Flexible polyhedra
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Flexible polyhedra are polyhedral surfaces which allow continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such polyhedra cannot be convex (this is also true in higher dimensions).
The first examples of flexible polyhedra (now called Bricard's octahedra) were discovered by Raoul Bricard in 1897. They are self-intersecting surfaces isometric to octahedra. The first examples of non-self-intersecting surface (Connelly shpere) were discovered by Robert Connelly in 1977.
[edit] Bellows Conjecture
In the late 1970's Connelly and other formulated the so called bellows conjecture stating that the volume of flexible polyhedra is invariant under flexing. This conjecture was proved by I.Kh. Sabitov in 1996 using the elimination theory first for polyhedra homeomorphic to a sphere, and then for in general orientable 2-dimensional polyhedral surfaces. Connelly et al. later found a simple proof using the theory of places.
[edit] Scissor congruence
Connelly conjectured that the Dehn invariant of flexible polyhedra is invariant under flexing. With the volume this would imply the scissor congruence under flexing. The special case of mean curvature is proved by Ralph Alexander.
[edit] References
- R. Connelly, "The Rigidity of Polyhedral Surfaces", Mathematics Magazine 52 (1979), 275-283
- R. Connelly, "Rigidity", in Handbook of Convex Geometry, vol. A, 223-271, North-Holland, Amsterdam, 1993.
- Eric W. Weisstein, Bellows Conjecture at MathWorld.
- R. Connelly, I. Sabitov, A. Walz, The Bellows Conjecture 38 (1997), 1-10.
- Ralph Alexander, Lipschitzian Mappings and Total Mean Curvature of Polyhedral Surfaces, Transactions of the AMS 288 (1985), 661-678
