Squiral

Discovered by

Info

This substitution arises from a reptile with infinitely many straight edges, cf. [GS87]. It answers the question ‘Are there selfsimilar substitution tilings where the prototiles have infinitely many straight edges?’ positively. The colours of the tiles indicate their chirality. The substitution rule is shown for the right handed tile only, the substitution of the left-handed tile is the reflected image. One can easily define a substitution using only the right handed tile, but this generates periodic tilings only.

By methods in [FS], one can show that these tilings are not cut and project tilings.

Substitution Rule

Rule Squiral

Patch

Patch Squiral

References

[FS]
Frettlöh, D and Sing, B
Computing modular coincidences
Preprint

[GS87]
Grünbaum, B and Shephard, G.C.
Tilings and Patterns
W.H. Freeman 1987, MR0857454