Space fillers of higher genus
type: Item
instance of: scholarly article
title: Space fillers of higher genus
zbMATH Open document ID: 0809.52029
DOI: 10.1016/0097-3165(94)90114-7
published in: Journal of Combinatorial Theory. Series A
publication date: 1994-10-30T00:00:00Z
review text: Based on regular skew polyhedra, isohedral (i.e., tile-transitive) tilings by polyhedral handlebodies of (euclidean, hyperbolic, and spherical) 3-space are constructed which exhibit the same symmetries as the underlying polyhedron. More precisely, the main result shows that to every regular skew polyhedron \(\{2l, 2m, \mid r\}\) with \((l - 2) (r - 2) < 4\) and \((m - 2)(r - 2) < 4\) there exists a tiling by polyhedral toroids such that the tiles bijectively correspond to the ``holes'' of the polyhedron. Each tile has \(5r\) vertices, \(8r\) edges, and \(3r\) 2-faces. Of the 2-faces, \(2r\) are convex pentagons (of two different kinds) and \(r\) are non-convex hexagons. From these tilings, tilings by spacefillers of a higher genus can be constructed. The symmetry group of these tiles is either \([r, m]\) or \([l, r]\) and coincides with the stabilizer of the tile (in the symmetry group of the tiling). Besides the general construction, the author discusses in detail some special cases, and a generalization of the construction to topological tilings. He concludes with some remarks on open problems.
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 52B70
zbMATH DE Number: 679884
zbMATH Keywords: toroidal spacefillers
zbMATH Keywords: spacefillers of higher genus
zbMATH Keywords: isohedral tilings of 3-space
zbMATH Keywords: regular skew polyhedra
author: Egon Schulte
MaRDI profile type: Publication
full work available at URL: 0097-3165(94)90114-7
OpenAlex ID: W2093483743
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description: scientific article; zbMATH DE number 679884