The Trilobite and Crab: a full explanation

TL;DR

This paper discusses the complexity behind proving that the trilobite and crab tile sets are among the simplest known aperiodic tilings, despite their minimal number of tiles.

Contribution

It provides a detailed explanation and proof of the aperiodicity of the trilobite and crab tile sets, highlighting their simplicity and the complexity of the proof.

Findings

The trilobite and crab are among the simplest aperiodic tile sets.

The proof of their aperiodicity is surprisingly complex.

These sets consist of only two tiles in eight translation classes.

Abstract

The trilobite and crab are among the very simplest aperiodic sets of tiles known: two tiles in eight translation classes. Yet the proof that they are an aperiodic set is surprisingly complex.