Martin Schlottmann

Discovered Tilings

Preview Square-triangle
Square-triangle

The substitution tilings which are most relevant as models for physical quasicrystals are 5-fold, 8-fold, 10-fold and 12-fold symmetric ones. In the 5-fold (resp. 10-fold) case, there are the variations of the Penrose rhomb tilings and the Tuebingen triangle, for the 8-fold case there are the Ammann-Beenker tilings. These examples use a minimal set of prototiles, in the sense that the number of prototiles is equal to the algebraic degree of the corresponding inflation factor.

Preview Tuebingen Triangle
Tuebingen Triangle

Beside the Penrose rhomb tilings (and its variations), this is a classical candidate to model 5-fold (resp. 10-fold) quasicrystals. The inflation factor is - as in the Penrose case - the golden mean, $\frac{\sqrt{5}}{2} + \frac{1}{2}$. The prototiles are Robinson triangles, but these tilings are not mld to the Penrose tilings. The relation is different: The Penrose rhomb tilings are locally derivable from the Tübingen Triangle tilings. These tilings were discovered and studied thoroughly by a group in Tübingen, Germany, thus the name [BKSZ90] .

Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution