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. Table of Geometry, from the 1728 Cyclopaedia. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...

Statement

Let 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.

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). Euclid (Greek: ), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 – April 10, 1813; b. ... Ernst Steinitz (June 13, 1871 – September 29, 1928) was an German mathematician. ... Aleksandr Danilovich Aleksandrov (Russian: Александр Данилович Александров, alternative transliterations: Alexandr or Alexander (first name), and Alexandrov (last name)) (August 4, 1912–July 27...

Generalizations and related results

• The result doe not hold on a plane or for non-convex polyhedra in . 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.

Aleksandr Danilovich Aleksandrov (Russian: Александр Данилович Александров, alternative transliterations: Alexandr or Alexander (first name), and Alexandrov (last name)) (August 4, 1912–July 27... Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ... Aleksandr Danilovich Aleksandrov (Russian: Александр Данилович Александров, alternative transliterations: Alexandr or Alexander (first name), and Alexandrov (last name)) (August 4, 1912–July 27... See: Intrinsic and extrinsic properties (philosophy) Intensive and extensive properties (physics) This is a disambiguation page: a list of articles associated with the same title. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... Stefan or Stephan Cohn-Vossen (28 May 1902-25 June 1936) was a German-Jewish mathematician, now best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie. ... A.V. Pogorelov. ... Flexible polyhedra are polyhedral surfaces which allow continuous non-rigid deformations such that all faces remain rigid. ... Flexible polyhedra are polyhedral surfaces which allow continuous non-rigid deformations such that all faces remain rigid. ...

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.