Cauchy's Theorem (geometry) - Generalizations and Related Results

Generalizations and Related Results

• The result does not hold on a plane or for non-convex polyhedra in : there exist non-convex flexible polyhedra that have one or more degrees of freedom of movement that preserve the shapes of their faces. In particular, Connelly' sphere, a flexible non-convex polyhedron homeomorphic to a 2-sphere was discovered by Robert Connelly in 1977.
• Although originally proven by Cauchy in three dimensions, the theorem was extended to dimensions higher than 3 by Alexandrov (1950).
• Cauchy's rigidity theorem is a corollary from Cauchy's theorem stating that a convex polytope cannot be deformed so that its faces remain rigid.
• In 1974 Herman Gluck showed that in a certain precise sense almost all (non-convex) polyhedra are rigid.
• Dehn's rigidity theorem is an extension of the Cauchy rigidity theorem to infinitesimal rigidity. This result was obtained by Dehn in 1916.
• Pogorelov's uniqueness theorem is a result by Pogorelov generalizing Alexandrov's uniqueness theorem to general convex surfaces.
  • Alexandrov's uniqueness theorem is a result by Alexandrov (1950), weakening conditions of the Cauchy theorem to convex polytopes which are intrinsically isometric.
  • The analogue uniqueness theorem for smooth surfaces was proved by Cohn-Vossen in 1927.
• Bricard's octahedra are self-intersecting flexible surfaces discovered by a French mathematician Raoul Bricard in 1897.