Tilings Encyclopedia | Aperiodic Monotile

Aperiodic Monotile

The shape of an Aperiodic Monotile alone (the so called “Einstein”) ensures that every tiling of the plane you can build from it will be an aperiodic tiling.

CAP

A substitution for the aperiodic hat monotile found in 2022 and published in [SMCG2024] , see also [BGS2023] . The “metatiles” in the first paper have some free parameter. Most of these parameters yield a substitution that is not selfsimilar. Here we show the selfsimilar version of the substitution rule. This tiling was used to derive the cut-and-project description of the hat tiling in [BGS2023] .

Aperiodic Monotile Self Similar Substitution

Hat

A pretty simple aperiodic monotile found in 2022 and published in [SMCG2024] . The tile is a tridecagon (13-gon) built from 16 basic triangles with edge lengths $1,2, \sqrt{3}$. The shape alone ensures that every tiling of the plane you can build from this tile will be aperiodic. In this sense it has the simplest matching rules imaginable. This one is not really a substitution tiling. However, its properties (e.g., being an aperiodic monotile) are deduced from related substitution tilings, see for instance the CAP tiling.

Aperiodic Monotile

Hexagonal Aperiodic Monotile

In 2009 Joan Taylor (Burnie, Tasmania) found a decoration of the hexagon, which - together with few local matching rules - allows only aperiodic tilings of the plane. This was probably the best example of an aperiodic monotile before the discovery of the Hat tiling. This decorated hexagonal tile, together with the local matching rule, is shown in Fig 1 of a joint paper by J. Socolar and J. Taylor: http://arxiv.org/pdf/1003.4279v2.

Aperiodic Monotile Self Similar Substitution

Spectre

The Spectre tiling was derived from the Hat tiling. While the aperiodic monotiles of the Hat tiling appear in two chiralities, the aperiodic monotiles of the Spectre tiling have the same chirality. As the Hat tiling this one is also not really a substitution tiling. The patch was generated with the Spectre Tile Generator, see [KAP2024] for details. A relative simple way to generic the Spectre tiling manually was published in [SSC2024] .

Aperiodic Monotile