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mAnasa-taraMgiNI supplement
- A further image of rAjAdhirAja from shailendra Java https://t.co/y29AaNz9Ir 15 hours ago
- A further gaNesha pendant from the champA kingdom in what's today Vietnam. https://t.co/amxnJg48VB 16 hours ago
- Interesting graphs in this article by Haidt: jonathanhaidt.substack.com/p/mental-healt… While Haidt himself likes to play equal-equal i… twitter.com/i/web/status/1… 16 hours ago
- There is an interesting parallel across steppe-derived civilizations that persisted across time & even post-settlem… twitter.com/i/web/status/1… 1 day ago
Top Posts
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Recent Posts
- 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
- Transcripts of conversations: the addiction principle:
- Phantom impressions-1
- A note on Śrī, Viṣṇu and śṛṅgāra
- Are civilizational cycles the norm?
- On the rise of the mātṛkā-s and the goddess Cāmuṇḍā
- Huns, Uralics, and empires of the steppe
- Some observations on the Lekkerkerker-Zeckendorf decomposition of integers
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Tag Archives: fractals
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 notes on the Henon-Heiles Hamiltonian system
Anyone familiar with dynamical systems knows of the Henon-Heiles (HH) system. What we are presenting here is well-known stuff about which reams of material have been written. However, we offer certain tricks for visualizing this system that make it easy … Continue reading
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
Chaotic behavior of some floor-squared maps
Consider the one dimensional maps of the form: , where is the fractional part of What will be evolution of a under this map when or ? We can see that for it will tend converge. However, the behavior is … Continue reading
Posted in Scientific ramblings
Tagged attractors, chaos, fractal, fractals, geometry, Golden Ratio, mathematics, numbers
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Sequences related to maps based on simple fractional functions
One of the pleasures of an unstructured youth in the pre-computer era was what we called calculator games. As our father took his prized calculator with him to work we only got a little time with it in the evenings. … Continue reading
Posted in Scientific ramblings
Tagged arithmetic, complex numbers, fibonacci, fractal, fractals, mathematics, nArAyaNa, numbers, sequence, trigonometry
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Chaos, eruptions and root-convergence in one-dimensional maps based on metallic-sequence generating functions
bronze_bouncer Over the years we have observed or encountered certain natural phenomena that are characterized by rare, sudden eruptive behavior occurring against a background of very low amplitude fluctuations. We first encountered this in astronomy: most remarkably, in the constellation … Continue reading
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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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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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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Division-multiplication parabolas, triplications, and quadratic residues
Introduction Many strands of our investigations on conic-generating integer sequences, word fractals and cellular automaton models for pattern formation came together in an unexpected manner while investigating a simple integer sequence. While some of these connections have have been known … Continue reading
Posted in Scientific ramblings
Tagged arithmetic, conics, fractal, fractals, geometry, mathematical entity, mathematics, Mephisto-Waltz, numbers, parabola, prime numbers, sequence
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The geometric principles behind discrete dynamical systems based on the generalized Witch of Agnesi
Consider the family of curves defined by the equation following parametric equation , where and It defines a family of probability distribution functions (PDFs): This can be seen from the above equations because Figure 1 Examples of these PDFs are … Continue reading
Posted in Scientific ramblings
Tagged arithmetic, chaos, chaotic flows, dynamics, fractal, fractals, geometry, Gumowski-Mira, mathematics, numbers, prime numbers
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Some novel observations concerning quadratic roots and fractal sequences
Disclaimer: To our knowledge we have not found the material presented here laid out here presented in completeness elsewhere. However, we should state that we do not follow the mathematical literature as a professional and could have missed stuff. Introduction … Continue reading
Posted in Scientific ramblings
Tagged floor, fractal, fractals, integers, quadratic equation, quadratic root, sequence, square root
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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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The Satija-Ketoja system
Satija and Ketoja discovered an interesting dynamical system in course of the study of the Schrödinger equation for one electron in a two dimensional periodic lattice on a uniform magnetic field. While this equation and its variants have several uses … Continue reading
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 maps-2 While the story of the chaotic 2D attractors began with the simple-looking maps of Henon and Lozi, by the early 1990s the high-point was … Continue reading
Posted in art, Scientific ramblings
Tagged attractors, chaos, chaotic flows, fractal, fractals, geometry, mathematics
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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, Mendes-France, 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 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
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
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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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Journeying through the fractal slopes of mount Meru with two-seeded 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
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Trigonometric tangles-3: the fractals
See also: https://manasataramgini.wordpress.com/2016/05/06/the-astroid-the-deltoid-and-the-fish-within-the-fish/ 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
Some visions of infinity from the past and our times
The great Hindu mathematician and astronomer Bhāskara-II’s work preserves a high-point 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 multi-parameter 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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Some reminiscences of our study of chaotic maps-2
Continued from part-1 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
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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 self-similarity over all scales could be generated by a rather … Continue reading
Posted in art, Life, Scientific ramblings
Tagged biology, developmental biology, evo-devo, evolution, fractal, fractals, geometry, mathematics, recreational geometry, recursion, science, temple
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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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Some lessons we learned from 3-color totalistic cellular automata
Cellular automata (CA) have attracted people’s attention to different degrees over the past several decades since the early work of pioneers like Ulam and von Neumann. Remarkably von Neumann played with his earliest versions of CA using a graph paper … Continue reading
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 gaṇaka-rāja
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
A journey through fractal objects
Posted in art
Tagged 3D Julia, fractal, fractals, geometry, Mandelbrot, recreational geometry, recursion
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The Mandel-diamond: crystals emerge from an amorphous background
A closer look at realms within this “Terra Mysterium” Crystals by the shore-line: Crystals in the crevice – realm of craters:
Virtual viruses and virtual cell lysis
Posted in art
Tagged art and science, cell lysis, fractals, icosahedron, IFS, Mandelbulb, simulating life, virus
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Computational driftwood
The above object is a gnarled piece of driftwood. Only that it never floated in the sea and exists only in the virtual realm – a pure mathematical abstraction – a combination of three functions. Some would say it is … Continue reading
Posted in art
Tagged driftwood, fractals, Mandelbulb, mathematical entity, Platonic ideals, virtual reality
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rudrAkSha-s from the shuddha-bhuvanAdhvan-s or Platonic viruses
Based on the Icosahedron mated with the Mandelbulb of power 8 Its dual, the Dodecahedron, mated with the Mandelbulb of power 8
Posted in art
Tagged dodecahedron, fractals, icosahedron, IFS, Mandelbulb, mental images, Platonic ideals, rudrAkSha, virus
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