Cauchy's theorem (geometry)

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Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes with congruent corresponding faces are congruent. This is a foundational result in rigidity theory.

1 Statement
2 History
3 Generalizations and related results
4 References

[edit] Statement

Let $P, Q \subset \Bbb R^3$ be combinatorially equivalent 3-dimensional convex polytopes, i.e. convex polytopes with isomorphic face lattices. Suppose corresponding faces are congruent, i.e. equal up to a rigid motion. Then $P$ and $Q$ are congruent.

[edit] History

The result originated in Euclid's Elements, where solids are called equal if the same holds for their faces. This version of the result was proved by Cauchy in 1813 based on earlier work by Lagrange. A technical mistake was found by Steinitz in 1920's and later corrected by him (1928) and Alexandrov (1950). A definitive modern version of the proof was given by Stoker (1968).

[edit] Generalizations and related results

The result does not hold on a plane or for non-convex polyhedra in $\Bbb R^3$ . It 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 rigidiry theorem to infinitesimal rigidity. This result was obtained by Dehn in 1916.
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 of Alexandrov's uniqueness theorem for smooth surfaces was proved by Cohn-Vossen in 1927.
Pogorelov's uniqueness theorem is a result by Pogorelov generalizing Alexandrov's uniqueness theorem to general convex surfaces.
Bricard's octahedra are self-intersecting flexible surfaces discovered by a French mathematician Raoul Bricard in 1897.
Connelly' sphere is a flexible non-convex polyhedron (embedded surface homeomorphic to a 2-sphere). It was discovered by Robert Connelly in 1977.

[edit] References

A.L. Cauchy, "Recherche sur les polyèdres - premier mémoire", Journal de l'Ecole Polytechnique 9 (1813), 66–86.
M. Dehn, "Über die Starreit konvexer Polyeder" (in German), Math. Ann. 77 (1916), 466-473.
A.D. Alexandrov, Convex polyhedra, GTI, Moscow, 1950. English translation: Springer, Berlin, 2005.
J.J. Stoker, "Geometrical problems concerning polyhedra in the large", Comm. Pure Appl. Math. 21 (1968), 119-168.
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.