Regular Polygonal Complexes in Space, I

Details Regular Polygonal Complexes in Space, I Regular Polygonal Complexes in Space, I

2009-06-05

arxiv.org/abs/0906.1178v1

Mathematics

Metric Geometry

Transactions American Mathematical Society (to appear), 41 pages, 9 figures

Scientific paper

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular if its symmetry group is transitive on the flags. The present paper and its successor describe a complete classification of regular polygonal complexes in 3-space. In particular, the present paper establishes basic structure results for the symmetry groups, discusses geometric and algebraic aspects of operations on their generators, characterizes the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in 3-space, and fully enumerates the simply flag-transitive complexes with mirror vector (1,2). The second paper will complete the enumeration.

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