Bricard octahedron
In geometry, a Bricard octahedron is a member of a family of flexible polyhedra constructed by Raoul Bricard in 1897. That is, it is possible for the overall shape of this polyhedron to 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. However, 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. However, 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.
The Bricard octahedra all have an axis of 180° rotational symmetry, and are formed from any three pairs of points such that each pair is symmetric around the same axis and there is no plane containing all six points. (For instance, the six points of a regular octahedron can be paired up in this way by an axial symmetry around a line through two opposite edge midpoints, although the Bricard octahedron resulting from this pairing would not be regular.) The octahedra have 12 edges, each of which connects two points that do not belong to the same symmetric pair as each other. These edges form the octahedral graph K2,2,2. Each of the eight triangular faces of these octahedra connects three points, one from each symmetric pair, in all of the eight possible ways of doing this.
It is also possible to think of the Bricard octahedron as a mechanical linkage consisting of the twelve edges, connected by flexible joints at the vertices, without the faces. Omitting the faces eliminates the self-crossings for many (but not all) positions of these octahedra. The resulting kinematic chain has one degree of freedom of motion, the same as the polyhedron from which it is derived.
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