Hexagonal Inflation Tilings and Planar Monotiles

Michael Baake,Uwe Grimm,Franz Gähler

doi:10.3390/sym4040581

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 focused 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.

Highlights

It is a face to face stone inflation, which means that each inflated tile is precisely dissected into copies of the prototile so that the final tiling is face to face
This rule defines an aperiodic tiling of the plane, but it does not originate from an aperiodic prototile set
Proposition 1 The inflation rule of Figure 2 defines a unique tiling LI class with perfect aperiodic local rules. The latter are formulated via an aperiodic prototile set, which consists of the three tiles from

Summary

Introduction

As is often the case in discrete geometry, following a proof might need some pencil and paper activity on the side of the reader; compare the introduction and the type of presentation in [1] This is true of arguments around local derivation rules, inflation properties and aperiodic prototile sets. When we prepared this manuscript, we rewrote known results on both tiling spaces in a way that emphasises their similarities, and mildly extended them, for instance by the percolation property of two derived parity patterns. The topological invariants and various other quantities for a comparison of the tilings are presented in Section 4, which is followed by some concluding remarks and open problems

Penrose’s Aperiodic Hexagon Tiling and Related Patterns

Taylor’s Inflation Tiling

Topological Invariants and the Structure of the Hulls

Outlook and Open Problems