
Pages
Categories
Search this Blog
Archives
mAnasataraMgiNI supplement
 The number of RTs the original twt shows the H in rather dim light twitter.com/goghritaM/stat… 7 hours ago
 A long story if you'd like to read: The dreadful apport manasataramgini.wordpress.com/2023/04/02/the… 7 hours ago
 BRICS to bricolage? 7 hours ago
 RT @blog_supplement: Some ruminations on asteroids and meteoritic falls manasataramgini.wordpress.com/2023/02/20/som… 8 hours ago
 It looks as though after a brief respite with the vigrahaNa by muShkavAn kastUrI, the "algorithm" has caught up with our account recently. 2 days ago
Top Posts

Recent Posts
 The dreadful apport
 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

Join 699 other subscribers
Meta
April 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 
Pages
Categories
Search this Blog
Archives
mAnasataraMgiNI supplement
 The number of RTs the original twt shows the H in rather dim light twitter.com/goghritaM/stat… 7 hours ago
 A long story if you'd like to read: The dreadful apport manasataramgini.wordpress.com/2023/04/02/the… 7 hours ago
 BRICS to bricolage? 7 hours ago
 RT @blog_supplement: Some ruminations on asteroids and meteoritic falls manasataramgini.wordpress.com/2023/02/20/som… 8 hours ago
 It looks as though after a brief respite with the vigrahaNa by muShkavAn kastUrI, the "algorithm" has caught up with our account recently. 2 days ago
Top Posts

Recent Posts
 The dreadful apport
 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

Join 699 other subscribers
Meta
April 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
Tag Archives: mathematics
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
Leave a comment
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
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
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
Posted in Heathen thought, History, Life, Scientific ramblings
Tagged consciousness, DMT, firstperson, ghosts, history of science, mathematics, objective, subjective
Leave a comment
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
Leave a comment
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
Leave a comment
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
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
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
Posted in art, Scientific ramblings
Tagged arithemetic, factorization, mathematical entity, mathematics, modulo, prime numbers, recreational geometry
Leave a comment
A guillochelike 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
Posted in art, Scientific ramblings
Tagged arithmetic, curves, mathematical entity, mathematics, prime numbers, recreational geometry, spirograph, trigonometry
Leave a comment
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
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
Leave a comment
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
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
Posted in Scientific ramblings
Tagged arithmetic, arithmetic mean, geometric mean, mathematical entity, mathematics, numbers, prime numbers, sequence
Leave a comment
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
Leave a comment
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
Leave a comment
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
Posted in Scientific ramblings
Tagged arithmetic, Brahmagupta, cube, Euler, Gauss, Hindu mathematics, mathematical entity, mathematics, numbers, prime numbers, Ramanujan, square
Leave a comment
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
Leave a comment
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
Leave a comment
Sequences related to maps based on simple fractional functions
One of the pleasures of an unstructured youth in the precomputer 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
Posted in Scientific ramblings
Tagged arithmetic, complex numbers, fibonacci, fractal, fractals, mathematics, nArAyaNa, numbers, sequence, trigonometry
Leave a comment
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
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
Posted in art, Life, Scientific ramblings
Tagged mathematics, polynomial, series sum, Story
Leave a comment
A modern glance at Nārāyaṇapaṇḍita’s combinatorics1
For improved reading experience one may use the PDF version. Students of the history of Hindu mathematics are wellacquainted with Nārāyaṇapaṇḍita’s sophisticated treatment of various aspects of combinatorics and integer sequences in his Gaṇitakaumudī composed in 1356 CE. In that … 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
Leave a comment
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
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
Leave a comment
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
Posted in Scientific ramblings
Tagged Bronze ratio, conic sections, conics, ellipse, Golden Ratio, mathematical entity, mathematics, right triangles, sequence, square, square root
Leave a comment
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
Leave a comment
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
Leave a comment
Newton’s cows
Cultures with an IndoEuropean background have had a long history of symbiosis with the bovine animal since they started herding on the steppes in the Black SeaCaspian region. Indeed, the very emergence of the modern steppes of Eurasia is likely … Continue reading
Posted in Scientific ramblings
Tagged cows, determinant, mathematics, Newton, simultaneous equations
Leave a comment
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
Leave a comment
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
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 4symbol nucleic acid … Continue reading
Posted in Life, Scientific ramblings
Tagged arithmetic, decimal, decimal cycle, Fermat little theorem, fractions, mathematics, prime numbers, sequence
Leave a comment
A layman’s overview of the arithmetic of encryption
Life as an encryptiondecryption 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
Posted in Scientific ramblings
Tagged arithmetic, code, encryption, key, mathematics, numbers, prime numbers, RSA
Leave a comment
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
Leave a comment
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
Leave a comment
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
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
Posted in Scientific ramblings
Tagged mathematical entity, mathematics, numbers, square root, squares, triangles
Leave a comment
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
Posted in Scientific ramblings
Tagged Gauss, mathematical entity, mathematics, prime numbers, sequence
Leave a comment
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
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
Posted in Scientific ramblings
Tagged arithmetic, mathematical entity, mathematics, palindrome, prime numbers
Leave a comment
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 BhāskaraI in his commentary on Āryabhaṭa’s Āryabhaṭīya): dvayādyaiḥ ṣaṭparyantair ekāgraḥ yo ‘vaśiṣyate rāśiḥ … Continue reading
Posted in Heathen thought, Scientific ramblings
Tagged AryabhaTa, Golden Ratio, Hindu mathematics, kuTTaka, mathematics, prime numbers
Leave a comment
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
Leave a comment
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
Leave a comment
Counting primes, arithmetic functions, Ramanujan and the like
We originally wished to have a tailpiece 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
Posted in History, Scientific ramblings
Tagged complex numbers, Euler, Gauss, mathematics, prime numbers, Ramanujan, Riemann
Leave a comment
Our autodiscovery 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
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Wisdom from a tag system
The case of the mathematician Emil Post, like that of several others, indicates how the boundary between mania and mathematics can be a thin one. Nevertheless, Post discovered some rather interesting things that were to have fundamental implications the theory … Continue reading
Posted in art, Scientific ramblings
Tagged Collatz, complex numbers, computation, fractal, fractals, mathematics, tag systems
Leave a comment
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
Leave a comment
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
Journeying through the fractal slopes of mount Meru with twoseeded recursive sequences
The Hindus have been fascinated by sequences and series from the beginning of their civilizational memory recorded in the Veda. This continues down to the medieval mathematician Nārāyaṇa paṇḍita, who discovered a general formula (sāmāsika paṅkti) that can be to obtain the … Continue reading
Posted in art, Scientific ramblings
Tagged fractal, fractals, Golden Ratio, Hofstadter, mathematics, recreational geometry, recursion
Leave a comment
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
Posted in art, Scientific ramblings
Tagged Geometric construction, geometry, mathematics, recreational geometry
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
Trigonometric tangles2
We had earlier described our exploration of the spirograph, hypocycloids, epicycloids and related curves. In course of our study of the śaiva tantras of the kaula tradition we started thinking about a remarkable piece of imagery mentioned in them. Tantras … Continue reading
Posted in art, Heathen thought, Life, Scientific ramblings
Tagged circles, circular waves, curves, epicycloid, Generative Art, hypocycloid, mathematics, religion, trigonometry
Leave a comment
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
Āryabhaṭa and his sine table
Everyone and his son have written about Āryabhaṭa and his sine table. Yet we too do this because sometimes the situation arises where you have to explain things clearly to a layman who might have some education but is unfamiliar … Continue reading
Euler and Ramanujan: primes, near integers and cakravāla
Mathematician Watson who worked on the famed notebooks said regarding some of Srinivasa Ramanujan’s equations: “a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me … Continue reading
Posted in Scientific ramblings
Tagged almost integers, diophantine, e, Hindu mathematics, mathematics, Monstrous moonshine, near integers, Pell, pi, Ramanujan
Leave a comment
Early Hindu mathematics and the exploration of some second degree indeterminate equations
The following is merely a record of our exploration as a nonmathematician/noncomputer scientist of a remarkable (at least to us) class of numerical relationships. An equation like can be solved to obtain specific solutions as: . However, if we have … 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
Leave a comment
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
Leave a comment
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
Posted in art, Life, Scientific ramblings
Tagged attractors, chaos, fractal, fractals, geometry, Gingerbread Man, Henon, logistic map, Lozi, mathematics, strange attractors, tent map
Leave a comment
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
Posted in art, Life, Scientific ramblings
Tagged biology, developmental biology, evodevo, evolution, fractal, fractals, geometry, mathematics, recreational geometry, recursion, science, temple
Leave a comment
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
Leave a comment
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
Leave a comment
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
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Posted in art, Life, Scientific ramblings
Tagged Geometric construction, geometry, mathematics, philosophy, triangles, trigonometry, van Aubel
Leave a comment
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
Leave a comment
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
Partitions, perforations and tilings
We are “geometric” in our thinking – perhaps, we are hence a little more Greek or the old type Arya than the later Hindu (who is more algebraic) in mentality. Long back in college we were fascinated by implicit trigonometric … Continue reading
Posted in art, Life
Tagged ancient Hindu thought, Generative Art, mathematics, trigonometry
Leave a comment
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
Leave a comment