An aperiodic hexagonal tile

TL;DR

This paper demonstrates that a single hexagonal prototile with specific matching rules can fill space without ever forming a repeating pattern, revealing new insights into aperiodic tilings and their structural properties.

Contribution

It introduces a new aperiodic hexagonal prototile with enforced local matching rules, providing two independent proofs of aperiodicity and exploring its relation to Penrose tilings.

Findings

The tiling forms a union of honeycombs with exponentially increasing lattice constants.

Two local isomorphism classes are consistent with the matching rules.

Alternative shapes enforce matching rules without connectedness or in three dimensions.

Abstract

We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three--dimensional prototile.