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 RT @Sapratha: List of identifiable H treatises on medicine, astronomy and mathematics translated under or before the Sui Dynasty [581–618 A… 20 hours ago
 The storming of the citadels of science, tech& medicine in the occident by navyonmAda is no surprise to its careful… twitter.com/i/web/status/1… 1 day ago
 schwa deletion is an abomination twitter.com/krithikasivasw… 1 day ago
 Interesting set of coins from what's today Burma with a triple li~Nga a fish. A shaiva trident symbol. garuDa. An I… twitter.com/i/web/status/1… 1 day ago
 RT @Rjrasva: Cultural History of Late Tokugawa Japan Trent Maxey, associate professor, Departments of History and Asian Languages and Civil… 1 day ago
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 Some ruminations on asteroids and meteoritic falls
 Comet C/2022 E3 (ZTF)
 The Vyomavyāpin in the Pāśupatatantra and a discursion on ninefold Rudramantras
 BhāskaraII’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 stotras, 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 Śaṃkarasaṃhitā
 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 IndoEuropean aspects of the Anatolian tradition
 The death of Miss Lizzie Willink
 IndoEuropean expansions and iconography: revisiting the anthropomorphic stelae
 Geopolitical summary: March 2022
 Human retroviruses, sociology of science, and biographical ruminations
 Transcripts of conversations: the addiction principle:
 Phantom impressions1
 A note on Śrī, Viṣṇu and śṛṅgāra
 Are civilizational cycles the norm?
 On the rise of the mātṛkās and the goddess Cāmuṇḍā
 Huns, Uralics, and empires of the steppe
 Some observations on the LekkerkerkerZeckendorf decomposition of integers

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March 2023 M T W T F S S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 
Pages
Categories
Search this Blog
Archives
mAnasataraMgiNI supplement
 RT @Sapratha: List of identifiable H treatises on medicine, astronomy and mathematics translated under or before the Sui Dynasty [581–618 A… 20 hours ago
 The storming of the citadels of science, tech& medicine in the occident by navyonmAda is no surprise to its careful… twitter.com/i/web/status/1… 1 day ago
 schwa deletion is an abomination twitter.com/krithikasivasw… 1 day ago
 Interesting set of coins from what's today Burma with a triple li~Nga a fish. A shaiva trident symbol. garuDa. An I… twitter.com/i/web/status/1… 1 day ago
 RT @Rjrasva: Cultural History of Late Tokugawa Japan Trent Maxey, associate professor, Departments of History and Asian Languages and Civil… 1 day ago
Top Posts

Recent Posts
 Some ruminations on asteroids and meteoritic falls
 Comet C/2022 E3 (ZTF)
 The Vyomavyāpin in the Pāśupatatantra and a discursion on ninefold Rudramantras
 BhāskaraII’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 stotras, 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 Śaṃkarasaṃhitā
 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 IndoEuropean aspects of the Anatolian tradition
 The death of Miss Lizzie Willink
 IndoEuropean expansions and iconography: revisiting the anthropomorphic stelae
 Geopolitical summary: March 2022
 Human retroviruses, sociology of science, and biographical ruminations
 Transcripts of conversations: the addiction principle:
 Phantom impressions1
 A note on Śrī, Viṣṇu and śṛṅgāra
 Are civilizational cycles the norm?
 On the rise of the mātṛkās and the goddess Cāmuṇḍā
 Huns, Uralics, and empires of the steppe
 Some observations on the LekkerkerkerZeckendorf decomposition of integers

Join 699 other subscribers
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March 2023 M T W T F S S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Tag Archives: geometry
BhāskaraII’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 premodern architecture and yantras 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 worldconquering strides of Viṣṇu, when he singlehandedly bridged the lacuna in Hindu mathematics from the days of the brāhmaṇas 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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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
Some observations on the LekkerkerkerZeckendorf decomposition of integers
In our youth, we learned of a nice arithmetic theorem of Lekkerkerker (more popularly known after Zeckendorf; hereinafter LZ) that relates to the famous Mātrāmeru 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 templetablet 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 longstanding problems in this regard is: what is the curve that best describes the shape of a dinosaurian egg? While … Continue reading
Self, nonself and segregation: a very basic look at agentbased 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
Posted in Life, Scientific ramblings
Tagged cellular automata, geometry, Ising, lattice models, mathematics, Schelling, self/nonself, society, sociobiology, sociology
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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
Posted in Scientific ramblings
Tagged geometry, Hindu mathematics, mathematical entity, mathematics, proof, sequence, series sum, sum
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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
Some notes on the HenonHeiles Hamiltonian system
Anyone familiar with dynamical systems knows of the HenonHeiles (HH) system. What we are presenting here is wellknown stuff about which reams of material have been written. However, we offer certain tricks for visualizing this system that make it easy … Continue reading
Bṛhaspatiśanaiścarayor yuddham2020 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 grahayuddhas in his Brihatsaṃhitā (chapter 17) thus: … Continue reading
Posted in Heathen thought, History, Scientific ramblings
Tagged astronomy, constellations, geometry, Greek, Greek thought, Hindu knowledge, Hindu science, planets, sky, stars
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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 bhūjākoṭikarṇanyāya 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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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
Conic conquests: biographical and historical
PDF file of same article Studying mathematics with our father was not exactly an easygoing experience; nevertheless, it was the source of many a spark that inspired fruitful explorations and lifelessons. We recount one such thread here, and reflect on … Continue reading
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
Posted in Heathen thought, Scientific ramblings
Tagged approximation for pi, circle, geometry, Greek, Greek thought, mathematical entity, mathematics, numbers, pi, prime numbers, sequence
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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
Rāsabhanyāyaśikṣā
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
Chaotic behavior of some floorsquared 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
Posted in Scientific ramblings
Tagged attractors, chaos, fractal, fractals, geometry, Golden Ratio, mathematics, numbers
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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
Posted in Scientific ramblings
Tagged arithmetic, cube, ellipse, geometry, history of science, Japan, mathematics, numbers, parabola, recreational geometry, square
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The Mātrāmeru 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
Posted in Scientific ramblings
Tagged fibonacci, geometry, Golden Ratio, mathematics, recreational geometry, sequence, triangles, trigonometry
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Some Nārāyaṇalike 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
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 worldchanging. One such event from our lifetime is the proving of the Last Theorem of Fermat … 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
Chaos, eruptions and rootconvergence in onedimensional maps based on metallicsequence 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
Posted in Scientific ramblings
Tagged Bronze ratio, chaos, chaotic maps, Copper Ration, curves, fractals, geometry, Golden Ratio, mathematics, recreational geometry, recursion, sequence, Silver Ratio
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Nārāyaṇa’s sequence, Mādhava’s series and pi
The cointoss problem and Nārāyaṇa’s sequence If you toss a fair coin times how many of the possible resultsequences 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 pentagonhexagondecagon 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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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
Posted in Scientific ramblings
Tagged Euler, geometry, Plato, Platonic ideals, polygons, polyhedra, polyhedron, space
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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
Posted in art, Scientific ramblings
Tagged chaos, complex numbers, fractal, fractals, geometry, Golden Ratio, IFS, polygons, recreational geometry
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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
Posted in art, Scientific ramblings
Tagged art and science, fractal, fractals, geometry, IFS, mathematics, recreational geometry
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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
Posted in Scientific ramblings
Tagged Bachet, cube, cubic, geometry, mathematical entity, mathematics, Mordell equation, numbers, prime numbers, quadratic, recreational geometry, square
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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
Posted in Scientific ramblings
Tagged attractors, chaos, chaotic flows, dynamics, fractal, fractals, geometry, mathematics, trigonometry
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Divisionmultiplication parabolas, triplications, and quadratic residues
Introduction Many strands of our investigations on conicgenerating 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
Posted in Scientific ramblings
Tagged arithmetic, conics, fractal, fractals, geometry, mathematical entity, mathematics, MephistoWaltz, numbers, parabola, prime numbers, sequence
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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: MethodI (Figure 1: this is based on an approach we described earlier) Figure 1 1) Mark point , which will … Continue reading
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
Posted in Scientific ramblings
Tagged arithmetic, chaos, chaotic flows, dynamics, fractal, fractals, geometry, GumowskiMira, mathematics, numbers, prime numbers
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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 yavanas is one of the foundations of science and mathematics as we know it. One remarkable class of numbers which they discovered were … 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 heartshaped curves, which are distinct from the … Continue reading
Posted in Scientific ramblings
Tagged circle, conic sections, conics, curves, ellipse, Geometric construction, geometry, heartcurve, 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
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
Posted in Scientific ramblings
Tagged biological conflict, biology, chaos, dynamics, ellipse, geometry, mathematical entity, mathematics, polygons, trigonometry
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The quotient triangle, the parabolahyperbola 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
Posted in Scientific ramblings
Tagged composite numbers, conic sections, conics, division, geometry, mathematics, Nicomachus, pi, prime numbers, Ramanujan
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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
Posted in art, Scientific ramblings
Tagged fractal, fractals, geometry, Hamiltonian, mathematical entity, mathematics, Oscillator, physics, recreational geometry, recursion
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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
The SatijaKetoja 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
Posted in art, Scientific ramblings
Tagged attractors, Cauchy distribution, Cauchy flights, Central Limit Theorem, chaos, fractal, fractals, geometry, Ketoja, mathematics, physics, Satija
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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 maps2 While the story of the chaotic 2D attractors began with the simplelooking maps of Henon and Lozi, by the early 1990s the highpoint was … Continue reading
Posted in art, Scientific ramblings
Tagged attractors, chaos, chaotic flows, fractal, fractals, geometry, mathematics
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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 BhāskaraII, but … Continue reading
The Meru and Nārāyaṇa’s cows: Words and fractals
The fractals described herein are based on and inspired by the work of the mathematicians Rauzy, MendesFrance, Monnerot and Knuth. Some their works, especially the first of them, are dense with formalism. Here we present in simple terms the means … Continue reading
Hofstadter and Nārāyaṇa: connections across space and time
The scientistphilosopher Douglas Hofstadter presents an interesting singleseeded 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 selfevident 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
The square root spiral and the Gamma function: entwined analogies
The topic discussed here is something on which considerable serious mathematical literature has published by P.J Davis, W. Gautschi and others. This partly historical narration is just a personal account of our journey through the same as a nonmathematician. As … Continue reading
Leaves from the scrapbook2
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
bhujākoṭikarṇanyāyaḥ koṭijyānyāyaś ca
bhujākoṭikarṇanyāyaḥ koṭijyānyāyaḥ
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Tagged Geometric construction, geometry, mathematics, recreational geometry
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Some personal reflections on Carl Gauss, Bernhard Riemann and associated matters
The biochemist Albert SzentGyörgyi had famously remarked that as he successively, journeyed for a better understanding of life from cell biology, to physiology, to pharmacology, to bacteriology, to biochemistry, to physical chemistry to quantum mechanics he lost life between his … Continue reading
Posted in History, Life, Scientific ramblings
Tagged Einstein, Euclid, Euler, Gauss, geometry, Greek thought, Noneuclidean geometry, Riemann, Thabit ibn Kurra
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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 mathematicianscientists 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: Selfevident 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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Sine rugs
Consider a square lattice with uniform vertical and horizontal spacing of a quantum . This can be represented as an array of complex numbers of the form: . For our purposes we chose . Thus the lattice comprises of all … Continue reading
Posted in art, Scientific ramblings
Tagged art, complex variable, geometry, recreational geometry, trigonometry
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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 straightedge 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 devaogdoad
Classical Hindu tradition holds that the ogdoad of devas 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
Trigonometric tangles3: the fractals
See also: https://manasataramgini.wordpress.com/2016/05/06/theastroidthedeltoidandthefishwithinthefish/ This exploration began in days of youth shortly after we learned about complex numbers. It culminated only much later in adulthood when we discovered for ourselves a class of fractal curves related to a celebrated curve discovered … Continue reading
Euler’s squares
On account of our fascination with the geometry of origami (albeit not wellendowed 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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Trigonometric tangles
Let us define a define the trigonometric tangle as the following parametric function: where can be a rational number or an irrational number. and are any real number. If is a rational number and then we get a tangle petals … Continue reading
Some visions of infinity from the past and our times
The great Hindu mathematician and astronomer BhāskaraII’s work preserves a highpoint of Hindu knowledge. His work contains ideas that are often seen as characterizing “modern” scientific understanding i.e. what in the west would seen as starting with Leibniz and Newton … Continue reading
Deliberations on richness and beauty: discovery of some multiparameter iterative maps
As we have explained in the earlier notes (1, 2, 3), the second major factor in our exploration of 2D strange attractors maps, IFS and other fractals was the aesthetic experience they produced. Around that time we came across a … Continue reading
Posted in art, Scientific ramblings
Tagged chaos, fractal, fractals, Generative Art, geometry, mathematics
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A strange Soviet construction
in our college days we used to visit the lālpustakbhaṇḍār 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 maps2
Continued from part1 The second two dimensional map we studied in our early days was that of Lozi: where and are constants. It becomes immediately evident that this map is conceptually similar to the Henon map, using the absolute value … Continue reading
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Tagged attractors, chaos, fractal, fractals, geometry, Gingerbread Man, Henon, logistic map, Lozi, mathematics, strange attractors, tent map
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Some elementary lessons from iterative fractal maps
The famous Sierpinski gasket was one of the first fractals we wrote code for when we got access to a computer. It impressed us enormously that an intricate object with selfsimilarity over all scales could be generated by a rather … Continue reading
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Tagged biology, developmental biology, evodevo, evolution, fractal, fractals, geometry, mathematics, recreational geometry, recursion, science, temple
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Some reminiscences of our study of chaotic maps1
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
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Tagged doubling cube, Geometric construction, geometry, Japan, origami, paperfolding, religion, Sundara Rao, trisection of angle
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Syllable, number and rules in the ideal realm
This note is neither meant to be complete exposition of this matter nor a complete view of all what we have realized in this regard. Nor can it be completely understood by those who are not insiders of the tradition. … Continue reading
Posted in Heathen thought, Life, Scientific ramblings
Tagged art, biology, curlicue, fractal, fractals, geometry, Hindu, mathematical entity, mathematics, pi, Plato, Platonic ideals, proteins, religion, science
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Chaos in the iterative Hindu square root method of the gaṇakarāja
For Hindus big numbers always mattered and our mathematics is quite reflection of this fascination. Since the earliest times, Hindus devised various methods to obtain square roots of numbers, especially approximations of irrational roots correct to multiple decimal places. The … Continue reading
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 stuff2
Related stuff: Golden Ratio0 Golden Ratio1 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 yavanas 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, LotkaVolterra, population dynamics, Story, triangles, trigonometry, trisection
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A journey through fractal objects
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Tagged 3D Julia, fractal, fractals, geometry, Mandelbrot, recreational geometry, recursion
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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
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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 NeoPlatonists of Harran described a figure they called the salinon, which was supposed to mean a “saltcellar”. This material was acquired by the Mohammedans from those NeoPlatonists 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 yogins would say ānanda … Continue reading
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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
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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 unmādatraya. 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 priestchiefs, … 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
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Tagged cardioid, Geometric construction, geometry, limacon, Mamikon, recreational geometry, visual calculus
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A personal discursion on conic sections
The brothers Menaechmus and Dinostratus, as per the last great yavana sage Proclus, were associates of the Plato, who made some of the most profound discoveries in the yavana knowledge systems. Indeed, their discoveries might be considered one of the … Continue reading
Posted in art, Heathen thought, Life
Tagged Altar, Apollo, conic sections, Delian, Delos, geometry, parabola, Plato
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Fancies of the parabola and hyperbola
Many revolutionsofthesun 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 insightgiving 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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