Three-dimensional orbifolds and space groups

J.H. Conway, O. Delgado Friedrichs, D.H. Huson, and W. Thurston
2000 (submitted)

Web links: http://www.mathematik.uni-bielefeld.de/~huson/papers.html

Tiling space by Platonic solids I

O. Delgado Friedrichs and D.H. Huson
Discrete and Computational Geometry21:299-315 (1999)

Web links: http://www.mathematik.uni-bielefeld.de/~huson/papers.html

Abstract

There exist precisely 914, 58 and 46 equivariant types of tile-transitive tilings of 3-dimensional Euclidean space by topological cubes, octahedra and tetrahedra, that fall in to 11, 3, and 9 topological families, respectively. Representatives are described for all topological families. A general method for obtaining results of this kind is introduced.

Orbifold Triangulations and Crystallographic Groups

O. Delgado Friedrichs and D.H. Huson
Per. Math. Hung., 34(1-2):29-55, 1997

Published in: Special Volume of Periodica Math. Hung. on Packing, Covering and Tiling

Web links:

Abstract

Given a triangulation of a 3-dimensional euclidean orbifold, e.g. in terms of the Delaney symbol of a periodic tiling, a method is discussed for identifying the isomorphism type of the corresponding space group. Of the 219 types of groups, 175 can be recognized solely by considering the orbifold graph associated with the given triangulation. Simple abelian invariants distinguish between the remaining 44 cases. The graphs and invariants are listed.

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