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REGULAR POLYGONAL COMPLEXES IN SPACE, I

DANIEL PELLICER and EGON SCHULTE
Transactions of the American Mathematical Society
Vol. 362, No. 12 (DECEMBER 2010), pp. 6679-6714
Stable URL: http://www.jstor.org/stable/40997221
Page Count: 36
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REGULAR POLYGONAL COMPLEXES IN SPACE, I
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Abstract

A polygonal complex in Euclidean 3-space E³ 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 ≥ 2 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 E³. 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 E³, and fully enumerates the simply flag-transitive complexes with mirror vector (1, 2). The second paper will complete the enumeration.

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