Fractal Images as Number Sequences I An Introduction

TL;DR

This paper introduces a method to represent fractal curves as integer sequences derived from grid walks, establishing a unique normalized sequence and creating an encyclopedia of fractals through ordered sequences.

Contribution

It presents a novel approach to encode fractal curves as unique integer sequences and translates grid morphisms into permutations, enriching fractal analysis.

Findings

Sequence normalization yields unique fractal representations

Grid morphisms correspond to signed permutations

Ordered sequences form an encyclopedia of fractals

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.