Tag Archives: geometry

The Platonic culmination of Euclid and the pentagon-hexagon-decagon identity

Why did great sage Pāṇini compose the Aṣṭādhyāyī? There were probably multiple reasons but often you hear people say that he wanted to give a complete description of the Sanskrit language. That was probably one of his reasons but was … Continue reading

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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

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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

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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

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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

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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

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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

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