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Re: Rigidity of Convex Polytopes
Posted:
Jun 20, 1994 9:29 AM


This is a long reply to the query about extensions of Alexandrov's Theorem to higher dimensions. There are published analogs  which I will reference below.
First there is a choice about which type of 'rigidity' you are interested in: (i) local uniqueness of the configuration (combinatorics and lengths, dimensions of faces), up to congruence; (ii) firstorder/static rigidity  strong local uniqueness of the configuration (implies (i)); (iii) global rigidity of the configuration, up to congruence; (iv) other local rigidities (second order ... ). (v) generic rigidity (Failure of firstorder rigidity is defined by a set of polynomial equations in the vertices of a simplicial polyhedron. If one choice of coordinates makes a polynomial nonzero then 'almost all' choices work  this rigidity is 'generic' and with probability 1 a random realization will be firstorder rigid.)
Cauchy's Theorem (1814) says that two triangulated strictly convex polyhedra, with the same combinatorics and the same edge lengths are congruent.
Alexandrov's Theorem says that this uniqueness remains true if you take a general strictly convex polyhedron, add any number of vertices along the 'natural' edges of this polyhedron, then triangulate each natural face with all its vertices.
(It is a discrete form of the uniqueness of convex realizations of an intrinsic metric on the surface.)
The proofs of these theorems also give proofs of the firstorder rigidity of the realizations  though the first explicit statement, for convex triangulated spheres is due to Max Dehn, using a different proof. (Buckminster Fuller had nothing to do with this  he just had good PR and emphasized the possibilities of using few edge lengths and few types of joints in construction.)
Infinitesimal Rigidity in higher dimensions:
The infinitesimal form of Cauchy's Theorem appears to be contained in a footnote of Efimov (of the same Russian School as Alexandrov). I have proven the analog of Alexandrov's Theorem in higher dimensions:
W. Whiteley, Infinitesimally rigid polyhedra I: Statics of frameworks, Trans. AMS 285 (1984), 431465.
Theorem 8.6 If a strictly convex dpolytope (d>2) is formed in dspace by (i) placement of a joint at each vertex; (ii) replacement of each 2face by a subframework which triangulates the polygon, then the resulting framework is statically (firstorder) rigid in dspace.
Theorem 8.7 If a strictly convex dpolytope (d>2) is formed in dspace by (i) placement of a joint at each vertex; (ii) placement of new joints along natural faces of dimensions <k<d; (b) replacement of each (k)face by a subframework which is statically (firstorder) rigid on the joints associated with that face, in their kspace, then the resulting framework is statically (firstorder) rigid in dspace.
I suspect that the uniqueness within the world of strictly convex polytopes also applies  but I have not looked at it. What follows immediately is that convexity implies the uniqueness of the geometry at each vertex. It is likely that a simple induction transports this to the entire structure.
A curious sidelight is that this rigidity result was used by Gil Kalai to prove the best lower bounds on the number of edges of a strictly convex dpolytope!
Generic Rigidity
For generic rigidity, something much stronger holds: A corollary of a theorem of Alan Fogelsanger (unpublished Ph.D. thesis, Cornell, 1988) says that all simplicial (d1) manifolds are generically rigid in dspace. So for example, triangulated tori, projective planes etc. are generically rigid in 3space. (There is no problem with selfintersection of the 'surface' when making the 'framework'.) I have a TeX file giving a rewrite of his proof, which uses 'minimal homology cycles'.
Nonconvex realizations:
Bob Connelly has an example of an immersed, nonconvex sphere which is flexible (violating any form of rigidity you might propose).
R. Connelly, A counter example to the rigidity conjecture for polyhedra, Inst. Haut. Etud. Sci. Publ. Math. 47 1978, 333335.
R. Connelly, A flexible sphere, Math. Intelligencer 1, 130131.
Therefore some hypothesis is required.
The following recent book has a good bibliography on rigidity, but addresses only some of the issues in generic rigidity:
J. Graver, B. Servatius, H. Servatius. Combinatorial Rigidity, AMS Graduate Studies in Math 1993.
As may be obvious, this is one of my areas of research. I would be happy to respond to any further queries on this topic, as there is a large literature on 'rigidity' (we have draft chapters for a three volume work on the subject).
Walter Whiteley Department of Mathematics and Statistics York University North York, Ontario whiteley@mathstat.yorku.ca



