Planar aperiodic tile sets: from Wang tiles to the Hat and Spectre monotiles

TL;DR

This paper reviews the history and recent breakthroughs in planar aperiodic tile sets, highlighting the discovery of simple monotiles like the Spectre that tile the plane without periodicity.

Contribution

It provides a historical overview and emphasizes recent advances, including the introduction of the simple Spectre monotile in 2023.

Findings

The Spectre tile is a simple monotile that enforces aperiodicity.

The development of aperiodic tiles spans over 60 years.

Recent discoveries include the Hat and Spectre monotiles.

Abstract

A brief history of planar aperiodic tile sets is presented, starting from the Domino Problem proposed by Hao Wang in 1961. We provide highlights that led to the discovery of the Taylor--Socolar aperiodic monotile in 2010 and the Hat and Spectre aperiodic monotiles in 2023. The Spectre tile is an amazingly simple monotile; a single tile whose translated and rotated copies tile the plane but only in a way that lacks any translational periodicity. We showcase this breakthrough discovery through the 60$+$ years that aperiodic tile sets have been considered.