Gruber P. Convex and Discrete Geometry

292 Convex Polytopes
17 Rigidity
Rigidity in the context of convex geometry can be traced back to Book XI of Euclid
[310], but the first proper result seems to be Cauchy’s [197] rigidity theorem for
convex polytopal surfaces in E 3 . Cauchy’s seminal result gave rise to a series of
developments.
First, to flexibility results for general closed polytopal surfaces, including the
result of Bricard [167] on flexible immersed octahedra, the examples of flexible embedded
polytopal spheres of Connelly [217] and Steffen [954] and the more recent
results of Sabitov [871]. Second, to rigidity in the context of differential geometry
and, much later, to rigidity in Alexandrov’s intrinsic geometry of closed convex surfaces,
the latter culminating in the rigidity theorem of Pogorelov [803] for closed
convex surfaces in E 3 , see Sect. 10.2. Third, to rigidity and infinitesimal rigidity
for frameworks starting with Maxwell [700] and with contributions by Dehn [252],
Alexandrov [16], Gluck [380] and Asimow and Roth [41].
In this section we present Cauchy’s rigidity theorem for convex polytopal surfaces
and a result of Asimov and Roth on the flexibility, resp. rigidity of convex
frameworks.
References to the voluminous literature may be found in the following books
and surveys: Alexandrov [16], Efimov [288, 289], Pogorelov [805], Ivanova-
Karatopraklieva and Sabitov [538, 539] (differential geometry, intrinsic geometry
of convex surfaces), Ivanova-Karatopraklieva and Sabitov [539], Connelly [218]
(non-convex polytopal surfaces), Roth [859], Graver, Servatius and Servatius [391],
Maehara [677], Graver [392] (convex and non-convex frameworks).
17.1 Cauchy’s Rigidity Theorem for Convex Polytopal Surfaces
In Book XI of the Elements of Euclid [310] Definition 10 is as follows.
Equal and similar solid figures are those contained by similar planes equal in multitude
and magnitude.
The intensive study of Euclid in modern times led to the question whether this was
a definition or, rather, a theorem saying that two polytopal surfaces are congruent
if their corresponding facets are congruent. Legendre [639] definitely thought that
it was a theorem, see the comment of Heath [310], but he was also aware that this
theorem could not hold without additional assumption. This is shown by the two
polytopal surfaces in Fig. 17.1. Legendre thought that convexity might be such an
additional assumption. He drew the attention of young Cauchy to this problem and
Cauchy [197] gave a positive answer. Minor errors in his ingenious proof were corrected
later on, see [218].
Cauchy won high recognition with this result. He submitted it for publication to
the Institute, as the Académie des sciences was called then. The referees Legendre,
Carnot, and Biot gave an enthusiastic report which concluded with the following
words: