Lots of Aperiodic Sets of Tiles

TL;DR

This paper significantly expands the catalog of explicit aperiodic tile sets by introducing hundreds of new constructions from a small set of tiles, enabling easier assembly and configuration for various applications.

Contribution

It presents generalized, robust techniques for creating a large variety of aperiodic tilings from minimal atomic subsets, increasing known examples by hundreds-fold.

Findings

Generated 25,380 distinct substitution tiling systems.

Identified three non-periodic, non-unique decomposition systems.

Provided easily assembled tile sets for complex tiling structures.

Abstract

Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational applications, but its study and use has been hindered by a lack of easily generated examples. Through readily generalized, robust techniques for controlling hierarchical structure, we increase the catalogue of explicit constructions of aperiodic sets of tiles hundreds-fold, in lots, easily assembled and configured from atomic subsets of 211 tiles, enforcing 25,380 distinct "domino" substitution tiling systems. Among these, we notice three non-periodic, non-unique decomposition substitution tiling systems.