Rigidity and Symmetry pp 253-278 | Cite as

# Rigidity of Regular Polytopes

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## Abstract

A (geometric) regular polygon { is a

*p*} in a euclidean space can be specified by a*fine Schläfli symbol*{*p*}, where$$\displaystyle{p = \frac{r} {s_{1},\ldots,s_{k}}}$$

*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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