Hexagonal inflation tilings and planar monotiles

TL;DR

This paper reviews and compares two hexagonal inflation tilings that serve as aperiodic monotiles, highlighting their structural properties, relation to classic tilings, and recent geometric and topological insights.

Contribution

It provides a comparative analysis of two inflation tilings as planar monotiles, including new results on their geometry and topology of tiling spaces.

Findings

Both tilings have aperiodic local rules and model set structures.

The tilings are related to the classic half-hex tiling.

New geometric and topological results are presented for the tiling spaces.

Abstract

Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focussed on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.