Bellows conjecture

Flexible polyhedron

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 . 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 .

Bellows conjecture

In the late 1970's Connelly and D. Sullivan formulated the Bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by .

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.

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.
  • Ralph Alexander, Lipschitzian Mappings and Total Mean Curvature of Polyhedral Surfaces, Transactions of the AMS 288 (1985), 661-678

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