Fundamental tilings with marked cubes in spaces of constant curvature
The author considers periodic tilings with a regular polyhedron \(P\) in a space of constant curvature. He analyses how \(P\) can occur as a fundamental domain for a space group up to equivalence. This is equivalent to decomposing the symmetry group of the tiling as a product of the symmetry group of \(P\) and of the space group. To do this the author sets up a computer program which determines the set of equivalence classes of all possible face identifications of \(P\). Moreover, if the tiling \(T\) with regular polyhedra \(P\) exists, then the dihedral angles of \(P\) are all equal to \({2\pi \over n}\). Thus, for each integer \(n\), the program checks further whether the word length of each cycle transformation divides \(n\). If this is the case a tiling with regular \(P\) exists and the group is a spherical, euclidean or hyperbolic space group depending on the value of \(n\). This program is used to enumerate all tilings by the cube in spherical space (dihedral angle \({2\pi \over 3})\), in euclidean space (dihedral angle \({2\pi\over 4})\), and hyperbolic space (dihedral angles \({2\pi \over 5}\) and \({2\pi \over 6})\).
scientific article; zbMATH DE number 897168
scientific article