A hierarchical strongly aperiodic set of tiles in the hyperbolic plane

We give a new construction of strongly aperiodic set of tiles in H2 , exhibiting a kind of hierarchical structure, simplifying the central framework of Margenstern’s proof that the Domino problem is undecidable in the hyperbolic plane [13]. Ludwig Danzer once asked whether, in the hyperbolic plane, where there are no similarities, there could be any notion of hierarchical tiling—an idea which plays a great role in many constructions of aperiodic sets of tiles in the Euclidean plane [1, 2, 4, 5, 6, 15, 17, 18]. It is an honor to dedicate this paper, which exposes a way to look at this question, to Herr Prof. Danzer in his 80th year. In 1966, R. Berger proved that the Domino Problem— whether a given set of tiles admits a tiling— is undecidable in the Euclidean plane, hanging his proof on the construction of an aperiodic set of tiles [2]. This first set was quite complex, with over 20,000 tiles; Berger himself reduced this to 104 [3] and in 1971, R. Robinson streamlined Berger’...

• Type: Journal
• Year: 2010
• Where: TCS
• Authors: Chaim Goodman-Strauss
• Keywords: Aperiodic Sets, Domino Problem, Hyperbolic Plane, TCS 2010, Theoretical Computer Science