhaps indicating a real Hindu tendency for the statement: na tasya pratimā asti yasya nāma mahad yaśaḥ। . We too seem to be resonant with this Hindu tendency and take it in a broad sense as was possibly understood by many of our greats of the past. In large part we believe that a man’s visual image or for that matter several details of his life should not matter at all. All that should matter are the words he leaves behind – do you find something in them or not – that’s all. Indeed, this is how it is for Āryabhaṭa-I or Bhāskara or Nīlakaṇṭha Somayājin. They are their words not their portraits or even biographies. Now one may ask: “Have you not said there is great value in studying the biographies of past intellectuals?” I still hold that view but have always held that not all of their biography really matters. Indeed most Hindu notables have left behind colophonic biographies that only stress the needful in the best case scenario. For instance, they are not shy of revealing their youthful genius, like bhaṭṭa Jayanta recording his advanced grammatical research as a kid or mathematician-astronomer Gaṇeśa recording his discoveries like the hyperbolic approximation of trigonometric functions in his teens or Vaṭeśvara his discovery of mathematical recipes for astronomy in his twenties.

This contrasts what we observed among the mleccha-s today and its back-flow to the Hindus. The person, his appearance and certain incidental aspects of his biography (e.g. ones sex, sexual escapades or sexual orientation) seem to matter a lot. Indeed, the appearance of a person has a serious correlation to his/her intellectual influence especially in a lokābhimukha sense. There is a drive to get people who look acceptable in a mleccha sense as the face of even an intellectual matter, like physics. I was amazed by how certain bhārata-s known to me converged to a similar style of appearance after hitting the talking circuit in the mleccha world. More than a person’s intellectual substance what matters is that they lie at one extreme of the bell-curve in terms of the mleccha-defined (partly universal) measure of appearance, appeal, etc. Almost as though out of guilt, the mleccha-s would then balance it with a few chosen tokens from the other extreme of the normal distribution (e.g. the diseased, the handicapped and the rare ethnicities) and pepper it with elements of what matters in the mleccha facade: e.g. aberrant sexuality. Finally, in the biographic sense, often what the person holds on matters well-beyond his/her expertise is of great importance in his/her projection as an icon: s/he better mouth liberal platitudes along the lines of how they are out to save the world, save the disadvantaged, and other feel-good messages quite removed from reality and sometimes even biology all while looking ‘cool’ at the same time. This would have even been half-understandable where it not for the concomitant insistence of the mleccha system being “equal opportunity”.

In face of this facade we think great father Manu had a point – the brāhmaṇa intellectuals are best off being low-key in such dimensions – the faceless intellectual whose biographic peculiarities matter little. In conclusion, perhaps keeping with the Hindu tepidity towards portraiture, we subscribe to the view that it is better for a man’s words than his portraits to linger on after Vaivasvata takes him away – indeed what use is it to be like the mysterious brāhmaṇa from what is today Afghanistan, the fragment of whose portrait is placed for auction in a mleccha market.

## loka-nīti-carcā

loka-nIticharchA

vijaya-nāma mahā-mlecchānām bahuprajāvān bahupatnīvāṃś ca vyāpārī gṛha-krayāc chailūṣa-pradarśanāc ca mahādhany abhavat । sa marūnmattair abhibhṛtāṃ pūrvatana-mleccha-rāja-patnīm atikrāntvā rāja-nirvācanam ajayata । so ‘bhavat mlecchādhipatiḥ । virodhakās tasya+anekāḥ । tasya vijayasya ca paṭṭābhiṣekasyānantaraṃ vṛṣṭy-ante puttikā ivo(u)tplavante bahavaḥ saṃkṣobhakāḥ । saṃkṣobhakeṣu gaṇeṣu santi śailūṣa-puṃścalyaḥ klībāḥ strīvādinī-striya āpaṇa-bhañjakā niyamollaṅgha-vadino ‘pasavya-mārgādhyāpakā vārttā-vikrayikāś ca (preto rākṣasa-putrasya+anuṣaṅginaḥ sadṛśā āsan) । te sarve sva-tantreṇa alpa-vivekinaḥ । tatas te kaiś codayanti ca vidhīyante ce(i)ti praśnaḥ । kecid vadanti so’sti soro-nāma-mahādhanī+iti । sa bejho-nāma+antar-jāla-mahāvyāpārī+ity anye vadanti । etāny uttarāṇi prayeṇa samucitāni kiṃ tu gūḍhaṃ viṣayaṃ na spṛśanti ॥

iyam asya viṣayasya mīmāṃsā । vijayopanāma-vyāpārī keṣāṃ pratinidhiḥ ? sva-dhana-vardhanañ ca sva-sāmarthya-bhogaś ca tasya pradhāne lakṣye । paraṃ tu pradhānā anuyāyinas tasya santi śveta-jāty-ādhipatyavādinaś ca vyatiropitāḥ śveta-karmakarāś ca viśiṣṭāḥ pretasādhakāś ca । kasmāt kāraṇāt te sarve vijayaṃ vyāpāriṇaṃ pratisevante? mlecchānāṃ madhye vartate ekaika-rākṣasa-nāma-saṃpradāyaḥ । prathamā anuyāyinas tasya+āṅgla-deśasya holadeśasya mulaikarākṣasavādinaś ca sāmarthya-sādhakāḥ । mahāmlecchavarṣasya vistārānantaraṃ yūropāt tasmin deśe jagmuḥ । te jana-bhāvebhyo rāṣṭra-sīmābhyo deśānusarebhyaś ca na cintayanti vyavasāyino ‘rthasādhakāḥ । teṣām ekarākṣasvatvaṃ matavihīna-veṣaṃ vā (āṅglānusāreṇa “secularism” iti) apasavya-mārga-veṣaṃ vā gṛhītvā janānāṃ madhye praviśati । etasya matasya bhāṇakānāṃ karmabhyaḥ sādhāraṇā janāḥ kupyanti । te dhana-kṣayaṃ kuṭumba-kṣatiṃ samāja-bhaṅgaṃ mūlya-nāśam anubhavanti । tasmāt pradhānāt ekaikarākṣasatva-viśiṣṭaikarākṣasatvayo raṇam ajāyata (viśiṣṭaikarākṣasatveṣu purāṇā bhedās santi) । tatas te prati-yoddhāraṃ mṛgyanti । vijayo vyāpāriṇo bhavena janasya kopaṃ sulabham abhijñātvā naya-yojanāṃ prāyuṅkta । anayañ janā vijayaṃ jayāya ॥

sanātana-deva-sādhakebhyo kiṃ mahattvam eṣa-viṣayasya ? ekaika-rākṣasatvam anyāny ekarākṣasāni+iva deva-dharmasya ripuḥ । “Secularism” iti mārgeṇa dharmasya nāśam anviṣati । sa sarvonmāda-samyukta-matavihīna-veṣa-bhṛt-bhūtaḥ । eṣa hi mārgo dharma-virodhāya mleccha-marūnmattābhi-sandheḥ ॥

paśya paśya sarvonmāda-samāyoga-puruṣaḥ । tasya prathamonmādaiva śiraḥ । bilmy anabrahmaṇā ca amahāmadena ca guptaḥ । tatra vartate abrahmā-nāmonmatta-janakasya duṣṭaḥ siddhāntaḥ । pretonmāda+ādadhāti tasya niguḍhaṃ dharma-niṣūdyamānaṃ kalevaram । rākṣasonmādas tasya bāhū dāruṇav asi-dhāriṇau । mahāmadasya vākyāni bhavanti tasya hastaghnāv aṅgulitre ca । rudhironmādo dṛḍhaḥ kavaco dātra-mudgarābhyām bhūṣitaḥ । adhobhāgas tasya janānāṃ mandatā । upahata-jana-vṛddhis tasya mahāmuṣkau । tasmin rakte vahati mūṣa-nāmonmattasya kutarka-pūrṇo durāgamaḥ । sa piparti navīkaroti ca sarvonmādāṃś ca teṣān tviṣaś ca । ayam eva deva-dharmasya ripur mahān । yady etañ jānāti tarhi sahasrāṇāṃ varṣāṇām ajñānaṃ layati ॥

## Deliberations on richness and beauty: discovery of some multi-parameter iterative maps

As we have explained in the earlier notes (1, 2, 3), the second major factor in our exploration of 2D strange attractors maps, IFS and other fractals was the aesthetic experience they produced. Around that time we came across a curious statement of Plato in the Timaeus: “Now, the one which we maintain to be the most beautiful of all the many figures of triangles (and we need not speak of the others) is that of which the double forms a third, the equilateral triangle.”(translation by Jowett via Birkhoff). Thus, for Plato the most beautiful of triangles is the so-called $\frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{2}$ (a.k.a. $30-60-90$) triangle. As we meditated upon this statement we realized that this triangle can be recursively used to make many aesthetic polygons beyond the equilateral triangle, like a rectangle, parallelogram, kite (all $2 \times$), square, rhombus (both $4 \times$), hexagon ( $6 \times$). In doing so we realized what Birkhoff had noticed in the 1930s. To us it also illustrated the second of the two basic principles behind aesthetic experience that had dawned on us namely, symmetry and recursion.

Starting with our study of the Henon and Lozi maps, we soon realized that there is more to this: All fractal objects produce a profound aesthetic experience in some people. I happen to be one of them, though I am cognizant of the fact that this is not universal — many people are neither wonder-struck nor aesthetically moved when I show them such objects. Nevertheless, the fascination for such objects is widespread across human cultures: In Hindu tradition, temple architecture increasingly converged towards fractional dimensions before its expression was terminated by the coming of the Meccan demons. We had briefly alluded to this earlier, pointing to a relationship between the floor plan of the central spire and the boundary curve produced by an IFS fractal using simple rotations. Other forms of fractal structures were also depicted in Hindu temple art, such as “vegetal motifs”. Similarly, Wolfram has documented examples of fractal objects in medieval Western art. Tendencies towards fractality might also be noted in Japanese Ukiyo-e, like in the famous “The Great Wave off Kanagawa”. All this suggested to us that complexity might have an important role in aesthetics.

An attempt to understand the role of complexity in aesthetic experience was made by in the 1930s by Birkhoff. For an aesthetic object he defined two values, “order” $(O)$ and “complexity” $(C)$, which led to the aesthetic measure, $M=\frac{O}{C}$. However, his $M$ measures for various polygons did not have a strong correlation with aesthetic experience they produced in us. This made us suspicious of the value of Birkhoff’s measure. The pioneer in the study of fractals, Mandelbrot, suggested that fractality of an object might be related to the aesthetic experience it produces. Exploration of fractal maps for aesthetics emerged from pioneering productions of physicists/mathematicians like Mira, Gumowski, Sprott, Pickover and Abraham among others. Sprott and Abraham carried out analyses of the aesthetic experience from fractal objects attempting to relate it to their fractal dimension. Based on their experiments they suggested that a fractal dimension in the middle of the range $1..2$ was probably a sweet spot for the best aesthetic experience in 2D maps. This generally corresponds to our own aesthetic evaluation of fractal objects.

Thus aesthetic experience was a major driver for us in the exploration of new maps for fractal objects. Thus, we studied the work of Sprott and Pickover among others, reproducing and exploring many of the strange attractors they had discovered using iterative maps. Our own experiments led us to the discovery of the map that produces the “butterfly attractor“, which we had described earlier. After obtaining it we realized that a certain Martin had also discovered a comparable class of maps which produced an attractor with a considerable richness of structure and aesthetic variety (also called hopalong or chip maps).

Martin’s maps are surprisingly simple but produces remarkable beautiful and complex structures. The simplest of these maps the “hopalong” is:
$x_{n+1} = y_n - sign(x_n) \sqrt{|b*x_n-c|}$
$y_{n+1} = a - x_n$

Where $a$, $b$, $c$ are three constants. An examples are shown in Figure 1 and the map is robust over a wide parameter range

Figure 1

We modified versions of the Martin map into a more complex set of maps. Map-1 works thus. The first mapping is similar to the Martin process:
$x_{n+1} = y_n - sign(x_n) \cos \left( \dfrac{\sqrt{|b x_n - c|} d}{2 \pi} \right) \arctan \left ( \left (log_l \left (|c x_n - b| \right ) \right)^2 \right)$
$y_{n+1} = a - x_n$
In place of $\cos()$ in the first x-mapping we could also have $\sin()$.
Then we subject $(x_{n+1},y_{n+1})$ to a further “affine” rotation before plotting it:
$x_{n+1}=x_{n+1} \cos(k) - y_{n+1} \sin(k)$
$y_{n+1}=x_{n+1} \sin(k) + y_{n+1} \cos(k)$

Here we thus have 6 constants: $a$, $b$, $c$, $d$, $l$, $k$

Map-2 works thus:
$x_{n+1}=y_n-sign(x_n) \cos \left ( \dfrac {log_{l1}(|b x_n-c|) d} {2 \pi} \right) \arctan \left(\left(log_{l2}\left(|c x_n-b|\right)\right)^2\right)$
$y_{n+1}=a-x_n$
In place of $\cos()$ in the first x-mapping we could also have $\sin()$.
Here again we subject the map to an “affine rotation” before plotting:
$x_{n+1}=x_{n+1} \cos(k) - y_{n+1} \sin(k)$
$y_{n+1}=x_{n+1} \sin(k) + y_{n+1} \cos(k)$

The constants here are similar to those in the above map but we have one extra one because we have two distinct logarithm terms. Finally in both these maps replacing $\arctan()$ by the hyperbolic function $\tanh()$ also produces interesting maps. Figures 2 and 3 respectively show examples of each of the above maps.

Figure 2 (First two with $\cos()$ and second two with $\sin()$

Figure 3 (all maps use $\tanh()$ in place of $\arctan()$ in Map-2; the third example uses $\sin()$.

These maps produce new forms of great beauty, diversity and richness of structure with considerable robustness — like what the old Hindus would term “nava-nava-camatkāra” or “muhur-muhur-āścaryāya kāraṇam”. Indeed the beauty in these maps relates to features beyond simple symmetry. They have a degree of rotational symmetry but it is not perfect. What is striking in however the fractional dimension and the tendency for doublings or multiplications (an element see in the evolution of the Hindu temple too). These features deeply touch the heart of beauty. Perhaps the order within chaos is the most beautiful of all.

The maps displayed here along with some additional ones can be see in this PDF file.

## “Like the vidyādhara’s sword”

In old Hindu tradition a man who attained siddhi in his mantra practice was believed to become a vidyādhara whose might was manifest in the form his beautiful female partner who flew beside him embodying the power of fertility and the sword he held in his hand embodying the essence of might itself. We had remarked to a clump of modern Hindus that unless his might and that of his nation is manifest like the vidyādhara’s sword which lays opponents low all their opulence will come to naught and not be set in history. They did not get anything of what we had said. We said: “Never mind. Nothing matters. Like the what happened to the archosaurs of the Mesozoic everything comes to an end so why care for anything at all?”

◊◊◊◊

A clump of Hindus were seated at a table on a somber afternoon in the big mleccha-land. The meal which was being consumed seemed so unremarkable that most would have not even realized that they had finished lunch. Yet most seemed to be quite contended. We do not know why that was the case for the rest but at least in our case it was simply because we do not think too much into the future and at that moment the gods had kept us free of pain. Some where positively exultant, talking of their successes with grants or businesses. Others were triumphantly talking of the monetary success story of the Hindus as a group in mleccha-land. They were duly comparing themselves with the prathamaikarāksasavādin-s. Then someone brought up the topic of politics in mleccha-land. Suddenly there was a bit of shift in the mood. Most expressed surprise and anguish over the crowning of the new mlecchādhipati. They went about the Russian conspiracy as though they were senior agents of the mleccha-spaśālaya. Some of them started the discussion of whether the outgoing ardhakṛṣṇa-mlecchendra was one of the or the greatest mleccheśvara-s. With three exceptions most in the clump seemed to settle on him being at least one of the greatest mlecchendra-s. Basking the pleasant warmth of that feeling their calm was restored a bit as though in the gentle twilight glow of the setting sun of the mlecchapa. Soothed, they began to talk again of the great monetary achievements of the Hindus of mleccha-land.

Some of them noticed our silence and asked what we had to say on the matter. After some attempts at deflection we simply presented our view of the reality. Most found it utterly unpalatable and were unable to come to terms with it. The only two in the room who had voted for the jayitṛ rather than the favored candidate turned to us and said: “You have a point. We have learned this the hard way after stumbling through the mleccha-maze for 35 years. How did you get there? We have always thought of you as not being a man of the world lost in impractical arcana.” One of them continued: “We have always felt bad for you given what people would think of you for your pursuit of the recondite. This vision you present is deeply depressing to some of us. How could live with it without going insane?” We simply smiled and said if it were to make them go insane then it was better they disregarded what we had said and move on. As for us we told them perhaps a bit too bluntly that their supposed commiseration was of utterly no use.

◊◊◊◊

To palaver about what we have said in many ways on these pages, even as our ancestors were said to speak of the same sat in many ways.
-Most of these Hindus do not get it that they will never become a śveta-tvaca-mleccha even though their chief desire seems to be to earn respect from such.

-Whatever great deeds the Hindu achieves on the academic front he is not going to be acknowledged for those and his conquests will be attributed to the mleccha. So if he is pursuing such conquests primarily to be recognized and awarded by the mleccha system he has little hope of getting there. This leads to tremendous frustration among those who do not get it and simultaneously an inability appreciate the genuine conquests of their own people because they are constantly using false yardsticks.

-In a subset it sparks the temptation to get into the “cartel” by cheating on matters of substance which usually results in even bigger damage.

-However good the Hindu is *on an average* he is going to be paid less and given fewer resources than a mleccha with lower or equal capacity to him. Unless he puts this in proper perspective, there will always be discontent and misunderstand regarding why he has not yet entered the “club”. This is because he has internalized the mleccha framework as the truth and is struck by why things are not working as enunciated within it. It is like tackling a problem in non-Euclidean geometry within a framework assuming the 5th postulate as true.

-Those who gloat over monetary success do not realize that king Vaiśravaṇa’s nidhi without his million-slaying antardhānāstra is of no consequence. The Hindu does not put his money into things that strengthen his memetic ramparts against the other. Instead, he pours it as āhuti-s of Triśiras Tvāṣṭra for the dānava-s and hopes that the deva-s would lift him up. The strengthening of the *Hindu* nation is very far from their minds. They content with their monetary success or of theirs as a group for the sake of boasting but do not translate that into efforts that will actually strengthen them as a nation. Instead they expend it either on mleccha causes or in plain hedonistic pursuits: do you really need a roomful of unused electronics etc etc…

-So is all this the whimpering of the losers? So let us take a look at the winners – may be we can learn something from them. There they are seated on their resplendent vāhana-s bearing niṣka-s and mudra-s conferred by great mleccha lords. How did they get there and what have the achieved? They got there by faithfully serving the mleccha or marrying a mleccha and the proud niṣka-s they sport are merely the biscuit Tim has tossed to his cur Tom. They might pass their whole life in great ease and in a cocoon of recognition from the mleccha. But at the end will they have set themselves in history? Not at all. The best they would be is a footnote in small print. What about their demography? As a part of the great “becoming” they will now be merged with mleccha and be swept away by the dysgenic memetic infections they eagerly inoculate themselves.

-Only he is an abhijit in the world of men who is backed by the possession of the mighty sword like that of that of the vidyādhara.

-ity alaṃ vistāreṇa

## A strange Soviet construction

in our college days we used to visit the lāl-pustak-bhaṇḍār in our city where Soviet books on science and mathematics were sold at a low price (alongside Marxian literature). They were a great resource that enormously contributed to our intellectual development. Among the books were some which contained nasty problems in mathematics and physics that were used for boasting rights in our circle. Some of the more curricularly oriented students in my class apparently used those books for entrance exams to certain Indian undergraduate institutes that we had no interest in joining. We were reminded of  the Soviet penchant for such problems recently by an acquaintance. She brought to my attention a Euclidean problem which was allegedly part of a set of difficult mathematical questions which were used by the Soviet authorities to prevent Jews and other “undesirables” from entering the Moscow State University (She learned of those from a paper containing a whole set of such problems and their solutions which was posted several years ago by Tanya Khovanova and Alexey Radul and was recently highlighted by Pickover on his well-known site). I am simply recording it here for I found it interesting and spent some time on it. I took a while to solve it but once achieved it looked so trivial that I felt like a fool. The yavana-s of yore and other ancients might have liked it.

Problem: “Construct with ruler and compass a square given one point from each side.”

1) Let A, B,C,D be the 4 points each from one side of the square.
2) Construct parallelogram ACDE using 3 of the points (A, C, D). Thus $\overleftrightarrow{DE}$ would be parallel to AC.
3) Drop a perpendicular from point D to the side of the parallelogram containing the other two points $\overline{AC}$.
4) Draw circle with center D and radius $\overline{DE}$ to cut above perpendicular at point F.
5) Draw $\overleftrightarrow{BF}$ and drop a perpendicular to $\overleftrightarrow{BF}$ from A to cut it at point J.
6) Drop a perpendicular to $\overleftrightarrow{AJ}$ from point D to cut it at point G.
7) Draw $\overleftrightarrow{GD}$ and drop a perpendicular to from point C to cut it at point H.
8) Draw $\overleftrightarrow{HC}$ and complete the desired square GHIJ via obtaining the intersection I between the above line and $\overleftrightarrow{BF}$.

A square is formed by intersection of two orthogonal, equidistant pairs of parallel lines. Thus, a segment formed by points lying on two opposite sides of a square if rotated by $90^o$ using one of the remaining two points as the pivot would define two points on the other pair of opposite sides. This is the principle enacted by the above construction to get the desired square.

We adduce below several other Soviet problems with the respective solutions without any detailed explanation.

*Given two intersecting lines on a plane find locus of points D such that the sum of the distances from D to each line is equal to a given number.

*Can you put six points on a plane, so that the distance between any two of them is an integer, and no three are colinear?

*Given two parallel segments (AB, CD) divide one of them (AB) into six equal parts using just a straight edge.

*Given a triangle ABC, construct, using a straight edge and compass, a point M
on AC and a point K on BC, such that
$\overline{AM}$ = $\overline{MK}$ = $\overline{BK}$

## Matters of religion: “he becomes Naravāhanadatta”

Somakhya’s mother (SM) and Lootika’s mother (LM) ran into each other during their visit to the shrine of Rudra beside the river on a Monday evening. They sat at the platform below the vast aśvattha tree beside the subsidiary Viṣṇu shrine to chat for some time.

LM: “How is it going with all the quiet at your place now that Somakhya has left?”
SM: “Well, I’ve returned to teaching the Mahābhārata and have been contemplating on the features of early Indo-Aryan. By the way I saw you brilliant daughter last evening at the clothes shop.”
LM: “Ah! You mean Vrishchika; have been worrying about her.”
SM: “Why? I heard that she has been invited to give a talk at some famous human genomics meeting regarding the paper she has just published. She seems to be following her stellar elder sister with this paper while just in the fourth year of med-school.”
LM: “That’s exactly the matter of worry. The conference is at Kṣayadrājanagara, a big bad city, where I am sure you heard that just a week ago there was a major attack at the train station by the marupiśāca-s in which several were killed. Vrishchika has set her mind on going and you know when that happens there’s no easy way to change it.”

SM: “You should let her go. After it will bring her yaśas. It would seem she is on course to out doing even Lootika. Moreover, you let Lootika go on multiple trips, which included that really scary day of the Uniform Civil Code riots. You also let Varoli go to the dreadful Visphotaka, which is an even worse city than Kshayadrajanagara.”
LM: “O dear, you over-estimate Vrishchika. Knowing my daughters well I can say that that Lootika is to the rest of them like Maghavan among the gods. Thus, even on that day, while I feared for her greatly, I knew deep within she’ll be back home: lūtikā bahuyutikā vā.”
SM: “But some day Vrishchika will be on her own. You can’t just keep her protected at home and unexposed to the big world. My brother lives in Kshayadrajanagara. I could give you his number. May be his daughter Saumanasa or son Mandara could pick Vrishchika up at the station and make sure she’s alright.”

LM: “That’s very kind of you. I will certainly take their number, it will be useful in more than one way. However, I understand that Somakhya’s friend, that kid Indrasena, who is her co-author and co-speaker in that conference, also lives there. I know she has gotten very pally with him and he would pick her up at the station. But at the same time I fear the two of them might be up to some mischief. Vrishchika let it slip that after their talk they were going to skip part of the meeting to go roaming in the city and make an excursion to Devaparvata.”
SM: “Ah, if Indrasena is there I’m sure he’ll ensure she’s OK. As for the mischief, I could give you his parents’ number. Call them and express your concerns; they would make sure that the two don’t do something too extravagant. But Vrishchika has to find her way in life: educate her but let her have her fun within limits. Let her meet my niece Saumanasa too: I am sure Saumanasa would benefit a lot from advice from Vrishchika.”

Just then they saw a mongoose scampering away into overgrowth adjacent to the northern wall of the temple. It paused for while giving them a good “darśana” and then vanished. LM : “That looks almost like an adbhuta to me. What might it mean?”
SM: “Well, since you saw it first possibly you are going to be favored by the mighty Kāmeśvara.”
LM : “Hope both of us are…”

◊◊◊◊◊

Vrishchika’s father had boarded her onto the train to Kshayadrajanagara, which soon got moving. Vrishchika felt tremendous happiness as the train started coursing on its way towards its destination – for the first time she was travelling alone, unlike her sister Lootika, and sensed that she had become an adult. She also felt nice to be in solitude – she remembered Lootika’s words on the importance of solitude: “When you have to spend all your time in close contact with mundane people you start thinking like them and soon lose your ability to see the parokṣa by which that which mystifies or even kills them can be apprehended. Hence, it is useful to have some time off in silent contemplation.” After a while she started preparing for her presentation. Here again she followed her sister’s method of carefully preparing her talk: laying out the time for each slide, planning what she was going say and when, and anticipate various questions and keep answers for them ready. At that point she wondered how their friend Somakhya had this uncanny knack of giving rather disparate talks back to back without preparing at all. She thought to herself: “Perhaps, that is tattvāveśa”. But then Lootika’s words came back to her that a good experimentalist must be methodical and well-planned in their approach to anything.

After a while, even as she was preparing to sleep, a woman passed by Vrishchika to her seat and as she did so dropped notes amounting to a few thousand rūpaka-s and her passport on the floor. Vrishchika called out to her and but she had her ears plugged with a headset and walked on oblivious of her lost property. Vrishchika gingerly gathered the stuff and walked up and gave it to her. She was very happy and gave Vrishchika a note as a reward. Vrishchika smiled saying that she did not need it as: “Dhaneśvara keeps me well!” The woman responded: “whoever that Dhaneśvara is may he continue to be good to you”, even as Vrishchika walked back to her seat.

◊◊◊◊◊

Indrasena and Vrishchika were rather disappointed that the former’s parents had poured water on their plan of roaming through certain parts of the city that night after the meeting was over. As they were dropping them off at the conference center they said: “Indra, if it were just you we would not even ask where you have been. But we have promised this charming young lady’s parents to keep her out of all trouble. While we have some confidence in your fighting skills, and even if you were to take your pistol with you, it is a wholly different matter of defending oneself and defending a young lady by your side. You know well that mahāmada’s prowlers and other assorted dasyu-s are constantly looking out for girls in places you want to go to.” Instead, they proposed that would pick them up at the conference center and bring them home for dinner.

As they were walking in Indrasena said somewhat sharply: “Gautamī, you should have been more careful with keeping secrets. If you want to have adventure you cannot announce everything to your parents.”
Vrishchika simpered and said: “I know. I just accidentally let it slip under pointed interrogation. But after all we might get to talk about the rahasya-s that I have been long wanting to know more about.”
Indrasena: “OK. Indeed, there are many exciting rahasya-s to talk about but remember that with such rahasya-s it is even more imperative to shut ones trap with the general public.”
Vrishchika: “I know, I know. I can be quite a Kunti when it comes to such matters but you know how it is when you are interrogated.”

Later that evening Indrasena’s parents picked them up and took them home for dinner. After dinner Indrasena took Vrishchika to his room. He offered Vrishchika a cushion as a seat and as she looked around his room for the first time taking in the sights she felt a strange sense of déjà vu. She saw a large painting of Naravāhana on the wall with his three wives Śriyā, Ṛddhi and Bhadrā hugging him and his son Nalakūbara seated beside him. The mantra “namo dhanadāya ca dhanasvāmine ca ||” was inscribed around the painting. Below it hung a Japanese painting with several figures of which she could only recognize one as being a yakṣa. Below that on a separate stool was a well-bound copy of the bṛhatkathā.

Vrishchika: “O ātreya, I see them all, the signs of Yakṣarāṭ everywhere. This is the indeed the sign of knower of rahasya-s. Who else today can even contemplate being on the path of Naravāhana-datta the son of Udayana. How was that most powerful path revealed to you? Most only sadly wish for it as losers do.”
Indrasena: “O Gautamī it is a long story but short of it goes thus. In my childhood as I was learning the Taittirīya-śruti I had dreams on many continuous nights of the great peak of Kailasa-parvata. Above it I saw hovering a great airplane, the Puṣpaka. By myself I began uttering the mantra-s starting with tirodhā-bhūḥ svāhā| Other mysterious mantra-s of Vaiśravaṇa started coming to my mind as the old Hindus would say as if from a past birth. But they were incomplete and I had wait till I could apprehend most of them due to the teachings of the vañga-siṃha.”

Vrishchika: “Who is this teacher – is it not remarkable that he obtained the Kauberī vidyā that lies concealed?”
Indrasena: “My father has a friend. He was known as the vaṅga-siṃho rājarāja-mata kaṇṭhīravaḥ: a vipra who was a lion among the vaṅga-s. He could be truly described as ‘dhanaṃ meru-tulyam’ and possessed of many kinds of vidyā-s. People looked at him and wondered ‘how could he be endowed with all of this wealth, health, beauty, intelligence, wife and children? He never seemed to age!’ After all it is almost a truism that: “indreṇa sarvāni śubhāny ekasmin puruṣe saṃyuktaṃ na sthāpitāni |’ Seeing him people got the hint that after all the advaitin-s were simply being losers when they asked ‘tataḥ kim?’ when confronted with dhanaṃ meru-tulyam| As though guided by Rājarāja, with youthful impropriety, I once jousted with him on a scientific topic very familiar to the two of us, as also your agrajā and Somakhya of quick discernment. I will tell you about that at a different point in time. While he did not openly admit it, despite being a widely acknowledged professor, he felt he had been defeated by a mere youth like me in that debate. It was then that it slipped out of his mouth: ‘Perhaps, this why the ācārya had said the totality of Dhanada-siddhi is not on me. It must be due to the mantra of the Bhṛgu-s I don’t have.’ I queried him on the Dhaneśvara-siddhi and he agreed to help me apprehend the Kaubera-śāsana whose mantra-s had briefly flashed before me saying if one mastered it ‘he becomes Naravāhanadatta.”

Vrishchika: “Most interesting, I see signs of a genuine Jambhala-vidvān in you. Indeed, why talk about the imaginary ‘eko brahmaṇa ānandaḥ’ when you cannot achieve ‘eko mānuṣa ānandaḥ’ in this world. It is more like wishful thinking coming from not applying oneself vigorously to the path of Dhaneśvara. But since there is no gain without pain I have long suspected that this path must have be arduous. What is the significance of that Nipponic painting below that of the Rājarāja-parivāra? It reminds me of the incident I have told you from childhood when I captured as a khārkhoḍa the bhūta of a prācya from those regions.”
Indrasena: “Ah that one is important to me as a reminder that, even though arduous, one who applies himself to the Vitteśa-siddhānta can be like a Naravāhanadatta even if he might be from the far-off land of the pītavarṇa-s”
Vrishchika: “How did you get it? Who are those figures depicted in it?”
Indrasena: “My father obtained it when visiting the prācya-s on matters relating to his work. He managed to with some difficulty befriend the phlegmatic prācya “deśika’’ of the temple on mount Shigi who eventually gave him this painting on mulberry paper. The figure you see to the left is a Nipponic mantra-vādin known as Myoren with immense Kaubera-siddhi. The figure to the right is Daigo the emperor of Japan. He was afflicted by an incurable disease. Myoren performed a Vitteśa-sādhanā and as result a guhyaka appeared – he is the central figure with the sword – the yakṣa then cured the emperor.”
Vrishchika: “Remarkable!”
Indrasena then walked up to the painting and turned it around: “Here on the back you can see the images of the great Vaiśravaṇa along with his wife Śriyā and son Nalakūbara. This a painting of their idols installed on Mt Shigi.”

Vrishchika: “O Atri, that painting of a sword with the siddham script also seems to be Nipponic. It too seems to be connected with śrī Naravāhana. It looks like the sword of a vidyādhara upon attaining siddhi. Could you please tell me more about it?”
Indrasena: “Gautamī that is the painting of the sword received by Sakanoue no Tamuramaro from Yakṣarāṭ before going to war with the Ainus. There is a long back story here. It is said that when Xuanzong, the Tang emperor of the cīna-s, was in deep fear of his enemies he was aided by Vaiśravaṇa and Nalakūbara due to their mantra-s deployed by Amoghavajra. A little later when the Silla Koreans backed by the cīna-s threatened the Nipponians the latter similarly invoked Vaiśravaṇa through an elaborate ritual that gave them total immunity from those mainland cousins of theirs. At that time a Nipponic mantravādin had a dream wherein a horse with a jeweled saddle led him to a holy site, Kuramadera, to the north of Kyoto, where he found a svāyambhuva idol of Kubera and installed it in a great temple there. Thereafter in the late 700s of the Common Era, the warrior Sakanoue no Tamuramaro worshiped Dhanada at that temple, attained siddhi much like our Naravāhanadatta, and obtained a sword from the god. He used this sword in his campaign against the Ainus and conquered their land. There he built a copy of the temple at Kuramadera depicting Yakṣarāṭ as being borne by guhyaka-s known as nara-s exactly as prescribed in our tradition.”
Vrishchika: “Thank you for the most interesting narrative. What is written in the siddham script on that painting?”
Indrasena: “It is the mantra Sakanoue used: oṃ vaiśravaṇa sahaparivāreṇa samāja jaḥ huṃ vaṃ hoḥ oṃ ve svāhā ||

Vrishchika: “Indra, I have so many things to talk to you about. But since time is short right now I would like to ask you regarding the Puṣpaka-vimāna yantra which is mentioned in the śruti of the Taittirīyaka-s as being engineered by the god Tvaṣṭṛ for Kubera. Could you please give me a darśana of the Puṣpaka-vimāna-yantra you worship and lead me to the Puṣpaka-vimāna-sādhanā?”
Indrasena: “Alinī. That’s a rahasya-prayoga. You would need to engage in much Kaubera-yogābhyāsa for any degree of success in that direction. Nevertheless let me show the Puṣpaka to you.” Indrasena led her to a closet in his room and opened the door to reveal the image of the vimāna in the center of which were the seated images of Rājarāja, Ṛddhi and Nalakūbara. His antardhānāstra was placed in front of him. In front of the vimāna was the image of yakṣa Maṅkanaka the gatekeeper of Kubera. There was also an image of the cow Sarvakāmadughā. There was a pot of the five-metal alloy with water and a glass pot with honey in it on the vimāna-pratimā. Indrasena: “Alinikā utter the following mantra-s after me gazing single-mindedly at the Puṣpaka-vimāna:
oṃ haṃ jambhalāya vaiśravaṇāya maṇibhadrāya pāṅcikāya pūrṇabhadrāya nalakūbaraya yakṣebhyo yakṣāṇāṃ patye namaḥ ।

Then he gave her some honey with a silver spoon and asked her to eat it with the mantra:
idaṃ jambhalasya madhu maghonaṃ madhunā prajāvatī payasvatī dhanavatī dhīmatī ayuṣmatī bhūyāsam |

Then with another spoon he gave Vrishchika the water with a small amount of cedar nut oil: “Apply this on your eyelids with the following mantra:
kuberasyodakam idam tenādṛṣṭaṃ dṛṣṭam bhavati huṃ nakulahastāya svāhā |

Vrishchika: “Ah this is the pratikṛti of the water sent by Viteśa to help Rāma and Lakṣmaṇa penetrate the māyā of Meghanāda!”

Indrasena: “Indeed. Then you shall close your eyes, visualize the great cave in the Himalayan heights where Kubera has placed the pot of golden kauberaka honey and meditate upon it doing japa of the mantra:
oṃ hrīṃ vaiśravaṇa! dhanaṃ puṣṭiṃ dehi me svāhā |
When you have become sthiramati emerge from your meditation uttering oṃ vaiśravaṇa arthatamom। 3 X huṃ phaṭ|

After Vrishchika did so she emerged from her dhyāna uttering: “apaśyaṃ tvā yakṣam ugraṃ tvāṣṭrīṃ vimānarūḍhaṃ puruścakrāṇi sahasra-vandhurāṇi । sa vaiśravaṇo rājarājā mahato mahīyān yasya pratigraheṇāpsyāmi dhanaṃ meru-tulyam । kāmam pratigacchāmi । dhanam pratigacchāmi । prajāḥ pratigacchami । īśvareṇa mahado3 o3 oṃ ||

Indrasena: “That’s good it appears you are on the path of īśvara-siddhi. I know you already have accomplishments as a mantravādinī but before proceeding I need to know if you have studied your earlier siddhānta-s well. Utter and expound the dvādaśa-nāmāni.”
Vrishchika: “I do know them from my sarvādhikāra-dīkṣa:
dhanadaś ca yakṣapatir vitteśo nidhipālakaḥ ।
rākṣasādhipatiś caiva piṅgalākṣo vimānagaḥ ॥
rudrasakhā kuberaś ca guhyakānāṃ patis tathā ।
vaiśravaṇeśvaraś caiva yakṣendraḥ parikīrtitaḥ ||

Dhanada: the giver of wealth; Yakṣapati: the lord of the yakṣa-s; Vitteśa: the lord of wealth; Nidhipālaka: the guardian of wealth; Rākṣasādhipati: the lord of the Rakṣa-s, this has been already stated in the Yajuṣ and the Atharvaṇa-śruti-s; Piṅgalākṣa: he whose eyes are of a golden tint; Vimānaga: one who can go anywhere on his Puṣpaka airplane; Rudrasakhā: friend of Rudra, this has been explained in our national epic; Rudra with his family often resides in the space station of Kubera; Kubera: One who has a frightful form; Guhyakānāṃ pati: lord of the hidden yakṣa-s who in the śruti are known as tirodhā; Vaiśravaṇa: the lord of the northern garden-land known as Viśravas; īśvara: the great lord; Yakṣendra: who is like Indra among the Yakṣa-s.”

Indrasena: “Great. At some future point when we are united after passing beyond the place known as Mlecchadigdvāravṛtti we shall perform together the great yāga as enjoined in the Taittirīya-śruti concluding with the mantra rājādhirājāya… By that time I would have mastered the full rahasya-s of this very mysterious mantra of the Bhṛgu-s from the Atharvaveda:

mānuṣaṃ vi gāhathāḥ |
virūpaḥ sarvasmā āsīt
saha yakṣāya kalpate ||

But for now you may do puraścaraṇa of the antardhānāstra-mantra. It will provide us with the fury needed withstand the assault of the ekarākṣasavādin-s who would seek to place in the museum in that great clash in the future.”

Vrishchika: “That sounds frightening. Tell me more of how one approaches this rahasya?”
Indrasena: “The full vidhi goes thus: One observes fast on the dvitīya. Then on tṛtīya one performs the rite having broken ones fast. One places an image of the gadā in a golden or silver vessel with ghee on which has been inscribed the 12 names of the yakṣa. One invokes Kubera with the following mantra enjoined by the Vaikhānasa-s:
rāyas-poṣāya āyuṣe prajāyai nīdhīśam āvāhayami ।
(For increasing prosperity, life and offspring I invoke the lord of wealth)

Then he does the visualization of Kubera thus:
atha dhyānam:
yakṣa-rākṣasa-sainyena guhyakānāṃ gaṇair api |
vimāna-yodhī dhanado vimāne puṣpake sthitaḥ ||
sa rājarājaḥ suśubhe yuddhārthī naravāhanaḥ |
prekṣamāṇaḥ śivasakhaḥ sākṣād iva śivaḥ svayam ||

(The lord of wealth, the overlord, is united with yakṣa, rakṣa-s and guhyaka hosts, as also Śaṅkha and Padmā. The lord of the king of kings, the wealthy one, the lord of riches holds a mace in his hand. The airplane-warrior, the giver of wealth is stationed in his Puṣpaka airplane. He the king of kings, residing in great auspiciousness, eager in combat, is borne by yakṣa-s known as Nara-s. Watching on, the friend of Śiva is himself like a second Śiva)

Then as ordained by the Vaikhānasa-s one offers the pūrvārghya of water in a receptacle with the gāyatrī:
rāja-rājāya vidmahe dhanādhyakṣyāya dhīmahi|
tan no yakṣaḥ pracodayāt ||

Then one performs japa and/or homa of the mantra:
asya mantrasya vadanya ṛṣiḥ । virāṭ chandaḥ । antardhāna-dhārin-ugra-kubero devatā ।
oṃ chaṇḍograyakṣāya huṃ phaṭ tirodhehi sapatnān naḥ svāhā ||

There after one concludes with the incantation of the Śānkhāyana-s:

nainaṃ rakṣo na piśāco hinasti na jambhako nāpy asuro na yakṣaḥ ||
(Neither rakṣa-s nor piśāca-s, nor jambhaka-s, i.e. Kubera’s agents, nor asura-s nor yakṣa-s harm him)

Then one offers the madhyamārghya as above with the mantra:
rudra-sakhāya vidmahe vaiśravaṇāya dhīmahi |
tan naḥ kuberaḥ pracodayāt ||

Then one recites the stuti:
dhanasya kāmasya praṇāyakas tvaṃ ।
vimānagas tvaṃ lokeśvaras tvaṃ ।
nāmāmi jiṣṇuṃ caṇḍogra-yakṣaṃ ॥
(You are the leader in wealth and desire;
You are the giver of happiness and profit;
You are the airplane-rider the lord of the world
I salute the conquering fierce formidable yakṣa)

Then one does tarpaṇa with the 12 names in the accusative case.
Additionally he also offers tarpaṇa to the parivāra:
ṛddhiṃ tarpayāmi ।
śriyāṃ tarpayāmi ।
nalakūbaraṃ tarpayāmi ।
śaṅkhaṃ tarpayāmi ।
maṅkanakaṃ tarpayāmi ।

One then offers the prasannārghya with the mantra:
tan no yakṣaḥ pracodayāt ||

One then makes offering of pure cooked food as bali with mantra ordained by the Vaikhānasa-s:
rāja-rājo dhanādhyakṣaḥ kubero viśravas-patiḥ |
prīyatāṃ nidhi-saṃyukta īśvarasya sakhā prabhuḥ ||
(The king of kings, the administrator of wealth, Kubera, the lord of the northern land of Viśravas, possessed of wealth, the friend of Rudra, may the lord be pleased)

Then he does upasthāna:
parivāreṇa saha kubero nakulahastaḥ suprīto suprasanno varado yathā sthānaṃ tiṣṭhatu ||
(With his retinue may Kubera, holding a mongoose, pleased and happy, boon-giving remain in his own region)”

Indrasena then continued: “One performs this vrata for an year observing restraints of not consuming alcohol and eating pure food and maintaining one’s body with discipline.”
Vrishchika: “I express my profuse gratitude to you for revealing the vidhi. I am hoping to perform the vidhi even as you have described it. But the only issue is how I might be able to obtain the gadā?”
Indrasena: “I will be duly sending you one. Again most only attain partial īśvarasiddhi. Only those born of good families for multiple generations attain that complete siddhi. In Gomūtra-nagara lived a man who saw all these vidyā-s but his discipline was incomplete so his lābha was also incomplete. Likewise with the vaṅgasiṃha. But fearlessly perform the puraścaraṇa. Someday all will come together.”
Vrishchika: “We will not be able see each other from some time in the coming future as our paths diverge but in the future we shall be united again and shall perform the great yāga.”