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- Looking back at the goroga attacks and forward at geopolitical developments
- Khitans and Mongols: A story of deep and persistent connections-1
- The rise of the psychopath oligarchy
- The rise of yajña and Kauśika exceptionalism
- Tricakra
- Yantrasiddhi
- The dreadful apport
- Some ruminations on asteroids and meteoritic falls
- Comet C/2022 E3 (ZTF)
- The Vyomavyāpin in the Pāśupata-tantra and a discursion on nine-fold Rudra-mantra-s
- Bhāskara-II’s polygons and an algebraic approximation for sines of pi by x
- Origins of the serpent cult and Bhāguri’s snake installation from the Sāmaveda tradition
- Two simple stotra-s, sectarian competition, and the Varāha episode from the archaic Skandapurāṇa
- The zombie obeys: a note on host manipulation by parasites and its ecological consequences
- Cārucitrābhisambodhi
- RV 10.78
- The turning of the yugacakra
- A sampler of Ramanujan’s elementary results and their manifold ramifications
- A catalog of attractors, repellors, cycles, and other oscillations of some common functional iterates
- The wink of the Gorgon and the twang of the Lyre
- Some poems
- The Kaumāra cycle in the Skandapurāṇa’s Śaṃkara-saṃhitā
- Some notes on the runiform “Altaic” inscriptions and the early Turk Khaghanates: Orkhon and beyond
- Vikīrṇā viṣayāḥ: India and the Rus
- Alkaios’ hymn to the Dioskouroi: Hindu parallels
- Some notes on the Indo-European aspects of the Anatolian tradition
- The death of Miss Lizzie Willink
- Indo-European expansions and iconography: revisiting the anthropomorphic stelae
- Geopolitical summary: March 2022
- Human retroviruses, sociology of science, and biographical ruminations
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Tag Archives: Geometric construction
Bhāskara-II’s polygons and an algebraic approximation for sines of pi by x
Unlike the Greeks, the Hindus were not particularly obsessed with constructions involving just a compass and a straightedge. Nevertheless, their pre-modern architecture and yantra-s from the tāntrika tradition indicate that they routinely constructed various regular polygons inscribed in circles. Of … Continue reading
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
Posted in Scientific ramblings
Tagged arithmetic, combinatorics, complex numbers, figurate numbers, Gamma function, Geometric construction, geometry, Hindu, Hindu mathematics, irrational numbers, mathematical entity, mathematics, numbers, prime numbers, recreational geometry, Riemann, sequence, series sum, zeta function
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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
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
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
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
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
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
Generating simple radially symmetric art
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 … Continue reading
Posted in art, Life
Tagged art, art and science, Generative Art, Geometric construction, geometry
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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
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
Posted in Heathen thought, Scientific ramblings
Tagged arithmetic, bhAskara, Euler, fibonacci, figurate numbers, Geometric construction, geometry, Hindu knowledge, Hindu mathematics, history of science, irrational numbers, line, mathematics, numbers, Pythagorean triples, recreational geometry, square, square root, squares
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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
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
The blue bottle and the talent show
This story itself is haunted. The deaths Seeing Vidrum intently gaze at something on his phone with an expression of near disbelief Kalakausha asked: “Vidrum, what are you looking at? We need to be leaving shortly. We have to buy … Continue reading
Posted in art, Life
Tagged Geometric construction, geometry, geometry box, recreational geometry, Story
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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
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
Posted in Heathen thought, History, Scientific ramblings
Tagged approximation for pi, coin toss, constant, Euclid, fibonacci, Geometric construction, geometry, Golden Ratio, hexagon, Hindu mathematics, history of science, mAdhava, mathematics, nArAyaNa, pentagon, pi, sequence, series sum, tribonacci
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The Platonic culmination of Euclid and the pentagon-hexagon-decagon identity
Why did great sage Pāṇini compose the Aṣṭādhyāyī? There were probably multiple reasons but often you hear people say that he wanted to give a complete description of the Sanskrit language. That was probably one of his reasons but was … Continue reading
Posted in Heathen thought, Life
Tagged Geometric construction, geometry, Golden Ratio, Greek, Hindu, Plato, polygons, polyhedra, recreational geometry, religion, triangles, trigonometry
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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
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
Posted in Scientific ramblings
Tagged circle, conic sections, conics, curves, ellipse, Geometric construction, geometry, heart-curve, hyperbola, ovals, parabola, recreational geometry
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The mathematics class
It was a dreary autumn day, the same year Lootika had joined their school. The apabhraṃśa class had just gotten over. Somakhya’s head was spinning with all the confusing genders of the vulgar apabhraṃśa that was dealt with in the … Continue reading
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
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
Hofstadter and Nārāyaṇa: connections across space and time
The scientist-philosopher Douglas Hofstadter presents an interesting single-seeded sequence H in his book ‘Gödel, Escher, Bach: An Eternal Golden Braid’. It is generated by the recurrence relation, where …(1) Working it out one can see that it takes the form: … Continue reading
Median and pedal triangles and derived fractals: an introductory account
It is rather easily seen that joining the midpoints of the sides of a triangle yields four congruent triangles that in turn are similar to the original triangle (Figure 1). This figure might be used to provided a self-evident geometric … Continue reading
Posted in art, Scientific ramblings
Tagged fractal, fractals, Geometric construction, geometry, mathematics, triangles
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Means and conics
By the time one reaches high school one learns that: (i) there are four means that one might find some use of in life (I know there are more though they are hardly used) – the arithmetic mean which is … Continue reading
Leaves from the scrapbook-2
As described here these entries are from the scrapbook of Somakhya. Entry 11; Arasa, year Pramādin of the first cycle: It was our first day at Kshayadrajanagara. I had exhausted all that was there to talk with my cousins Mandara … Continue reading
Posted in Heathen thought, Life, Scientific ramblings
Tagged ellipse, Geometric construction, geometry, recreational geometry, Story
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Cobwebs on the golden hyperbola and parabola
The material presented here is rather trivial to those who have spent even a small time looking at chaotic systems. Nevertheless, we found it instructive when we first discovered it for ourselves while studying conics. Hence, as part of recording … Continue reading
bhujā-koṭi-karṇa-nyāyaḥ koṭijyā-nyāyaś ca
bhujā-koṭi-karṇa-nyāyaḥ koṭijyā-nyāyaḥ
Posted in art, Scientific ramblings
Tagged Geometric construction, geometry, mathematics, recreational geometry
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The two squares theorem
I do not know who might have discovered this simple relationship first. I stumbled upon it while drawing figures in the notebook during a seminar. Take any two squares such that they are joined at one side and the two … Continue reading
Posted in art, Scientific ramblings
Tagged cuboid, Geometric construction, geometry, linear, mathematics, recreational geometry, square
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Constructing a regular heptagon with hyperbola and parabola
There is little doubt that Archimedes was one of the greatest yavana intellectuals. He would also figure in any list of the greatest mathematician-scientists of all times. His work on the construction of a regular heptagon has not survived the … Continue reading
Posted in art, Scientific ramblings
Tagged complex numbers, conics, Gauss, Geometric construction, geometry, Greek, heptagon, hyperbola, mathematics, parabola, polygons, Thabit ibn Kurra
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Infinite bisections required for trisection of an angle
Figure 1: Self-evident demonstration of Figure 2: Application of the same as serial bisections to trisect the angle. In the example chosen here we have . In ten steps we get to which is a pretty close, though in principle … Continue reading
Posted in art, Scientific ramblings
Tagged Geometric construction, geometry, mathematics, recreational geometry, series sum, triangles, trisection
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Doubling the cube with ellipses
The problem of doubling of the cube which emerged in the context of the doubling of the cubical altar of the great god Apollo cannot be solved using just a straight-edge and a compass. It needs one to construct a … Continue reading
Posted in art, Scientific ramblings
Tagged conic sections, conics, Delian, Delos, doubling the cube, ellipse, Geometric construction, geometry, mathematics
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The magic of the deva-ogdoad
Classical Hindu tradition holds that the ogdoad of deva-s corresponding to their directions is: Indra: East; Agni: Southeast; Yama: South; Nirṛti: Southwest; Varuṇa: West; Vāyu: Northwest; Kubera: North; Īśāna: Northeast. The central position might be occupied in certain traditions by … Continue reading
Euler’s squares
On account of our fascination with the geometry of origami (albeit not well-endowed in mathematical capacity) we discovered for ourselves shortly after our father had taught us trigonometry that, We had earlier shown the origami proof for that. But it … Continue reading
Posted in Scientific ramblings
Tagged Euler, Geometric construction, geometry, Golden Ratio, mathematics, pi, recreational geometry, square, trigonometry
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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 … Continue reading
Posted in Scientific ramblings
Tagged Geometric construction, geometry, mathematics, recreational geometry, Russia, square
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Some reminiscences of our study of chaotic maps-1
Starting in our teens, we began our exploration of chaotic (strange) attractors emerging from simple iterative maps, numerical solution of ordinary differential equations and other fractals inspired by the work of Benoit Mandelbrot. It led us in two directions. First, … Continue reading
Posted in art, Scientific ramblings
Tagged art, attractors, beauty, chaos, fractal, fractals, Geometric construction, geometry, Henon map, iterative systems, mathematics, recreational geometry
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Paper folding, Sundara Rao and geometrical constructions
Paper folding, which we shall hereinafter refer to as origami without further discussion on the correctness of the usage, is believed to have had a long history in Japan. Some believe that some primitive form of origami might have been … Continue reading
Posted in art, Life, Scientific ramblings
Tagged doubling cube, Geometric construction, geometry, Japan, origami, paper-folding, religion, Sundara Rao, trisection of angle
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Ramanujan’s second construction for the approximate squaring of a circle
To experience the greatness of great men one has to relive or redo some acts of theirs to the best of ones ability. In ones youth such enactments might inspire one to make a bid for greatness. Whether this happens … Continue reading
Some meanderings among golden stuff-2
Related stuff: Golden Ratio-0 Golden Ratio-1 If the golden ratio can fascinate erudite men of high IQ then what to say of simpletons like us. Hence, we shall here talk about some more trivia in this regard. The golden ratio … Continue reading
Knotting a string: line, parabola, conchoid and knot
The basic construction In course of studying various methods of constructing conics we stumbled upon a means of using the relationship between uniform circular motion (UCM) and simple harmonic motion (SHM) to construct four distinct loci with common procedure. They … Continue reading
The Apollonian parabola
Some say that Archimedes and Apollonius of Perga (modern Murtina in Turkey; the center of the great yavana temple of the goddess Artemis in the days of Apollonius) were the two great yavana-s who might have rivaled Karl Gauss or … Continue reading
Posted in History, Scientific ramblings
Tagged Apollonius, conic sections, conics, Geometric construction, geometry, history, mathematics, parabola
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Leaves from the scrapbook
There were extensive memoirs in the form of electronic scrapbooks of Somakhya, Lootika and some members of their circle. Those in the know read the available excerpts due to matters of considerable interest being recorded in them. Other parts were … Continue reading
Posted in art, Life
Tagged chaos, Geometric construction, geometry, ghost, ghosts, Lotka-Volterra, population dynamics, Story, triangles, trigonometry, trisection
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Some meanderings among golden stuff
There are some angles that we often encounter in the construction of the golden ratio and its use in religious art. The first is the most obvious is the angle which is the angle made by the diagonals connecting a … Continue reading
A golden construction
Anyone with even a small fancy for geometrical matters would have at some point in their lives played with the golden ratio (). Indeed, we too have had our share of fun and games with the golden ratio. In course … Continue reading
Posted in art, Scientific ramblings
Tagged Generative Art, Geometric construction, geometry, Golden Ratio, mathematical entity, mathematics, vectors
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The first three squareable Lunes
For the sake of some readers we shall first define a lune: A lune is a concave closed region bounded by two circular arcs respectively with radii and and distance between their centers as , where . This region looks … Continue reading
Posted in art, Life, Scientific ramblings
Tagged area, circle, circumcenter, Geometric construction, geometry, lune, mathematics, quadrature of circle, squaring a circle
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The salinon
The yavana Archimedes or some later commentator of his among the Neo-Platonists of Harran described a figure they called the salinon, which was supposed to mean a “salt-cellar”. This material was acquired by the Mohammedans from those Neo-Platonists from whom … Continue reading
Posted in Life, Scientific ramblings
Tagged Archimedes, Geometric construction, geometry, mathematics, Neoplatonism, salinon
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van Aubel’s theorem
The van Aubel’s theorem is a simple theorem which is comparable to the theorem attributed to the French conqueror Napoleon Bonaparte regarding triangles. It is easy to prove once you know the upāya, even as the yogin-s would say ānanda … Continue reading
Posted in art, Life, Scientific ramblings
Tagged Geometric construction, geometry, mathematics, philosophy, triangles, trigonometry, van Aubel
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Ovals, drops, tops, eights, pears and the like
Ovals, drops, tops, eights, pears and the like This piece may be seen as a continuation of the earlier one on our journey through the world of ovals. As it needed a lot of figures and some mathematical notation it … Continue reading
Posted in art, Life, Scientific ramblings
Tagged curves, evolutes, Geometric construction, geometry, leminiscates, mathematics, ovals, pedal curves, piriform
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The astroid, the deltoid and the fish within the fish
As this article needed a lot of figures with some mathematical notation it is being presented as a PDF file: The astroid, the deltoid and the fish within the fish
A biographical journey from conics to ovals
As this article needed a lot of figures with some mathematical notation it is being presented as a PDF file: A biographical journey from conics to ovals This may be read a continuation of earlier notes such as: Ovals, drops, tops … Continue reading
Iamblichus, quadratures, trisections and the lacuna of the cycloid
Today Syria has been turned into a hellhole by the unmāda-traya. However, just before the irruption of the second Abrahamism which ended the late Classical world, it was home to great men like Iamblichus. Hailing from a clan of priest-chiefs, … Continue reading
The cardioid and the arbelos: the scimitar and the axe
The arbelos of Archimedes, an object most wondrous; it brought pleasure to us, when stalked by enemies, as the old yavana by Romans, who ended for good his days. What is the mystery of the scimitar and the axe which … Continue reading
Posted in art, Life, Scientific ramblings
Tagged cardioid, Geometric construction, geometry, limacon, Mamikon, recreational geometry, visual calculus
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Idiosyncratic synesthetic experiences in some trivial trigonometric identities
Mathematical objects, despite existing in a purely abstract “Platonic” realm, have the ability (perhaps by the very virtue of their Platonic idealism) to produce synesthetic experiences. We have often wondered in our life as to why the realization or experience … Continue reading
Posted in art, Heathen thought, Life
Tagged beauty, Euler, Geometric construction, Hindu, mathematics, philosophy, Platonic ideals, trigonometry
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Squaring of the circle by Srinivasa Ramanujan
Geometric constructions have a special effect on us because they depict visually the Platonic abstractions that often underlie the laws of nature. So in a sense a construction is a mapping of natural laws on the canvas of Platonic space. … Continue reading
Posted in art, Heathen thought, Scientific ramblings
Tagged Geometric construction, pi, quadrature of circle, squaring a circle
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Fancies of the parabola and hyperbola
Many revolutions-of-the-sun ago we were seeking a device that could construct a parabola. We made a discovery in this connection that is exceedingly elementary Euclid for mathematicians, but for us it was an insight-giving philosophical revelation. A parabola can be … Continue reading
Posted in Life, Scientific ramblings
Tagged conic sections, conics, Geometric construction, geometry, hyperbola, parabola, recreational geometry
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