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Petrie Schemes

Published online by Cambridge University Press:  20 November 2018

Gordon Williams
Gordon Williams*
Affiliation:
Moravian College, Department of Mathematics and Computer Science, Room 219 Priscilla Payne Hurd Academic Complex, Bethlehem, Pennsylvania 18018, U.S.A., e-mail: gordon@moravian.edu
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Abstract

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Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum–Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

Keywords

Primary: 52B15secondary: 52B05Petrie polygonpolyhedronpolytopeabstract polytopeincidence complexregular polytopeCoxeter group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 4 , 01 August 2005 , pp. 844 - 870
Copyright
Copyright © Canadian Mathematical Society 2005

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