Chaim Goodman-Strauss

Discovered Tilings

Preview Chaim's Cubic PV
Chaim's Cubic PV

Part of an infinite series, where most tilings in this series are not flc, this one is the exception. The reason is that the inflation factor is a - real - PV number. By an argument in [PR] this forces flc. Interestingly, the shape of the tiles can vary. That is, there is one free parameter $l$ , $0 < l < 1+s$, and the smallest prototile is the triangle with sides $1,s,l$ ($s$ the largest root of $x^{3}-x-1$).

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution

Preview Triangle Duo
Triangle Duo

Infinite Rotations Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Squeeze

One possible version of a substitution rule with a free parameter: The upper tip of the three triangles can be shifted arbitrarily in horizontal direction, the result is always a self-similar substitution. The inflation multiplier is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.

Self Similar Substitution Polytopal Tiles Plastic Number

Preview Non Unique Decomposition
Non Unique Decomposition

Preview Goodman-Strauss 7-fold rhomb
Goodman-Strauss 7-fold rhomb

Whereas it is simple to generate rhomb tilings with n-fold symmetry by the cut and project method, it can be hard to find a substitution rule for such tilings. Here we see a rule for n=7. This one was later generalized by E. Harriss to arbitrary n.

Finite Rotations Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

Preview Chaim's Square Tiling
Chaim's Square Tiling

A simple non-periodic substitution tiling with just one decorated prototile mentioned in the extended version of [Goo98] . Find here the vector graphic.