Flexible polyhedron

In geometry, a flexible polyhedron is a polyhedral surface that 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 (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 (1977).

Bellows conjecture

In the late 1970s 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 I. Kh. Sabitov (1995) using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by Robert Connelly, I. Sabitov, and Anke Walz (1997).

Scissor congruence

Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. This is known as the strong bellows conjecture. Preservation of the Dehn invariant is known to be equivalent to scissors congruence of the enclosed region under flexing. The special case of mean curvature has been proved by Ralph Alexander.

Generalizations

Flexible polyhedra in the 4-dimensional Euclidean space and 3-dimensional Lobachevsky space were studied by Hellmuth Stachel. In November 2009 it was not known whether flexible polyhedra exist in the Euclidean space of dimension $n\geq 5$.

References

• Bricard, R. (1897), "Mémoire sur la théorie de l'octaèdre articulé", J. Math. Pures Appl. 5 (3): 113–148, http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1897_5_3_A5_0 
• Connelly, Robert (1977), "A counterexample to the rigidity conjecture for polyhedra", Publications Mathématiques de l'IHÉS 47 (47): 333–338, doi:10.1007/BF02684342, ISSN 1618-1913, MR0488071, http://www.numdam.org/item?id=PMIHES_1977__47__333_0 
• 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.
• Weisstein, Eric W., "Bellows conjecture" from MathWorld.
• Weisstein, Eric W., "Flexible polyhedron" from MathWorld.
• Connelly, Robert; Sabitov, I.; Walz, Anke (1997), "The bellows conjecture", Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry 38 (1): 1–10, ISSN 0138-4821, MR1447981, http://www.mat.ub.es/EMIS/journals/BAG/vol.38/no.1/1.html 
• Ralph Alexander, Lipschitzian Mappings and Total Mean Curvature of Polyhedral Surfaces, Transactions of the AMS 288 (1985), 661–678
• Sabitov, I. Kh. (1995), "On the problem of the invariance of the volume of a deformable polyhedron", Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 50 (2): 223–224, ISSN 0042-1316, MR1339277 
• H. Stachel, Flexible octahedra in the hyperbolic space, in Non-Euclidean geometries. János Bolyai memorial volume(Eds. A. Prékopa et al.). New York: Springer. Mathematics and its Applications (Springer)581, 209–225 (2006). ISBN 0-387-29554-2.
• H. Stachel, Flexible cross-polytopes in the Euclidean 4-space, J. Geom. Graph. 4, No.2 (2000), 159–167.

Popular level

• D. Fuchs, S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics