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Transactions of the American Mathematical Society

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Regular polygonal complexes in space, I

Authors: Daniel Pellicer and Egon Schulte
Journal: Trans. Amer. Math. Soc. 362 (2010), 6679-6714
MSC (2010): Primary 51M20; Secondary 52B15, 20H15
Published electronically: July 14, 2010
MathSciNet review: 2678991
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Abstract: A polygonal complex in Euclidean $ 3$-space $ \mathbb{E}^3$ is a discrete poly- hedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number $ r\geq 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 $ \mathbb{E}^3$. 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 $ \mathbb{E}^3$, and fully enumerates the simply flag-transitive complexes with mirror vector $ (1,2)$. The second paper will complete the enumeration.

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Additional Information

Daniel Pellicer
Affiliation: Department of Mathematics, York University, Toronto, Ontario, Canada M3J 1P3
Address at time of publication: Instituto de Matematicas, Unidad Morelia, CP 58089, Morelia, Michoacan, Mexico

Egon Schulte
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Regular polyhedron, regular polytope, abstract polytope, complex
Received by editor(s): December 15, 2008
Received by editor(s) in revised form: June 3, 2009
Published electronically: July 14, 2010
Additional Notes: The second author was supported by NSA-grant H98230-07-1-0005
Dedicated: With best wishes for Branko Grünbaum
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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