Priebe, Natalie M.; Solomyak, B. Characterization of planar pseudo-self-similar tilings. (English) Zbl 0997.52012 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) Cited in 1 ReviewCited in 14 Documents 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 PDF BibTeX XML Cite \textit{N. M. Priebe} and \textit{B. Solomyak}, Discrete Comput. Geom. 26, No. 3, 289--306 (2001; Zbl 0997.52012) Full Text: DOI OpenURL