Tilings Encyclopedia

The tilings encyclopedia aims to become a useful reference for things tiling related. The first goal is to give a database of known substitution rules. We welcome all feedback.

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Preview Kolakoski-(3,1) variant B, with dual
Kolakoski-(3,1) variant B, with dual

The substitution $a \rightarrow abcc, b \rightarrow a, c \rightarrow bc$ is a member of the MLD class of the Kolakoski-(3,1) sequence. The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other. The dual substitution scales by about 1.485, and rotates clockwise by about 81.22°.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Kolakoski-(3,1) variant A, with dual
Kolakoski-(3,1) variant A, with dual

The substitution $a \rightarrow bcc, b \rightarrow ba, c \rightarrow bc$ is a member of the MLD class of the Kolakoski-(3,1) sequence. As the reversed substitution generates the same hull, it is mirror symmetric. The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other. The dual substitution scales by about 1.485, and rotates clockwise by about 81.22°.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Kolakoski-(3,1) symmmetric variant, dual
Kolakoski-(3,1) symmmetric variant, dual

The substitution $a \rightarrow aca, b \rightarrow a, c \rightarrow b$ has palindromic and thus mirror symmetric variant of the Kolakoski-(3,1) substitution, which is in the same MLD class, along with the further variants A (mirror symmetric) and B (with its mirror image). The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Kolakoski-(3,1), with dual
Kolakoski-(3,1), with dual

The substitution $a \rightarrow abc, b \rightarrow ab, c \rightarrow b$ is closely related to the Kolakoski-(3,1) sequence, and is one of the examples whose windows (dual tiles, Rauzy fractals) have been analysed in detail [BaS04] . It is MLD to the mirror symmetric variant given by the palindromic substitution $a \rightarrow aca, b \rightarrow a, c \rightarrow b$. As a consequence, the Kolakoski-(3,1) substitution is MLD to its mirror image, even though it is not mirror symmetric itself.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution