Regular polygonal complexes in space, I
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- by Daniel Pellicer and Egon Schulte PDF
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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 $r\geqslant 2$ 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.References
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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
- Email: pellicer@matmor.unam.mx
- Egon Schulte
- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
- MR Author ID: 157130
- ORCID: 0000-0001-9725-3589
- Email: schulte@neu.edu
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 6679-6714
- MSC (2010): Primary 51M20; Secondary 52B15, 20H15
- DOI: https://doi.org/10.1090/S0002-9947-2010-05128-1
- MathSciNet review: 2678991
Dedicated: With best wishes for Branko Grünbaum