Rigidity of Regular Polytopes

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.