Cauchy's theorem (geometry)
Encyclopedia
Cauchy's theorem is a theorem in geometry
, named after Augustin Cauchy
. It states that
convex polytope
s in three dimensions with congruent
corresponding faces must be congruent to each other. That is, any 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 together 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: 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.
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).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, named after Augustin Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
. It states that
convex polytope
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...
s in three dimensions with congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
corresponding faces must be congruent to each other. That is, any polyhedral net
Net (polyhedron)
In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron...
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 together 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: 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 lattices. 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'sEuclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, 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
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
. A technical mistake was found by Steinitz
Ernst Steinitz
Ernst Steinitz was a German mathematician.- Biography :Steinitz was born in Laurahütte , Silesia, Germany , the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D...
in 1920's and later corrected by him (1928) and Alexandrov
Aleksandr Danilovich Aleksandrov
Aleksandr Danilovich Aleksandrov , and Alexandrov ) , was a Soviet/Russian mathematician, physicist, philosopher and mountaineer.- Scientific career :...
(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
: 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 ConnellyRobert ConnellyRobert Connelly is a mathematician specializing in discrete geometry and rigidity theory. He is best known for discovering embedded flexible polyhedra. One such polyhedron is in the National Museum of American History....
in 1977. - Although originally proven by Cauchy in three dimensions, the theorem was extended to dimensions higher than 3 by AlexandrovAleksandr Danilovich AleksandrovAleksandr Danilovich Aleksandrov , and Alexandrov ) , was a Soviet/Russian mathematician, physicist, philosopher and mountaineer.- Scientific career :...
(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 DehnMax DehnMax Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
in 1916. - Pogorelov's uniqueness theorem is a result by PogorelovAleksei PogorelovAleksei Vasil'evich Pogorelov , was a Soviet and Ukrainian mathematician. He was most famous for his contributions to convex and differential geometry...
generalizing Alexandrov's uniqueness theorem to general convex surfaces.- Alexandrov's uniqueness theorem is a result by AlexandrovAleksandr Danilovich AleksandrovAleksandr Danilovich Aleksandrov , and Alexandrov ) , was a Soviet/Russian mathematician, physicist, philosopher and mountaineer.- Scientific career :...
(1950), weakening conditions of the Cauchy theorem to convex polytopes which are intrinsicallyIntrinsic and extrinsic propertiesAn intrinsic property is an essential or inherent property of a system or of a material itself or within. It is independent of how much of the material is present and is independent of the form the material, e.g., one large piece or a collection of smaller pieces...
isometricIsometryIn mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
. - The analogue uniqueness theorem for smooth surfaces was proved by Cohn-VossenStephan Cohn-VossenStefan or Stephan Cohn-Vossen was a mathematician, now best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie, translated into English as Geometry and the Imagination. The Cohn-Vossen transformation is also named for him.He was born in Breslau...
in 1927.
- Alexandrov's uniqueness theorem is a result by Alexandrov
- Bricard's octahedra are self-intersecting flexible surfaces discovered by a French mathematician Raoul BricardRaoul BricardRaoul Bricard is a French engineer and a mathematician. He is best known for his work in work in geometry, especially descriptive geometry and scissors congruence, and kinematics, especially mechanical linkages.- Biography :...
in 1897.