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In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions). The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces
isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...
to an octahedron. The first example of a flexible non-self-intersecting surface in \mathbb^3, the Connelly sphere, was discovered by .
Steffen's polyhedron In geometry, Steffen's polyhedron is a flexible polyhedron discovered (in 1978) by and named after . It is based on the Bricard octahedron, but unlike the Bricard octahedron its surface does not cross itself.. With nine vertices, 21 edges, and 14 ...
is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra.


Bellows conjecture

In the late 1970s Connelly and
D. Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate ...
formulated the bellows conjecture stating that the volume of a flexible polyhedron is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under flexing. This conjecture was proved for polyhedra
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to a sphere by using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by . The proof extends Piero della Francesca's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges. Since the edge lengths cannot change as the polyhedron flexes, the volume must remain at one of the finitely many roots of the polynomial, rather than changing continuously.


Scissor congruence

Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. This was known as the strong bellows conjecture or (after it was proven in 2018) the strong bellows theorem. Because all configurations of a flexible polyhedron have both the same volume and the same Dehn invariant, they are scissors congruent to each other, meaning that for any two of these configurations it is possible to
dissect Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause of ...
one of them into polyhedral pieces that can be reassembled to form the other. The total mean curvature of a flexible polyhedron, defined as the sum of the products of edge lengths with exterior dihedral angles, is a function of the Dehn invariant that is also known to stay constant while a polyhedron flexes.


Generalizations

Flexible
4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
s in 4-dimensional Euclidean space and 3-dimensional hyperbolic space were studied by . In dimensions n\geq 5, flexible polytopes were constructed by .


See also

* Flexagon *
Rigid origami Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain ...


References


Notes


Primary sources

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Secondary sources

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External links

* {{Mathematics of paper folding Nonconvex polyhedra Mathematics of rigidity