Rigidity and polyhedral combinatorics
December 3 to December 7, 2007
American Institute of Mathematics,
Palo Alto, California
and Igor Pak
This workshop will be devoted to polyhedral objects in Euclidean spaces.
Specifically, which metric or combinatorial properties must change or
remain constant under certain classes of deformations, such as
bending, folding, stretching, or flexing? When it is impossible to
carry out one or another of these operations in some prescribed
manner, an object is said to possess some sort of "rigidity". This
can occur, for example, when too many metric or combinatorial
properties are forced to remain constant under a given operation. The
past several years have seen a number of significant advances, such as
a proof of the bellows conjecture, constructions of unfoldings for
polytopes of all dimensions, and an algorithm for Alexandrov's theorem
on 3-polytopes. But with those advances has come increasing
recognition that many basic questions remain largely open,
particularly in higher dimensions.
Geometry and algebra of flexible objects.
In 1977, R. Connelly showed that there exists a surface in
Euclidean three-space composed of rigid triangles, hinged along
their edges, that is flexible. In 1995 I. Sabitov showed the very
remarkable "bellows property" for such flexible surfaces: the
volume they bound is constant during the motion; thus there is no
perfect mathematical bellows. But even more remarkable is that
there is a monic polynomial, whose coefficients are themselves
polynomials in the edge lengths, which is satisfied by the volume.
This is a new fundamental property of triangulated surfaces in
Euclidean three-space; it is an algebraic identity, but it does not
seem to be part of what is known in algebraic geometry. Another
example is the area of a cyclic polygon in the plane, which is
integral over the ring generated by the edge lengths (a favorite
problem studied by David Robbins just before his death, and solved
just after his death by Fedorchuk-Pak and others). These suggest
problems such as the following, some of which could benefit from
key insights using algebraic geometry of configuration spaces --
perhaps the kind of algebraic geometry considered by Develin,
Martin, and Reiner in their work on rigidity.
- Is there a higher-dimensional analogue of the Sabitov
polynomial? Only partial results are known here.
- Are there other geometric quantities that are integral over
rings generated by other geometric quantities? For example, is the
n-dimensional volume of an n-simplex integral over the ring
generated by the k-dimensional volumes of its k-faces for k=2, n=5?
- Are there non-trivial examples of flexible triangulated
surfaces for dimensions greater than 4?
- With the example of flexible surfaces, which are known,
not only is the volume constant, but the surface seems to be the sum
of rigid surfaces and flexible surfaces of zero volume, where the
surfaces are regarded as singular simplicial cycles in the sense of
algebraic topology. Is this always true?
- For flexible surfaces, is the Dehn invariant constant during
the motion? If so, this would imply that the regions bounded by
the surfaces are equivalent by dissection, as in Hilbert's third
Folding and bending.
There is a tradition, centering around the school of A. D. Alexandrov
in Russia, of interest in the rigidity of convex polyhedral surfaces
in three-space. The issues are related to the classical theorem of
Cauchy in 1813 that says there is a unique convex realization of any
convex polyhedron in three-space once the the faces and their
adjacencies are determined. Alexandrov himself showed a basic
existence result: any intrinsically convex polyhedral surface can be
realized as an extrinsically convex polyhedron or doubly covered
polygon in three-space. Recently, Bobenko and Izmestiev have shown
how to implement this result by an effective new proof of Alexandrov's
theorem. Their methods produce an explicit algorithm to obtain any
three-dimensional polytope by appropriately gluing any given foldout
("development") of its boundary in the plane. Many intriguing
questions related to this theory remain.
- Given a convex polyhedral surface, can one cut along
some of the edges and unfold it into a polyhedral disk in the plane?
The analogous question makes sense in higher dimensions, but even for
polyhedra in three-space it is open. Miller and Pak have related
results in all dimensions, but the cuts are not restricted to lie
along ridges (edges, in the case of surfaces).
- Can one unfold polyhedral surfaces in three-space
expansively (continuous blooming)? Does the source unfolding bloom in
- J-M Schlenker has a conjecture that extends Cauchy's
result to the case when the vertices only are in convex position, and
the interior can be triangulated without adding more vertices. It
would be interesting to look at some cases to check the conjecture.
- What is the maximum volume bounded by a surface
submetric to a given polyhedral surface? Polyhedral surfaces never
bound the maximum volume, by a theorem of I. Pak, but can approximate
it; what does the limiting surface look like? For example, is it
smooth except at finitely many points?
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
Papers arising from the workshop: