This Wanderer tiling is the first of an infinite series of substitution tilings by Joan Taylor based on paper-folding sequences. It uses a single square tile with four distinct rotations and two distinct reflections. Here we use colours to distinguish left-handed (brown) from right-handed (white) tiles. In the substitution rule the orientation of the tiles is indicated by a line in the interior of the tiles. In the large patch below these lines and all edges are omitted since the interesting feature are the patterns produced by white resp.

Limitperiodic Polytopal Tiles Rep Tiles Self Similar Substitution

This Wanderer tiling is one in an infinite series of substitution tilings by Joan Taylor based on paper-folding sequences. It uses a single square tile with four distinct rotations and two distinct reflections. Here we use colours to distinguish vertical (blue) from horizontal (ochre) tiles. In the substitution rule the orientation of the tiles is indicated by a line in the interior of the tiles, the chirality (left-handed vs right-handed) is indicated by a point.

Limitperiodic Polytopal Tiles Rep Tiles Self Similar Substitution

It is hard to find substitution tilings with dense tile orientations (like the pinwheel tiling) that uses a single prototile with fractal boundary. The fractal version of he pinwheel tiling by Frank and whittaker uses 13 different prototiles. An extensive computer search revealed that there are only two substitutions producing tilings with dense tile orientations using five copies of a single fractal prototile. Both substitutions use the same prototile. One substitution is shown here, the other one is obtained by dissecting the blue two tile patch in the supertile in exactly the opposite way into two of the tiles.

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