An aperiodic monotile

TL;DR

This paper introduces a new class of aperiodic monotile shapes, including the 'hat' polykite, demonstrating their ability to tile the plane without periodicity through substitution rules and geometric incommensurability, solving a longstanding problem.

Contribution

The authors construct a continuum of aperiodic polygons, including the 'hat' shape, and prove their aperiodicity using novel geometric and computational methods.

Findings

The 'hat' polykite can form hierarchical, aperiodic tilings.

A continuum of aperiodic polygons is demonstrated.

A new geometric incommensurability argument proves generic aperiodicity.

Abstract

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.