In geometry

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 ....

, a flexible polyhedron is a polyhedral

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 that allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem

Cauchy's theorem (geometry)

Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states thatconvex polytopes in three dimensions with congruent corresponding faces must be congruent to each other...

 shows that in dimension 3 such a polyhedron cannot be convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

 (this is also true in higher dimensions).

The first examples of flexible polyhedra, now called Bricard's octahedra, were discovered by . They are self-intersecting surfaces isometric

Isometry

In 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...

 to an octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

. The first example of a non-self-intersecting surface in R³, the Connelly sphere, was discovered by .

Bellows conjecture

In the late 1970s Connelly and D. Sullivan formulated the Bellows conjecture stating that the volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 by
using elimination theory

Elimination theory

In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables....

, and then proved for general orientable

Orientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

 2-dimensional polyhedral surfaces by .

Scissor congruence

Connelly conjectured that the Dehn invariant

Hilbert's third problem

The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...

 of a flexible polyhedron is invariant under flexing. This is known as the strong bellows conjecture. Preservation of the Dehn invariant is known to be equivalent to scissors congruence of the enclosed region under flexing. The special case of mean curvature

Mean curvature

In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....

 has been proved by Ralph Alexander.

Generalizations

Flexible polyhedra in the 4-dimensional Euclidean space and 3-dimensional Lobachevsky space

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

 were studied by Hellmuth Stachel

Hellmuth Stachel

Hellmuth Stachel is an Austrian mathematician, a Professor of Geometry at the Technical University of Vienna, who is known due to his contributions to geometry, kinematics and computer-aided design.- Biography :...

. In November 2009 it was not known whether flexible polyhedra exist in the Euclidean space of dimension .

Popular level

• D. Fuchs, S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics