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Abstract:
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This dissertation sets forth a method for classifying, up to euclidean similarity, fully-transitive polyhedra which have crystallographic symmetry groups. Branko Grunbaum's definition of polyhedron is used, in which the faces are "hollow" - i.e., a face is a collection of edges. The approach is algorithmic: in a euclidean space $R\sp{n}$, assume we have classified the crystallographic groups up to conjugation by similarity transformations. For each similarity class of groups G, we follow a procedure which will generate all polyhedra in $R\sp{n}$ on which G acts fully-transitively. The method also assures that there is no duplication of examples.
Following the presentation of this method, there is a discussion of the topology of these polyhedra. It is shown how to associate a non-compact surface with each polyhedron, and how to determine where this surface fits in the classification of non-compact 2-manifolds. Following this discussion, the method described is applied to the case of 2-dimensional crystallographic groups, and a complete classification of fully-transitive tilings is obtained. |
