Flexible polyhedron
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In geometry, a flexible polyhedron is a polyhedral surface which allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron 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 an octahedron. The first example of a non-self-intersecting surface in R3, the Connelly sphere, was discovered by Robert Connelly in 1977.
[edit] Bellows conjecture
In the late 1970's Connelly and others formulated the Bellows conjecture stating that the volume of a flexible polyhedron 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 valuations.
[edit] Scissor congruence
Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. If this is true then it would lead to a scissor congruence under flexing for the volume together with the Dehn invariant. The special case of mean curvature has been 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
