Fundamental tilings with marked cubes in spaces of constant curvature
type: Item
instance of: scholarly article
title: Fundamental tilings with marked cubes in spaces of constant curvature
zbMATH Open document ID: 0858.52009
DOI: 10.1007/BF00052190
published in: Acta Mathematica Hungarica
publication date: 1996-07-07T00:00:00Z
review text: 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 \({2\pi \over 6})\).
reviewed by: Charles Leytem
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 897168
zbMATH Keywords: cube tiling
zbMATH Keywords: periodic tilings
zbMATH Keywords: regular polyhedron
zbMATH Keywords: space of constant curvature
author: I. Prok
MaRDI profile type: Publication
cites work: Q3973296
cites work: Q3855356
cites work: On Poincaré's theorem for fundamental polygons
cites work: Q3679824
cites work: Polyhedron complexes with simply transitive group actions and their realizations
cites work: Q4328199
cites work: Q4006285
cites work: Q4328200
cites work: Q4285990
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description: scientific article; zbMATH DE number 897168