Characterization of planar pseudo-self-similar tilings. (English) Zbl 0997.52012

Priebe, Natalie M.; Solomyak, B.

Discrete Comput. Geom. 26, No. 3, 289-306 (2001).

A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a non-exact substitution rule: the substitution for a tile is not geometrically similar to itself. An example is the Penrose tiling drawn with ‘thick’ and ‘thin’ rhombi.
The authors prove that a nonperiodic repetitive tiling of the plane is pseudo-self-similar if and only if it has a finite number of derived Voronoi tilings up to similarity. To establish this characterization, the authors settle a conjecture of E. A. Robinson by providing an algorithm which converts any pseudo-self-similar planar tiling into a self-similar tiling with some useful topological properties.

Reviewer: Charles Leytem (Cruchten)

MSC:

52C20	Tilings in \(2\) dimensions (aspects of discrete geometry)
37A30	Ergodic theorems, spectral theory, Markov operators
37B15	Dynamical aspects of cellular automata
68R10	Graph theory (including graph drawing) in computer science

Keywords:

dynamical system; complex Perron number; pseudo-self-similar tiling; substitution rule; Voronoi tilings