Fractal Images as Number Sequences I An Introduction

Arie Bos

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arXiv · 2207.12942

Fractal Images as Number Sequences I An Introduction

Arie Bos

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Abstract

In this article, we considered a fractal image as a fractal curve, that is, as a walk on a grid in Euclidean space $\R^d$. We placed integers on the generating vectors of a grid, such that opposite directions have opposite numbers. This numbering system converts a curve on that grid into a sequence of integers, corresponding with the curve's edges. The corresponding sequence contains the same fractal structure, i.e., an approximant of the curve corresponds to that of the sequence. We introduced a normalized sequence which is unique for a curve. The morphisms of the grid generators were translated into signed permutations on the alphabet of all the numbers used. By ordering the fractal sequences, we obtained an encyclopedia of fractals. A variety of examples and images enriched the text.

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Arie Bos. 2022-07-26. Fractal Images as Number Sequences I An Introduction. https://arxiv.org/abs/2207.12942

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