Category Archives: Scientific ramblings

Cārucitrābhisambodhi

Chakkalal and Mundalal saw that Gannaram Dakiya, the owner of the little eatery, had taken a bit too much of an ethanolic beverage and had forgotten to lock the safe with his phone, cards and some cash. They broke into … Continue reading

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A sampler of Ramanujan’s elementary results and their manifold ramifications

As we have remarked before, Ramanujan seemed as if channeling the world-conquering strides of Viṣṇu, when he single-handedly bridged the lacuna in Hindu mathematics from the days of the brāhmaṇa-s of the Cerapada to the modern era. Starting around the … Continue reading

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A catalog of attractors, repellors, cycles, and other oscillations of some common functional iterates

One of the reasons we became interested in functional iterates was from seeking an analogy for the effect of selective pressure on the mean values of a measurable biological trait in a population. Let us consider a biological trait under … Continue reading

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The wink of the Gorgon and the twang of the Lyre

The discovery of the archetypal eclipsing binary Algol The likes of Geminiano Montanari are hardly seen today. This remarkable Italian polymath aristocrat from the 1600s penetrated many realms of knowledge spanning law, medicine, astronomy, physics, biology and military technology. Having … Continue reading

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Human retroviruses, sociology of science, and biographical ruminations

We learnt via a recent obituary that the French researcher Luc Montagnier died a month or so ago after living for nearly 90 years. He along with his compatriot and erstwhile colleagues, Françoise Barré-Sinoussi and Jean-Claude Chermann discovered HIV-1. Subsequently, … Continue reading

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Transcripts of conversations: the addiction principle:

A friend recorded some of our verbalizations and made transcripts of them. He sent them to us to and we decide to edit them and post them as and when we get the chance — not out of a narcissistic … Continue reading

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Are civilizational cycles the norm?

Nearly two and half decades back, we used to have several conversations with a late śūlapuruṣīya professor, mostly on topics with a biological angle. While not a mathematician, he had a passing interest in dynamical systems, for he felt that … Continue reading

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Huns, Uralics, and empires of the steppe

A map by Savelyev et al. for the geographic orientation of the reader of the below article. The Huns of Europe “The lord of the Huns, King Attila, born of his father Mundzuk, lord of the bravest tribes, who with … Continue reading

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Some observations on the Lekkerkerker-Zeckendorf decomposition of integers

In our youth, we learned of a nice arithmetic theorem of Lekkerkerker (more popularly known after Zeckendorf; hereinafter L-Z) that relates to the famous Mātrā-meru sequence : 0, 1, 1, 2, 3, 5, 8… defined by the recurrence relationship . … Continue reading

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Subjective and objective insight

The black American scientist Sylvester Gates mentioned a curious personal anecdote in a talk. To paraphrase him, when he was in college, he had to take a calculus course. He mentioned how he could cut through differentiation as it was … Continue reading

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On the passing of E.O. Wilson

E.O. Wilson, one of the great biologists of the age, has fallen to the noose of the king, the black son of Vivasvān. He lived a long, productive, and eventful life, just 8 years shy of a century. He was … Continue reading

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Relationships between incircles of the “equilateral triangles in a square” system

This note relates to geometric relationships that may be likened to the Japanese temple-tablet problems. The inspiration for discovering and exploring it came from an origami construction presented by the pioneer in that field, Sundara Rao of Kumbhaghoṇa, in the … Continue reading

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The birth defects of Dhṛtarāṣṭra and Pāṇḍu and related matters

This note has its origin in a conversation with Sharada. We originally intended to incorporate the core of it into one of our usual fantastical stories. However, following a second conversation with her, we decided that it might be best … Continue reading

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Some biographical reflections on visualizing irrationals

In our childhood, our father informed us that, though the school told us that , it was not valid. However, he added that for “small fractions” [Footnote 1] it was a great approximation. Moreover, the numerical problems, which we would … Continue reading

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Turagapadādi

This note stems from a recent conversation with a friend, where he pointed out that the graph representing all possible positions the horse (knight) can take on the chessboard from a given starting square produces interesting graphs. It struck us … Continue reading

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Asians and Pacific Islanders: The triangle

In our youth, we read with great excitement old books on anthropology obtained from a library with considerable difficulty. The excitement was primarily from learning about the osteology of extinct apes and monkeys, including the closest sister groups of Homo … Continue reading

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The shape of dinosaur eggs

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 … Continue reading

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

In our college days, a part time physics teacher from an old and respected V 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 … Continue reading

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Pandemic days: bālabodhana

As the pandemic grinds to a close or at least to a pause in some parts of the world, there is a certain fear from new mutants threaten that threaten break current the status quo. The strain that arose in … Continue reading

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The cicadas return

Seventeen years after we first saw them emerge, like a great horde of Cīna-s invading Tibet, the cicadas of Brood X reemerged. 17 years is a good amount of time, making one pause to reflect on what has passed by … Continue reading

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Two exceedingly simple sums related to triangular numbers

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 … Continue reading

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Pandemic days: Vaccines and war

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 … Continue reading

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

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 … Continue reading

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Johannes Germanus Regiomontanus and his rod

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, … Continue reading

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A great statistician, and biographical, numerical musings on ancient game

Recently my friend brought it to my attention that C. Radhakrishna Rao had scored a century. Born in 1920 CE to Doraswamy Nayadu and A. Laxmikanthamma from the Andhra country, he is one of the great mathematical thinkers and statisticians … Continue reading

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Modulo rugs of 3D functions

Consider a 3D function . Now evaluate it at each point of a integer lattice grid. Compute corresponding to each point and plot it as a color defined by some palette that suits your aesthetic. The consequence is a what … Continue reading

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A guilloche-like trigonometric tangle

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 … Continue reading

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Some notes on the Henon-Heiles Hamiltonian system

Anyone familiar with dynamical systems knows of the Henon-Heiles (HH) system. What we are presenting here is well-known stuff about which reams of material have been written. However, we offer certain tricks for visualizing this system that make it easy … Continue reading

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Bṛhaspati-śanaiścarayor yuddham-2020 ityādi

The below is only for information. Parts of it should not be construed as any kind of prognostication on our part. The great Hindu naturalist Varāhamihira describes various kinds of planetary conjunctions or grahayuddha-s in his Brihatsaṃhitā (chapter 17) thus: … Continue reading

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Prakīrṇaviṣayāḥ: Life, brains, warfare and society

1 On big brains An occidentally conditioned person remarked that “we were making bad use of the great brains we have evolved. Instead of using it for human betterment, we were expending it on killing each other with sophisticated weapons.” … Continue reading

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Bhāskara’s dual square indeterminate equations

PDF for convenient reading Figure 1. Sum and difference of squares amounting to near squares. In course of our exploration of the bhūjā-koṭi-karṇa-nyāya in our early youth we had observed that there are examples of “near misses”: . Hence, we … Continue reading

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Pandemic days-6: Genetic risk factors

The coronavirus that made its way to humans aided by the Cīna-s at Wuhan has now been with us for nearly an year. Right from the early days of this outbreak, one thing has been notable about this virus: some … Continue reading

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Counting pyramids, squares and magic squares

Figure 1. Pyramidal numbers The following note provides some exceedingly elementary mathematics, primarily for bālabodhana. Sometime back we heard a talk by a famous contemporary mathematician (M. Bhargava) in which he described how as a kid he discovered for himself … Continue reading

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Ruminations on meteorites, organics and water

In our times the Christian Anglo-Saxons were famous for their “war on drugs”. However, in the 1800s, when they lorded over India, they were famous as global drug dealers. On the morning of August 25, 1865 CE around 9:00 AM, … Continue reading

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An arithmetic experiment and an unsolved problem

We realized that a simple arithmetic experiment we had performed in our youth is actually related to an unsolved problem in number theory. It goes thus: consider the sequence of natural numbers Then find the distance of to nearest prime … Continue reading

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Conic conquests: biographical and historical

PDF file of same article Studying mathematics with our father was not exactly an easy-going experience; nevertheless, it was the source of many a spark that inspired fruitful explorations and life-lessons. We recount one such thread here, and reflect on … Continue reading

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Generalizations of the prime sieve and Pi

PDF version for better reading Eratosthenes, the preeminent yavana philosopher of early Ptolemaic Egypt [footnote 1], composed a hymn to the god Hermes of which only some fragments have come down to us. This connection to Hermes is evidently related … Continue reading

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A simple second order differential equation, ovals and chaos

In our youth as a consequence of our undying fascination with ovals we explored many means of generating them. In course of those explorations we experimentally arrived at a simple second order differential equation that generated oval patterns. It also … Continue reading

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Pandemic days-4: viruses get new hosts

That we have come to be in these pandemic days evokes some wonderment or even disbelief among laypeople. The general thinking of a large section of the populace is that this event is something completely unexpected or out of the … Continue reading

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Rāsabha-nyāya-śikṣā

Vrishchika had been seeing several kids of patients affected by the chemical leak that had happened sometime ago. While she saw some purely for routine clinical practice, she was also particularly interested in the several cases exhibiting heterotaxy and had … Continue reading

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Chaotic behavior of some floor-squared maps

Consider the one dimensional maps of the form: , where is the fractional part of What will be evolution of a under this map when or ? We can see that for it will tend converge. However, the behavior is … Continue reading

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The culmination of Galtonism or pandemic days-3

We saw a list of famous elderly people whom the virus has already placed in the abode of Vaivasvata at the time of writing. We recognized at least two names: (1) Robert Carroll the paleontologist, whose hard to find book … Continue reading

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The culmination of Galtonism or pandemic days-2

Ensconced in the apparent safety of the 4 walls the mind looks out into the completely silent streets with hardly a soul or even a passing ratha — a mere 120 days have made the world look and sound different. … Continue reading

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The Plague: historical, biographical and current: a brief roundup

Globalization is not a new thing. The Indo-European empire of the steppes was perhaps the first one. In addition to having a serious component of our genetic ancestry and most of our memetic inheritance in it, we can still see … Continue reading

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Two squares that sum to a cube

Introduction This note records an exploration that began in our youth with the simple arithmetic question: Sum of the squares of which pair integers yields a perfect cube? Some obvious cases immediately come to mind: . In both these cases … Continue reading

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Difference of consecutive cubes, conics and a Japanese temple tablet

Introduction In our part of the world, someone with even a nominal knowledge of mathematics might be aware of the taxicab number made famous by the conversation of Ramanujan and Hardy: the smallest number that can be expressed as the … Continue reading

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The Mātrā-meru and convergence to a triangle

What is presented below will be elementary for someone with even just the mastery of secondary school mathematics. Nevertheless, even simple stuff might present points of interest to people who see beauty in such things. Consider the following question: Given … Continue reading

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Sequences related to maps based on simple fractional functions

One of the pleasures of an unstructured youth in the pre-computer era was what we called calculator games. As our father took his prized calculator with him to work we only got a little time with it in the evenings. … Continue reading

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Some Nārāyaṇa-like convergents and their geometric and trigonometric connections

While playing with an iterative geometric construction in our youth we discovered for ourselves a particular right triangle whose sides are in the proportion , where is the Golden Ratio. This triangle is of course famous as being the basis … Continue reading

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Some notes on rational sector triangle triples

Rational points on a unit circle There are some events that happen in the course of ones life that might be considered historical or world-changing. One such event from our lifetime is the proving of the Last Theorem of Fermat … Continue reading

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The aftermath: A polynomial equation

This is merely the tailpiece to the last tale of the strange hauntings. A reader may wonder why expend so many words on a high school problem. While the ball could have fitted into the socket, it rolled away. As … Continue reading

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The minimal triangle circumscribing a semicircle

Consider a fixed semicircle with center at and radius . Let be the isosceles triangle which circumscribes it (Figure 1). Figure 1 What will be the characteristics of the minimal form of the said triangle, i.e. triangle with minimum perimeter, … Continue reading

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A modern glance at Nārāyaṇa-paṇḍita’s combinatorics-1

For improved reading experience one may use the PDF version. Students of the history of Hindu mathematics are well-acquainted with Nārāyaṇa-paṇḍita’s sophisticated treatment of various aspects of combinatorics and integer sequences in his Gaṇita-kaumudī composed in 1356 CE. In that … Continue reading

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Chaos, eruptions and root-convergence in one-dimensional maps based on metallic-sequence generating functions

bronze_bouncer Over the years we have observed or encountered certain natural phenomena that are characterized by rare, sudden eruptive behavior occurring against a background of very low amplitude fluctuations. We first encountered this in astronomy: most remarkably, in the constellation … Continue reading

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Big fruits and dead giants

The fruits of the elephants nāgo bilvam ivākramya pothayiṣyāmy ahaṃ śiraḥ | alabhyām icchatas tasya kīcakasya durātmanaḥ || Like an elephant seizing a bilva [fruit] I will crush the head of that evil Kīcaka who desires that which ought not … Continue reading

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Pearl necklaces for Maheśvara

Śrīpati’s pearl necklace for Maheśvara The brāhmaṇa Śrīpati of the Kāśyapa clan was a soothsayer from Rohiṇīkhaṇḍa, which is in the modern Buldhana district of Maharashtra state. Somewhere between 1030 to 1050 CE he composed several works on mathematics, astronomy … Continue reading

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Nārāyaṇa’s sequence, Mādhava’s series and pi

The coin-toss problem and Nārāyaṇa’s sequence If you toss a fair coin times how many of the possible result-sequences of tosses will not have a successive run of 3 or more Heads? The same can be phrased as given tosses … Continue reading

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Discovering bronze in the characteristic ellipse of right triangles

The arithmetic mean square of a right triangle An entire family of right triangles that includes all the different forms of right triangles defined in terms of the proportion of their legs can be obtained by setting their altitude to … Continue reading

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From Plato to Euler and back

This is primarily meant as an educational handout on some very basic theorems of geometry that one might have studied in school. Some educated adults whom we asked about these had either forgotten them or claimed to have never studied … Continue reading

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Packing constants for polygonal fractal maps

Among the very first programs which we wrote in our childhood was one to generate the famous Sierpinski triangle as an attractor using the “Chaos Game” algorithm of Barnsley. A couple of years later we returned to it generalize it … Continue reading

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Cows, horses, sheep and goats

It goes without saying that humans are what they are today because of cows, horses, sheep and goats. Hindu civilization in particular is in the very least the product for the first two, while remaining two contributed to it from … Continue reading

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Fraudulent science by Indians: some really bad news

Around 2011 we were approached by a researcher of Indian origin for a collaboration in biochemistry regarding a family of proteins whose biochemical functions we had uncovered. After more than an year of dealing with him, it became clear that … Continue reading

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Liṅga-kāmādi-sūtrāṇi

Devanagari PDF version .. Sūtrapāṭhaḥ .. atha liṅga-kāmādi-sūtrāṇi vyākhyāsyāmaḥ . jīvasūtrāṇunām anukrameṣu parimeyā vikārā jīvā-paramparāyā+avaśyam . jantvoḥ saṃgrāmas tasya paramaṃ kāraṇam . tasmād ajāyata jīvasūtrāṇunāṃ vyūḍhīkaraṇam . RecA-nāma jīvakāryāṇu-kulaṃ jīvasūtrāṇunāṃ vyūḍhīkaraṇaṃ karoti . mukhyaśo ‘nagnijīvasūtrāṇunām . anagnijīvasūtrāṇu-mithunayor maithunāt . idam … Continue reading

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Creating patterns through matrix expansion

People who are seriously interested in emergent complexity and pattern formation might at some point discover matrix expansion for themselves. It is a version of string rewriting that allows one to create complex patterns. For me, the inspiration came from … Continue reading

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An apparition of Mordell

Consider the equation: where is a positive integer 1, 2, 3… For a given , will the above equation have integer solutions and, if yes, what are they and how many? We have heard of accounts of people receiving solutions … Continue reading

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Newton’s cows

Cultures with an Indo-European background have had a long history of symbiosis with the bovine animal since they started herding on the steppes in the Black Sea-Caspian region. Indeed, the very emergence of the modern steppes of Eurasia is likely … Continue reading

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A novel discrete map exhibiting chaotic behavior

The map proposed by R. Lozi over 40 years ago is one of the simplest two dimensional maps that exhibits chaotic behavior and generates a wide range of interesting structures. The map may be defined thus: where are real parameters. … Continue reading

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1859 CE and beyond: Some reflections

The yuga between 1800 CE and 1900 CE saw a remarkable change in our understanding of the world at many levels. It is not that some of these ideas did not exist long before that time but they came together … Continue reading

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Visualizing the Hindu divisibility test

Prologue This article continues on the themes covered by the last two (here and here) relating to factorization and the primitive root modulo of a prime number. Early in ones education one learns the divisibility tests for the first few … Continue reading

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Fermat’s little theorem and the periods of the reciprocals of primes

From the genetic code to the proof of Fermat’s little theorem Nucleic acids encode the 20 amino acids found in the sequence of a protein using just 4 bases: A, G, T, C in DNA. Thus, the 4-symbol nucleic acid … Continue reading

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A layman’s overview of the arithmetic of encryption

Life as an encryption-decryption cycle Encryption is a concept as old as life itself. The sequence of proteins, the primary purveyors of function in life as we know it, is encrypted within nucleic acids. It is decrypted by this remarkable … Continue reading

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Division-multiplication parabolas, triplications, and quadratic residues

Introduction Many strands of our investigations on conic-generating integer sequences, word fractals and cellular automaton models for pattern formation came together in an unexpected manner while investigating a simple integer sequence. While some of these connections have have been known … Continue reading

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The mean hyperbola and other mean functions

Let be two numbers such that, We use to construct a specific rectangular hyperbola using one of the following methods: Method-I (Figure 1: this is based on an approach we described earlier) Figure 1 1) Mark point , which will … Continue reading

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The geometric principles behind discrete dynamical systems based on the generalized Witch of Agnesi

Consider the family of curves defined by the equation following parametric equation , where and It defines a family of probability distribution functions (PDFs): This can be seen from the above equations because Figure 1 Examples of these PDFs are … Continue reading

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Reflections on our journey through the aliquot sums and sequences

The numerology of aliquot sums and perfect numbers The numerology of the Pythagorean sages among the old yavana-s is one of the foundations of science and mathematics as we know it. One remarkable class of numbers which they discovered were … Continue reading

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The ghost in the tattered Gattermann

Vidrum had dropped by to see Somakhya and Lootika when they had just started their household together. They had reconstituted a fairly elaborate lab in the biggest room of their home. They had also completely set up their fire room, … Continue reading

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The hearts and the intrinsic Cassinian curve of an ellipse

Introduction This investigation began with our exploration of pedal curves during the vacation following our university entrance exams in the days of our youth. It led to us discovering for ourselves certain interesting heart-shaped curves, which are distinct from the … Continue reading

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Residues of squares, sequence curiosities and parabolas galore

Squares and their residues This is an exploration of number triangles in the same vein as some other such we have previously described . It resulted in some observations that seemed interesting to us. Some are perhaps trivial but some … Continue reading

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Some words on mathematical truth, scientific conviction and the sociology of science

Sometime in the bronze age more than one group of humans, including our own Aryan ancestors, discovered that the squares of the two legs of a right triangle sum up to the square of the hypotenuse. This is the famed … Continue reading

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A note on the least prime divisor sequences of 2p plus or minus 1

Let be the sequence of prime numbers: 2, 3, 5, 7… Define the sequences such that . Then sequence is defined such that is the lowest prime divisor (LPD) of and sequence is defined so that is the LPD of … Continue reading

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A sequence related to prime counting

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

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Convergence to a palindrome

This is a brief account of a sequence we constructed inspired by Dattatreya Ramachandra Kaprekar. It is not known to us if he had discovered it in one of his obscure publications from a small town in the Maharatta country. … Continue reading

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

Around 600 CE in the examinations of one of the Hindu schools of mathematics and astronomy one might have encountered a problem such as below (given by Bhāskara-I in his commentary on Āryabhaṭa’s Āryabhaṭīya): dvayādyaiḥ ṣaṭ-paryantair ekāgraḥ yo ‘vaśiṣyate rāśiḥ … Continue reading

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

When we wrote a previous article on this matter we had stated that new data will alter the details of our understanding of picture discussed therein. Indeed, two new manuscripts which were deposited in the past month by McColl et … Continue reading

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The remarkable behavior of a map displaying derived from a simple model for a biological conflict

One of the simplest yet profound mathematical models for biological growth emerged sometime in the middle of the 1800s due to the work of Verhulst. It describes population growth thus: let be the population of the organism at time . … Continue reading

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The quotient triangle, the parabola-hyperbola sequence, the remainder triangle and perfect numbers

The quotient triangle Consider a positive integer . Then for all do the floor operation . Say , we get , a sequence of quotients of the division . If we do this for all we get the quotient triangular … Continue reading

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Counting primes, arithmetic functions, Ramanujan and the like

We originally wished to have a tail-piece for our previous note that would describe more precisely the relationship between the Möbius function and the distribution of prime numbers. However, since that would have needed a bit of a detour in … Continue reading

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Our auto-discovery of the Möbius and Mertens sequences

Recently, we were explaining to our friend the Möbius and the Mertens functions and their relationship to the prime number distribution. We also heard with some wonder from a physicist of a theoretical model where multiparticle states behave as bosons … Continue reading

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Astronomical experiments: a preliminary look at globular clusters

While Alexander von Humboldt was exploring South America he observed a fuzzy object in the constellation of Centaurus and thought that it was a new comet. However, it turned out that it was the globular cluster Centauri. It had already … Continue reading

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Frustrations and ramblings ensuing from Cretaceous amber

Time and again I have been frustrated by the inability of Hindus to make the most of the riches that are available in their own land or right next to them. One such case is that of Cretaceous amber from … Continue reading

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Pattern formation in coupled map lattices with the circle map, tanh map, and Chebyshev map

The coupled map lattices (CMLs), first defined by Kunihiko Kaneko around the same time Wolfram was beginning to explore cellular automata, combine features of cellular automata with chaotic maps. The simplest CMLs are defined on a one dimensional lattice with … Continue reading

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Some novel observations concerning quadratic roots and fractal sequences

Disclaimer: To our knowledge we have not found the material presented here laid out here presented in completeness elsewhere. However, we should state that we do not follow the mathematical literature as a professional and could have missed stuff. Introduction … Continue reading

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The incredible beauty of certain Hamiltonian mappings

In our teens we studied Hamiltonian functions a little bit as part of our attempt to understand classical and quantum physics. A byproduct of it was a superficial interest in the geometry of some of the mappings arising from such … Continue reading

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Triangles, Hexes and Cubes

One philosophical question which we have often ponder about is: Are numbers “real”? One way to approach this question is via figurate numbers, where numbers directly manifest as very tangible geometry. This idea has deep roots in our tradition: as … Continue reading

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Godhūmo vijayate

As we have noted several times before, the Ṛgveda represents a very early stage of the evolution of Indo-Aryan. Its agricultural terms relate to a major puzzle in Indo-European history. There is an unmistakable agricultural vocabulary including words such as … Continue reading

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A note on tales of fratricide, warfare, cannibalism and incest

The Osman conqueror of Constantinople, Mehmet-II, bothered by the civil wars his predecessors had to fight to take the throne, institutionalized the system of fratricide. In this system, the rival brothers of the sultan, who took the throne, were all … Continue reading

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The Satija-Ketoja system

Satija and Ketoja discovered an interesting dynamical system in course of the study of the Schrödinger equation for one electron in a two dimensional periodic lattice on a uniform magnetic field. While this equation and its variants have several uses … Continue reading

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Some simple maps specifying strange attractors

This note may be read a continuation of: Some reminiscences of our study of chaotic maps-2 While the story of the chaotic 2D attractors began with the simple-looking maps of Henon and Lozi, by the early 1990s the high-point was … Continue reading

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Citrabhānu’s cubes

The Hindus unlike their yavana cousins preferred algebra to geometry. Yet on occasions they could indulge in geometric games for demostrating proofs of algebraic relations. We see a bit of this in the Āryabhaṭa school and the great Bhāskara-II, but … Continue reading

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Make your own sky map

We live in the age of photo-realism, be it for maps of the world or of the sky – it is a realism that not long ago was beyond the reach of even our vision. That is why even though … Continue reading

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