en>ChrisGualtieri: General Fixes using AWB

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General Fixes using AWB

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'''Cauchy's theorem''' is a theorem in [[geometry]], named after [[Augustin Louis Cauchy|Augustin Cauchy]]. It states that
[[convex polytope]]s in three dimensions with [[congruence (geometry)|congruent]] corresponding faces must be congruent to each other. That is, any [[Net (polyhedron)|polyhedral net]] formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape.

This is a foundational result in [[rigidity theory (structural)|rigidity theory]]: one consequence of the theorem is that, if one makes a physical model of a [[convex polyhedron]] by connecting together rigid plates for each of the polyhedron faces with flexible hinges along the polyhedron edges, then this ensemble of plates and hinges will necessarily form a rigid structure.

==Statement==
Let ''P'' and ''Q'' be ''combinatorially equivalent'' 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic [[face lattice]]s. Suppose further that each pair of corresponding faces from ''P'' and ''Q'' are congruent to each other, i.e. equal up to a rigid motion. Then ''P'' and ''Q'' are themselves congruent.

==History==
The result originated in [[Euclid|Euclid's]] ''[[Euclid's Elements|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 [[Joseph Louis Lagrange|Lagrange]]. A technical mistake was found by [[Ernst Steinitz|Steinitz]] in 1920's and later corrected by him (1928) and [[Aleksandr Danilovich Aleksandrov|Alexandrov]] (1950). A definitive modern version of the proof was given by Stoker (1968).

==Generalizations and related results==
* The result does not hold on a plane or for non-convex polyhedra in <math>\Bbb R^3</math>: 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 [[Aleksandr Danilovich Aleksandrov|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 [[Max Dehn|Dehn]] in 1916.
* '''Pogorelov's uniqueness theorem''' is a result by [[Aleksei Pogorelov|Pogorelov]] generalizing Alexandrov's uniqueness theorem to general convex surfaces.
** '''Alexandrov's uniqueness theorem''' is a result by [[Aleksandr Danilovich Aleksandrov|Alexandrov]] (1950), weakening conditions of the Cauchy theorem to convex polytopes which are [[Intrinsic and extrinsic properties|intrinsically]] [[isometry|isometric]].
** The analogue uniqueness theorem for smooth surfaces was proved by [[Stephan Cohn-Vossen|Cohn-Vossen]] in 1927.
* ''Bricard's octahedra'' are self-intersecting [[Flexible polyhedra|flexible surfaces]] discovered by a French mathematician [[Raoul Bricard]] in 1897.

== References ==
* A.L. Cauchy, "Recherche sur les polyèdres - premier mémoire", ''Journal de l'Ecole Polytechnique'' '''9''' (1813), 66–86.
* M. Dehn, [http://dz-srv1.sub.uni-goettingen.de/cache/toc/D37460.html "Über die Starreit konvexer Polyeder"] (in German), ''Math. Ann.'' '''77''' (1916), 466-473.
* A.D. Alexandrov, ''Convex polyhedra'', GTI, Moscow, 1950. [[English language|English]] translation: Springer, Berlin, 2005.
* J.J. Stoker, "Geometrical problems concerning polyhedra in the large", ''[[Communications on Pure and Applied Mathematics|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.

[[Category:Theorems in discrete geometry]]
[[Category:Polytopes]]
[[Category:Mathematics of rigidity]]
[[Category:Euclidean geometry]]
[[Category:Theorems in convex geometry]]

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en>ChrisGualtieri: General Fixes using AWB