Cauchy's theorem is a theorem in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, named after

Augustin Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...

. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any

polyhedral net In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra a ...

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 fundamental result in rigidity theory: one consequence of the theorem is that, if one makes a physical model of a

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...

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 A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

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. To see that convexity is necessary, consider a

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

. One can "push in" a vertex to create a nonconvex polyhedron that is still combinatorially equivalent to the regular icosahedron. Another way to see it, is to take the pentagonal pyramid around a vertex, and reflect it with respect to its base.

History

The result originated in Euclid's ''

Elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of ...

'', 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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaErnst Steinitz,

Isaac Jacob Schoenberg Isaac Jacob Schoenberg (April 21, 1903 – February 21, 1990) was a Romanian-American mathematician, known for his invention of splines. Life and career Schoenberg was born in Galați. He studied at the University of Iași, receiving his ...

, and

Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...

. The corrected proof of Cauchy is so short and elegant, that it is considered to be one of the

Proofs from THE BOOK ''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mat ...

Generalizations and related results

* The result does not hold on a plane or for non-convex polyhedra in

\mathbb R^3

: 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, the Bricard octahedra are self-intersecting flexible surfaces discovered by a French mathematician Raoul Bricard in 1897. The ''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 Alexandrov (masculine, also written Alexandrow) or Alexandrova (feminine) may refer to: * Alexandrov (surname) (including ''Alexandrova''), a Slavic last name * Alexandrov, Vladimir Oblast, Russia * Alexandrov (inhabited locality), several inhabite ...

(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''

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

closed surfaces 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. *

Alexandrov's uniqueness theorem The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each oth ...

is a result by

(1950), generalizing Cauchy's theorem by showing that convex polyhedra are uniquely described by the

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

s of

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s on their surface. The analogous uniqueness theorem for smooth surfaces was proved by Cohn-Vossen in 1927. Pogorelov's uniqueness theorem is a result by Pogorelov generalizing both of these results and applying to general convex surfaces.

References

{{Reflist * A. L. Cauchy, "Recherche sur les polyèdres – premier mémoire", ''Journal de l'École Polytechnique'' 9 (1813), 66–86. *

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. H ...

"Über die Starrheit konvexer Polyeder"
(in German), ''Math. Ann.'' 77 (1916), 466–473. *

, ''

Convex polyhedra A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wor ...

'', GTI, Moscow, 1950.

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national id ...

translation: Springer, Berlin, 2005. *

James J. Stoker James Johnston Stoker (March 2, 1905 – October 19, 1992) was an American applied mathematician and engineer. He was director of the Courant Institute of Mathematical Sciences and is considered one of the founders of the institute, Courant ...

, "Geometrical problems concerning polyhedra in the large", ''

Comm. Pure Appl. Math. ''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited b ...

'' 21 (1968), 119–168. * Robert Connelly, "Rigidity", in ''Handbook of Convex Geometry'', vol. A, 223–271, North-Holland, Amsterdam, 1993. Augustin-Louis Cauchy Theorems in discrete geometry Polytopes Mathematics of rigidity Euclidean geometry Theorems in convex geometry

Statement

History

Generalizations and related results

See also

References