Tilings Encyclopedia

The tilings encyclopedia shows a wealth of examples of nonperiodic substitution tilings.
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D. Frettlöh, F. Gähler, E. Harriss: Tilings encyclopedia, https://tilings.math.uni-bielefeld.de/

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Preview Tiling with Transcendental Inflation Multiplier
Tiling with Transcendental Inflation Multiplier

An one-dimensional substitution rule that uses an infinite number of proto tiles and yields a transcendental inflation multiplier. The inflation factor is approximately $2.7899$. The substitution rules are given by: $T_{0}\rightarrow T_{0},T_{1}$ $T_{1}\rightarrow 3T_{0},T_{2}$ $T_{2}\rightarrow 2T_{0},T_{1},T_{3}$ $T_{3}\rightarrow T_{0},T_{2},T_{4}$ $T_{4}\rightarrow 2T_{0},T_{3},T_{5}$ $T_{5}\rightarrow T_{0},T_{4},T_{6}$ $T_{6}\rightarrow T_{0},T_{5},T_{7}$ $T_{k}\rightarrow (1+f\left(k\right))T_{0},T_{k-1},T_{k+1}$ with $f\left(k\right)$ as the Thue-Morse sequence. The corresponding substitution matrix can be written as: $1 3 2 1 2 1 1 2 2 …$

One Dimensional Self Similar Substitution

Preview CAP

A substitution for the aperiodic hat monotile found in 2022 and published in SMCG2023, 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.

Preview Hat

A pretty simple aperiodic monotile found in 2022 and published in SMCG2023. 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.

Preview Millars n-fold
Millars n-fold

J. Millar discovered a set of tilings with patches of dihedral symmetry $D_2n$ and inflation multiplier $\sqrt{2 + 2 \cos(\frac{\pi}{n})}$, which is the same inflation multiplier as of the Generalized Godreche-Lancon-Billard Binary. All interior angles of all prototiles are integer multiples of $\frac{\pi}{n}$. All prototiles have sides with unit length. All tilings have a prototile in the shape of a rhomb with interior angle $\frac{\pi}{n}$. The longer diagonal also defines the inflation multiplier.

Finite Rotations Polytopal Tiles Finite Local Complexity