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Concept
Bricard octahedron
Summary
In geometry, a Bricard octahedron is a member of a family of flexible polyhedra constructed by Raoul Bricard in 1897. The overall shape of one of these polyhedron may change in a continuous motion, without any changes to the lengths of its edges nor to the shapes of its faces.
These octahedra were the first flexible polyhedra to be discovered.
The Bricard octahedra have six vertices, twelve edges, and eight triangular faces, connected in the same way as a regular octahedron. Unlike the regular octahedron, the Bricard octahedra are all non-convex self-crossing polyhedra. By Cauchy's rigidity theorem, a flexible polyhedron must be non-convex, but there exist other flexible polyhedra without self-crossings. Avoiding self-crossings requires more vertices (at least nine) than the six vertices of the Bricard octahedra.
In his publication describing these octahedra, Bricard completely classified the flexible octahedra. His work in this area was later the subject of lectures by Henri Lebesgue at the Collège de France.
A Bricard octahedron may be formed from three pairs of points, each symmetric around a common axis of 180° rotational symmetry, with no plane containing all six points. These points form the vertices of the octahedron. The triangular faces of the octahedron have one point from each of the three symmetric pairs. For each pair, there are two ways of choosing one point from the pair, so there are eight triangular faces altogether. The edges of the octahedron are the sides of these triangles, and include one point from each of two symmetric pairs. There are 12 edges, which form the octahedral graph K2,2,2.
As an example, the six points (0,0,±1), (0,±1,0), and (±1,0,0) form the vertices of a regular octahedron, with each point opposite in the octahedron to its negation, but this is not flexible. Instead, these same six points can be paired up differently to form a Bricard octahedron, with a diagonal axis of symmetry.
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