A chiral aperiodic monotile

TL;DR

This paper introduces a new class of chiral aperiodic monotiles, including the Spectres, which only admit non-periodic tilings with a fixed chirality, advancing the understanding of aperiodic tiling shapes.

Contribution

It demonstrates the existence of strictly chiral aperiodic monotiles, including the Spectres, which only produce non-periodic tilings with a specific chirality, a novel finding in tiling theory.

Findings

The equilateral shape related to the 'hat' is a weakly chiral aperiodic monotile.

Modified shapes called Spectres are strictly chiral aperiodic monotiles.

Spectres admit only chiral non-periodic tilings based on hierarchical substitution.

Abstract

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.