Two exceedingly simple sums related to triangular numbers

mAnasa-taraMgiNI

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 $latex z=x^2+y^2$.

matrixmod01_318Figure 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.

matrixmod06_318Figure 2: $latex z=x^4-x^2-y^2+y^4, n=318$

matrixmod08_315Figure 3: $latex z=x^4-x^2-y^2+y^4, n=315$

matrixmod13_309Figure 4: $latex z= x^6-x^4-y^4+y^6, n=309$

matrixmod12_318Figure 5: $latex z=x^6-x^2-y^2+y^6, n=318$

matrixmod07_312Figure 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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