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Rigidity of Regular Polytopes

  • Peter McMullenEmail author
Chapter
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Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

A (geometric) regular polygon {p} in a euclidean space can be specified by a fine Schläfli symbol {p}, where
$$\displaystyle{p = \frac{r} {s_{1},\ldots,s_{k}}}$$
is a generalized fraction; here, \(0 \leq s_{1} < \cdots < s_{k} \leq \frac{1} {2}r\). This means that {p} projects onto planar polygons \(\{ \frac{r} {s_{j}}\}\) (reduced to lowest terms) in orthogonal planes, with \(\infty = \frac{1} {0}\) giving the linear apeirogon and 2 the digon (line segment). More generally, it may be possible to specify the shape or similarity class of a geometric regular polytope by means of a fine Schläfli symbol, whose data contain information about certain regular polygons occurring among its vertices in terms of generalized fractions. If so, then the fine Schläfli symbol is called rigid. This paper gives various criteria for rigidity; for instance, the classical regular polytopes are rigid. The theory is also illustrated by several examples. It is noteworthy, though, that a combinatorial description of a regular polytope – a presentation of its symmetry group – can differ considerably from its fine Schläfli symbol.

Keywords

Polytope Abstract Regular Realization Fine Schläfli symbol Rigidity 

Subject Classifications

51M20 52C25 

Notes

Acknowledgements

This paper expands on original material contained in the mini-course The Geometry of Regular Polytopes given by me during a workshop in the Thematic Program on Discrete Geometry and Applications at the Fields Institute in October 2011. My grateful thanks go to the Institute for its support and hospitality during that visit.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University College LondonLondonUK

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