Abstract
Every regular polytope has the remarkable property that it inherits all symmetries of each of its facets. This property distinguishes a natural class of polytopes which are called hereditary. Regular polytopes are by definition hereditary, but the other polytopes in this class are interesting, have possible applications in modeling of structures, and have not been previously investigated. This paper establishes the basic theory of hereditary polytopes, focussing on the analysis and construction of hereditary polytopes with highly symmetric faces.
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Acknowledgements
A substantial part of this article was written while we visited the Fields Institute for extended periods of time during the Thematic Program on Discrete Geometry and Applications in Fall 2011. We greatly appreciated the hospitality of the Fields Institute and are very grateful for the support we received. Mark Mixer was Fields Postdoctoral Fellow in Fall 2011. Egon Schulte was also supported by NSF-grant DMS–0856675, and Asia Ivić Weiss by NSERC. Finally, we wish to thank Peter McMullen and Barry Monson for helpful comments.
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Mixer, M., Schulte, E., Weiss, A.I. (2014). Hereditary Polytopes. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_14
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