Folklore
Discovered Tilings
A simple rule to generate nonperiodic tilings with one prototile, a triangle with angles 30°, 60°, 90°. It looks pretty much periodic: the hexagonal patches cover 75% of the plane, and this part is clearly periodic. The triangles in between the hexagons destroy the periodicity. But, by the …
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 …
Finite Rotations Rep Tiles
The three letter substitution rule whose scaling is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.
Though it might not look it at first glance, the Rauzy fractal is connected.
This can be shown using the method of A. Siegel described in [Sie04].
The Rauzy fractal:
One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles Plastic Number
A member of an infinite family of substitution rules for similar quadrangles possessing two right interior angles at opposite vertices. A big copy of such a quadrangle can be divided into (smaller) similar quadrangles in several ways. Some of them are compatible with a substitution rule. This one is …
A classic simple substitution rule with Rauzy Fractal:
One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles