Fractal dual substitution tilings

TL;DR

This paper introduces a method to generate infinitely many fractal boundary substitution tilings from a given tiling, revealing new geometric and combinatorial features while maintaining local derivability.

Contribution

It presents a novel approach to construct fractal boundary tilings using graph iterated function systems, enabling analysis of their geometric and topological properties.

Findings

Constructed new substitution tilings with fractal boundaries

Established mutual local derivability with original tilings

Computed fractal dimensions and Čech cohomology of tilings

Abstract

Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct border-forcing substitution rules. For a large class of examples we show that the combinatorial dual tiling has a realization as a substitution tiling. Since the boundaries of our new tilings are fractal we are led to compute their fractal dimension. As an application of our techniques we show how to compute the \v{C}ech cohomology of a (not necessarily border-forcing) tiling using a graph iterated function system of a fractal tiling.