The shape of dinosaur eggs

mAnasa-taraMgiNI

Readers of these pages will know that we have a special interest in the geometry of ovals. One of the long-standing problems in this regard is: what is the curve that best describes the shape of a dinosaurian egg? While all archosauromorphs hatch from eggs outside their mother’s body, the form of their eggs is rather variable; crocodylians and turtles may lay either leathery or hard-shelled eggs. The dinosaurs almost always lay hard-shelled eggs that tend to be rather uniform in shape in the wild. Being hard-shelled, the shape of a dinosaurian egg can be described by the characteristic curve of its maximal (area) cross-section. The egg itself will be the solid of rotation of this curve around its longest axis. Using this definition, the noted morphometrician and student of Aristotelian zoology, D’arcy Thompson, classified bird eggs into various forms in his famous book “On growth and form”. More…

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Twin Āditya-s, twin Rudra-s

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This note originated as an intended appendix to the article on Rudra and the Aśvin-s we published earlier. The first offshoot from that work, which we published separately, explored the links between Rudra, Viṣṇu and the Aśvin-s in the śrauta ritual. We finally found the time to fully write down the intended appendix and present it as a separate note. To rehash, we noted an intimate connection between the primary Rudra-class deity (typically in his manifestation as the great heavenly Asura, the father of the worlds) and the twin deities (the Aśvin-class) of the ancestral Indo-European religion. This is preserved in multiple descendants of our ancestral religion, such as in the śruti, the para-Vedic material in the aitihāsika-paurāṇika corpus, in the Roman religion relating to Castor and Pollux, and probably the non-Zoroastrian strains of the Iranian religion. It was definitely there in at least some branches of the Germanic…

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Self, non-self and segregation: a very basic look at agent-based lattice models

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In our college days, a part time physics teacher from an old and respected V$latex _1$ clan used to chat with us about issues of mutual interest that were beyond that of the rest of the class (or for that matter the rest of the teachers) and well out of the scope of the syllabus. He was the only one among the physics staff with an interest in science for science’s sake. We always felt he had it in him to be a scientist and he was indeed was pursuing a doctoral program at his own pace on the side. However, he clarified to us that he was the big fish in the small pond and that every man’s ambition is like a rocket set off on a Dīpāvalī night — drawing out a parabola on the board he declared with his characteristic smirk: “It will come down; hence, why…

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Matters of religion: Varuṇāvasiṣṇavam, Agnavasiṣṇavam and the vyahṛti-s

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Like the clouds lifting after the monsoonal deluge to unveil the short-lived comforts of early autumn, the metaphorical pall over the nation cast by the engineer’s virus was lifting. Somakhya and Lootika were at the former’s parents’ house, relieved that they had survived and overcome the tumultuous events. Somakhya’s parents asked them to offer the Varuṇāvasiṣṇava and associated oblations as ordained by the Bhṛgu-s and Āṅgirasa-s of yore. Vrishchika and Indrasena were also present as observers of the rite. Somakhya donned his turban and identified himself with the god Indra to initiate the rite, for indeed the śruti has said: tad vā etad atharvaṇo rūpaṃ yad uṣṇīṣī brahmā । — that brāhmaṇa who is turbaned is indeed of the form of the Atharvan. He explained to Indrasena that the śruti holds the Indra took the shape of the Atharvaveda in his turbaned form to protect the ritual of the gods…

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Two exceedingly simple sums related to triangular numbers

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This note records some elementary arithmetic pertaining to triangular numbers for bālabodhana. In our youth we found that having a flexible attitude was good thing while obtaining closed forms for simple sums: for some sums geometry (using methods of proofs pioneered by Āryabhaṭa which continued down to Nīlakaṇṭha Somayājin) was the best way to go; for others algebra was better. The intuition was in choosing the right approach for a given sum. We illustrate that with two such sums.

Sum 1 Obtain a closed form for the sum: $latex displaystyle sum_{j=1}^{n} (2j-1)^3$

These sums define a sequence: 1, 28, 153, 496, 1225…
Given that we can mostly only visually operate in 3 spatial dimensions, our intuition suggested that a cubic sum as this is best tackled with brute-force algebra with the formulae for individual terms derived by Āryabhaṭa and his commentators. Thus we have:

$latex displaystyle sum_{j=1}^{n} (2j-1)^3 = sum_{j=1}^{n}…

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Pandemic days: Vaccines and war

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In American history-writing we come across various attempts to the justify the use of nuclear weapons on Japan in the closing phase of WW2. We often hear the claim that by using the nukes they avoided a large number of casualties that they would have suffered in a long-drawn conventional war to conquer Japan. Neutral outsiders who have studied the matter realize that this is merely the American narrative to justify and positively spin something, which many of their own people (some leaders included) found rather disturbing. A closer look indicates that the Japanese were brought to the brink of surrender by the demolition they faced at the hands of the Rus in Manchuria. Indeed, the Rus were poised to invade the main islands and probably kill the emperor of Japan. Faced with this, the Japanese calculated that surrendering to the Americans might help them save the emperor and perhaps…

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Generating simple radially symmetric art

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Many people experience beauty in structures with bilateral, radial and rotational symmetries with or without recursion. The recursive or nested structure are the foundation of the beauty in fractal form, the generation of which has become increasingly easy for the lay person with ever-improving computing power. One could generate beautiful fractal structures using a range of open source software; however, there is no substitute for writing ones own code and taking in some of the mathematics behind the beauty — truly fractal structures provide the clearest bridge between mathematics and beauty. While we have presented some discussion on such structures on these pages, that is not the topic of this note. Here, we shall talk about stuff that is mostly art for art’s sake (We fully understand that what constitutes art can have some subjectivity) that is generated based on simple repeats of certain motifs with an emphasis on radial…

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Making an illustrated Nakṣatra-sūkta and finding the constellation for a point in the sky

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The illustrated Nakṣatra-sūkta

Towards the latter phase of the Vedic age, multiple traditions independently composed sūkta-s that invoked the pantheon in association with their home nakṣatra-s as part of the śrauta Nakṣatreṣṭi or related gṛhya homa-s. Of these oldest and the most elaborate is seen in the form of the Nakṣatra-sūkta of the Taittirīya brāhmaṇa. From the time we first learned this in our youth, it has been a meditative experience that compensates for the bane of urban existence — bad skies. Passing from nakṣatra to nakṣatra, we could bring to our mind the various glorious celestial bodies that we had been recording since the 10th year of our life. Thus, the desire arose in us to create an illustrated Nakṣatra-sūkta that would aid in bringing them to mind as we recited it in an indoor urban setting. We have been making our own star maps for a while, each…

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Johannes Germanus Regiomontanus and his rod

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Even before we had become acquainted with the trigonometric sum and difference formulae or calculus are father had pointed to us that there was an optimal point at which one should stand to observe or photograph features on vertical structures, like on a tall gopura of a temple or a tree. That point can be calculated precisely with a simple Euclidean construction. Hence, we were rather charmed when we encountered this question in a German book on historical problems in mathematics. It was posed in 1471 CE by Johannes Germanus Regiomontanus to a certain professor Roderus of Erfurt (Figure 1): At what point on the [flat ground] does a perpendicularly suspended rod appear the largest (i.e. subtends the largest angle)? Let the rod be of length $latex a$ and it is suspended perpendicularly at height $latex h$ from the ground. The question is then to find the point $latex P$…

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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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