Classification of dodecahedral space forms
type: Item
instance of: scholarly article
title: Classification of dodecahedral space forms
zbMATH Open document ID: 0926.52021
author: I. Prok
published in: Beiträge zur Algebra und Geometrie
publication date: 1999-11-21T00:00:00Z
full work available at URL: https://eudml.org/doc/231491
review text: This paper is concerned with regular tilings of spherical or hyperbolic 3-space, by (congruent regular) dodecahedra, and their classification by systematic combinatorial search. The author describes the theoretical background, the algorithm, and presents the results obtained by a computer-aided enumeration which is based on the number \(n=3,5,6\) of dodecahedra along each edge and a characteristic pairing of the twelve faces of a dodecahedron, induced by groups of (fixed point free) symmetries of the tiling. There are 147~essentially different (\textit{non-equivariant}) pairings (for each of which a million cases has to be checked). For \(n=3\) two distinct orientable spherical manifolds are found; for \(n=5\) the enumeration yields 12~compact orientable hyperbolic manifolds, of which at least 7 are distinct because they have distinct first homology groups; and for \(n=6\) there are 78~non-compact hyperbolic manifolds (10~orientable and 68~non-orientable), of which at least~38 (7~and 31, respectively) are distinct (due to distinct homology groups or distinct cusp structures). The details of these results are presented in several tables. They confirm and extend previous enumerations. A typical sample entry is illustrated and discussed in detail. This paper can be viewed as part of a series of papers (on related topics) by the author and Emil Molnár.
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1208022
zbMATH Keywords: (regular) dodecahedron
zbMATH Keywords: regular tilings
zbMATH Keywords: (classification) hyperbolic space
zbMATH Keywords: spherical space
zbMATH Keywords: polyhedral manifolds
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1208022