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Transactions of the American Mathematical Society

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Regular polygonal complexes in space, II


Authors: Daniel Pellicer and Egon Schulte
Journal: Trans. Amer. Math. Soc. 365 (2013), 2031-2061
MSC (2010): Primary 51M20; Secondary 52B15
DOI: https://doi.org/10.1090/S0002-9947-2012-05684-4
Published electronically: October 31, 2012
MathSciNet review: 3009652
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Abstract: Regular polygonal complexes in euclidean $ 3$-space $ \mathbb{E}^3$ are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The present paper and its predecessor describe a complete classification of regular polygonal complexes in $ \mathbb{E}^3$. In Part I we established basic structural results for the symmetry groups, discussed operations on their generators, characterized the complexes with face mirrors as the $ 2$-skeletons of the regular $ 4$-apeirotopes in $ \mathbb{E}^3$, and fully enumerated the simply flag-transitive complexes with mirror vector $ (1,2)$. In this paper, we complete the enumeration of all regular polygonal complexes in $ \mathbb{E}^3$ and in particular describe the simply flag-transitive complexes for the remaining mirror vectors. It is found that, up to similarity, there are precisely 25 regular polygonal complexes which are not regular polyhedra, namely 21 simply flag-transitive complexes and $ 4$ complexes which are $ 2$-skeletons of regular $ 4$-apeirotopes in $ \mathbb{E}^3$.


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Additional Information

Daniel Pellicer
Affiliation: 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
Email: schulte@neu.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05684-4
Keywords: Regular polyhedron, regular polytope, abstract polytope, complex, crystallographic group
Received by editor(s): May 5, 2011
Received by editor(s) in revised form: August 4, 2011
Published electronically: October 31, 2012
Additional Notes: This research was supported by NSF grant DMS–0856675
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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