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Topic: Rigidity of Convex Polytopes
Replies: 6   Last Post: Jun 20, 1994 9:29 AM

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Walter Whiteley

Posts: 418
Registered: 12/3/04
Re: Rigidity of Convex Polytopes
Posted: Jun 20, 1994 9:29 AM
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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) first-order/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 first-order rigidity is defined by a set of
polynomial equations in the vertices of a simplicial polyhedron.
If one choice of coordinates makes a polynomial non-zero then 'almost all'
choices work - this rigidity is 'generic' and with probability 1 a random
realization will be first-order 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 first-order 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), 431-465.

Theorem 8.6 If a strictly convex d-polytope (d>2) is formed in d-space by
(i) placement of a joint at each vertex;
(ii) replacement of each 2-face by a subframework which triangulates
the polygon, then the resulting framework is statically (first-order) rigid
in d-space.

Theorem 8.7 If a strictly convex d-polytope (d>2) is formed in d-space 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
(first-order) rigid on the joints associated with that face, in their k-space,
then the resulting framework is statically (first-order) rigid in d-space.

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 d-polytope!

Generic Rigidity

For generic rigidity, something much stronger holds:
A corollary of a theorem of Alan Fogelsanger (unpublished Ph.D. thesis,
1988) says that all simplicial (d-1) manifolds are generically rigid
in d-space. So for example, triangulated tori, projective planes etc.
are generically rigid in 3-space. (There is no problem with
self-intersection of the 'surface' when making the 'framework'.)
I have a TeX file giving a rewrite of his proof, which uses
'minimal homology cycles'.

Non-convex realizations:

Bob Connelly has an example of an immersed, non-convex 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, 333-335.

R. Connelly, A flexible sphere, Math. Intelligencer 1, 130-131.

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

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