Modulo rugs of 3D functions

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Consider a 3D function $latex z=f(x,y)$. Now evaluate it at each point of a $latex n times n$ integer lattice grid. Compute $latex z mod n$ corresponding to each point and plot it as a color defined by some palette that suits your aesthetic. The consequence is a what we term the “modulo rug”.
For example, below is a plot of $latex z=x^2+y^2$.

Figure 1: $latex z=x^2+y^2, n=318$

We get a pattern of circles around a central circular system reminiscent of ogdoadic arrangements in various Hindu maṇḍala-s. From the aesthetic viewpoint, the best modulo rugs are obtained with symmetric functions higher even powers — this translates into some pleasing symmetry in the ru. Several examples of such are shown below.

Figure 2: $latex z=x^4-x^2-y^2+y^4, n=318$

Figure 3: $latex z=x^4-x^2-y^2+y^4, n=315$

Figure 4: $latex z= x^6-x^4-y^4+y^6, n=309$

Figure 5: $latex z=x^6-x^2-y^2+y^6, n=318$

Figure 6: $latex z=x^4-x^2+y^2-y^4, n=310$

All the above $latex…

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A guilloche-like trigonometric tangle

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Coprimality, i.e., the situation where the GCD of 2 integers is 1 is one of the fundamental expressions of complexity. In that situation, two numbers can never contain the other within themselves or in multiples of them by numbers smaller than the other. In other words, their LCM is the product of the 2 numbers. There are numerous geometric expressions of this complexity inherent in coprime numbers. One way to illustrate it is by the below class of parametric curves defined by trigonometric functions:

$latex x=a_1cos(c_1t+k_1)+a_2cos(c_2t+k_2)[5pt]
y=b_1sin(c_3t+k_3)+b_2sin(c_4t+k_4)$

The human mind perceives symmetry and certain optimal complexity as the hallmarks of aesthetics. Hence, we adopt the following conditions:
1) $latex a_1, a_2, b_1, b_2$ are in the range $latex tfrac{3}{14}$..1 for purely aesthetic considerations.
2) $latex k_1, k_2, k_3, k_4$ are orthogonal rotation angles that are in the range $latex [0, 2pi]$
3) $latex c_1$, for aesthetic purposes relating to optimal…

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Huntington and the clash: 21 years later

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This note is part biographical and part survey of the major geopolitical abstractions that may be gleaned from the events in the past 21 years. Perhaps, there is nothing much of substance in this note but an uninformed Hindu might find a sketch of key concepts required for his analysis of geopolitics as it current stands. The biggest players in geopolitics are necessarily dangerous entities; hence, things will be in part stated in parokṣa — this goes well with the observation in our tradition that the gods like parokṣa.

In closing days of 1999 CE, we had our first intersection with Samuel Huntington and his hypothesis of the clash of civilizations. We found the presentation very absorbing because it lent a shape to several inferences, we had accumulated over the years both in Bhārata and on the shores of the Mahāmleccha land. The firsthand experience on shores of the Mahāmlecchadeśa…

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It is that time again …

NOEMAYA

It is that time again, to give the master a visit.

Drained by the drag of this swamp, starving for the holy company

I must return once again to that blessed deathbed, so that I may sneak through the sliding door, and the clashing rocks, and re-emerge in the field where I am no more.

Yes, that’s what I long for the most these days, a sip of eternity, where the hands of time drop and I become an innocent child again.

I’ve been in the state of emergency long before this pandemic. I’ve been wearing masks and social distancing and putting up walls since the Berlin Wall fell.

I’m a fortress, a fortress protecting mere air, protecting a mere idea, a big fat idea, a figment of imagination called “me.”

I’m weary of ideas, the cheap fridge magnets of a frightened ego. I entertain so many of them and…

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The phantoms of the bone-pipe

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As Vidrum was leafing through some recent case studies to gather the literature for his own production, he received a call from his chauffeur. He had fetched Vidrum’s new car. Vidrum went out to take a look at it. As he saw it gleaming in the mellow light of the parkway lamp he thought of his old friends for some reason: “Clever Lootika or Vrishchika would have said that it looks like a work of the Ṛbhu-s. That triplet of deities meant a lot to the four sisters, but I had never heard of them before I came to know them. May be after all there is a reason why they say the brāḥmaṇa-s are the conduit for communicating with the gods. No wonder this new car looks good but for some reason I experience no thrill of the kind I experienced when I got my first bicycle or for…

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Some notes on the Brahmayajña brāhmaṇa and Uttama-paṭala of the Atharvaṇ tradition

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The Brahmayajña brāhmaṇa (1.1.29 of the Gopatha-brāhmaṇa) of the Atharvaveda provides a glimpse of the Vedic saṃhitā canon as known to the brāhmaṇa authors of the AV tradition. The Brahmayajña might be done as part of the basic rite as done by dvija-s of other śākhā-s or as part of the more elaborate AV tradition of the annual Veda-vrata. The annual vrata-s of the Atharvaṇ brāhmaṇa-s include the Sāvitrī-vrata, Veda-vrata, Kalpa-vrata, Mitra-vrata, Yama-vrata and Mṛgāra-vrata. The kṣatriya-s and vaiśya-s should do at least 3 and 2 of them respectively, with the first 2 being obligatory. During these vrata-s the ritualist follows certain strictures like not consuming butter milk nor eating kidney beans, common millets, or the masura lentils at the evening meal, bathing thrice a day and wearing woolen clothing. Before performing Brahmayajña, he performs the ācamana as per the vidhi which states:
sa ācamanaṃ…

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The Taxidermist

NOEMAYA

Is it a gift or a curse, self-awareness! The endless expanse of this Self is at times terrible and frightening to look at. At times I cannot take up its infinitude. I am more comfortable when the Self breastfeeds me the mundane, and I am more at ease when placed among the herd and in the swamp of everydayness. But there is an irresistible impulse to look up and cut loose, to turn to the life of solitude. Alas, even there, even in the seemingly serene solitude where the gods are supposed to dwell, even there I fall into another swamp, of a filth of another kind, into the dark chambers of the mind.

The naked divinity is a terrible sight and a source of extreme uneasiness. It is not meant for the human to encounter the divine alone. What is of practical use with the divinity lies in its…

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Yajus incantations for the worship of Rudra from the Kāṭhaka ritual manuals

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This article is available as a pdf document. The notes from it are appended below.

The loss of the northern and northwestern Kṛṣnayajurveda traditions due to the Mohammedan depredations of Northern India (aided an abetted by the predatory Anglospheric regimes) has been one the great tragedies faced by Hindudom. Hence, it is rather important to collate and restore whatever remains of these traditions, namely those belonging to the Kaṭha and Kapiṣṭhala schools, which were once dominant in the greater Panjab and Kashmir. In the 1940s, vidyābhāskara, vedāntaratna Sūryakānta, saṃskṛtācārya of the Pañjāba-viśvavidyālaya, Lavapura (modern Lahore) had collated several Kaṭha fragments that came from lost texts outside of the relatively well-preserved saṃhitā. These came from the lost brāhmaṇa and the surviving āraṇyaka, as well as the lost mantrapāṭha of the Kaṭha-s that went with the sūtra-s of Laugākṣi. Notable in this regard, were the following manuscripts that Sūryakānta found in…

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Mahabharata Tatparya Nirnaya: Chapter 29: Part 1

Anandatirtha Prathishtana

• 1) The two Krishnas – Vasishta Krishna and Yadava Krishna – consecrated Dharmaraja as the Emperor and Bhimasena as the crown-prince in the presence of Vipras. They made them happy through their blessings.
• 2) At that moment, an extremely evil Rakshasa, known as Charvaka, appeared as a Tridandi Sanyasi and condemned Yudhisthira saying – “Brahmanas are berating you”.
• 3) Listening to those words, Dharmaraja became sorrowful. Seeing this, Vipras cursed that Charvaka saying – “You are saying that we berated him even when we have not done so. Therefore may you be burnt to ashes immediately”. As soon as they said thus, that great sinner was burnt to ashes.
• 4-5) When that charlatan in the guise of a Sanyasi was burnt thus, Sri Krishna, the best amongst the Vrishnis, told a sorrowful Dharmaraja – “He was not a Yati. He was a great sinner who was always engaged in cruel…

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Mahabharata Tatparya Nirnaya: Chapter 28: Part 6

Anandatirtha Prathishtana

• 201) Bhima’s body was tougher than a diamond. If Dhritharashtra had hugged him directly, he would have suffered no deformity and Dhritharashtra’s defect (of wishing evil) would never have surfaced. Hence Sri Krishna deceived him in this way.
• 202-203) The King hugged all of them with affection and blessed them. “I am the sinner who has caused the destruction of our clan. I am deserving of your curse” – when Dharmaraja said thus to Gandhari, she became angry and stared at his beautiful toe-nails through the edge of the cloth tied to her eyes. Due to the fire of her anger, his toe-nails were burnt. Thereafter, he became one with black toe-nails.
• 204) With anger and great pain, when Bhima approached in order to bow to her, she asked him – “O Bhima! How could you have killed my son through Adharma?”.
• 205) When she asked thus, desiring to extinguish…

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