The Tiling of the Hyperbolic 4D Space by the 120-cell is Combinatoric

Maurice Margenstern (Laboratoire d'Informatique Théorique et Appliquée, EA 3097, Universite de Métz, France)

Abstract: The splitting method was defined by the author in [Margenstern 2002a], [Margenstern 2002d]. It is at the basis of the notion of combinatoric tilings. As a consequence of this notion, there is a recurrence sequence which allows us to compute the number of tiles which are at a fixed distance from a given tile. A polynomial is attached to the sequence as well as a language which can be used for implementing cellular automata on the tiling.

The goal of this paper is to prove that the tiling of hyperbolic 4D space is combinatoric. We give here the corresponding polynomial and, as the first consequence, the language of the splitting is not regular, as it is the case in the tiling of hyperbolic 3D space by rectangular dodecahedra which is also combinatoric.

Keywords: cellular automata, hyperbolic plane

Categories: F.1.1, F.1.3