Space fillers of higher genus
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.
scientific article; zbMATH DE number 679884
scientific article