Gruber P. Convex and Discrete Geometry

17 Rigidity 297
in E k+1 with small spheres centred at the vertices and applying the result in S k ,
the polytopal surface turns out to be rigid at each vertex. This immediately yields
Cauchy’s theorem in E k+1 . Since the rigidity theorem holds in E 3 it thus holds in
E d for all d ≥ 3. See Alexandrov [16] and Pogorelov [804]. For more information
compare Connelly [218].
Flexible Non-convex Polytopal Spheres and the Bellows Conjecture
Cauchy’s rigidity theorem implies the following. A closed convex polytopal surface
in E 3 with rigid facets and hinges along the edges cannot be flexed. Also the nonconvex
example of Legendre of Fig. 17.1 does not admit a flexing. More generally,
Euler [311] conjectured that
a closed spatial figure allows no changes as long as it is not ripped apart.
Bricard [167] gave an example of a flexible octahedral surface in E 3 . Unfortunately,
it suffers from the defect that the surface is not embedded, that is, it has
self-intersections. The next step of this story is due to Gluck [380] who showed
that the closed, simply connected polytopal embedded surfaces in E 3 are generically
rigid, that is, at least the large majority is rigid. Finally, Connelly [217], surprisingly,
specified a flexible, embedded polytopal sphere in E 3 . A simpler example is due to
Steffen [954].
The so-called bellows conjecture asserts that the volume of a flexible polytopal
sphere does not change while flexing. Sabitov’s [871] affirmative answer is based on
an interesting formula in which the volume is expressed as a polynomial in terms of
edge-lengths. This formula may be considered as a far-reaching extension of formulae
of Heron for the area of a triangle and Euler for the volume of a tetrahedron. For
a survey see Schlenker [889].
17.2 Rigidity of Frameworks
A framework is a system of rods in E d with joints at common endpoints such that
the rods can rotate freely. A basic question is to decide whether a given framework
is rigid or flexible.
Early results on frameworks in the nineteenth century are due to Maxwell [700],
Peaucellier [787], Kempe [573], Bricard [167]. Throughout the twentieth century
and, in particular, in the last quarter of it, a multitude of results on frameworks were
given. Amongst others, these results deal with rigidity and infinitesimal rigidity, with
stresses and self-stresses.
In this section a result of Asimow and Roth [41] will be presented, showing that
a framework consisting of the edges and vertices of a proper convex polytope in E 3
is rigid if and only if all facets are triangular. We follow Roth [859].
Definitions
An abstract framework F consists of two sets, the set of vertices V = V(F) =
{1,...,v} and the set of edges E = E(F), the latter consisting of two-element subsets