Combinatorics of a fractal tiling family

TL;DR

This paper explores the enumeration of distinct fractal tiling configurations generated from a recursive binary tiling process, revealing a finite set of unique fractals for specific initial tiles using combinatorial methods.

Contribution

It introduces a combinatorial approach to count all possible fractal configurations from recursive tiling, including exact counts for certain initial tiles and partial results for symmetric cases.

Findings

Exactly 232 unique fractals for a 2x2 initial tile.

Finite number of configurations for each initial tile.

Partial enumeration results for symmetric 3x3 initial tiles.

Abstract

In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For each initial tile choice the number of possible configurations is finite. This combinatorial problem recalls the famous Escher tiling problem [2]. By using the Burnside lemma we show that there are exactly 232 really different fractals when the initial tile is a particular 2x2 matrix. Partial results are also presented in the 3x3 case when the initial tile presents some symmetry properties.