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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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: