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 selfsimilarity of the tilings, one finds larger periodic subsets in the tiling, covering 93,75%, 98,44%… of the plane. Thus, the tiling is limitperiodic.
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.
Finite Rotations Rep Tiles
The three letter substitution rule whose scaling is the smallest PV number, 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
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 the smallest possibility, using three prototiles. Examples with more prototiles are Chord-Quadrangle-4-3, Chord-Quadrangle-5-3, Chord-Quadrangle-5-5, and so on. (Search for chord-quadrangle.)