Global InformationBricard octahedron information
In geometry, a Bricard octahedron is a member of a family of flexible polyhedra constructed by Raoul Bricard in 1897.[1] 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.[2] These octahedra were the first flexible polyhedra to be discovered.[3]
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,[3] 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.[4]
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.[5]
- ^ Bricard, Raoul (1897), "Mémoire sur la théorie de l'octaèdre articulé", Journal de mathématiques pures et appliquées, 5e série (in French), 3: 113–150. Translated as "Memoir on the Theory of the Articulated Octahedron" by E. A. Coutsias, 2010.
- ^ Cite error: The named reference
rcwas invoked but never defined (see the help page). - ^ a b Stewart, Ian (2004), Math Hysteria: Fun and games with mathematics, Oxford: Oxford University Press, p. 116, ISBN 9780191647451.
- ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "23.2 Flexible polyhedra", Geometric Folding Algorithms: Linkages, origami, polyhedra, Cambridge University Press, Cambridge, pp. 345–348, doi:10.1017/CBO9780511735172, ISBN 978-0-521-85757-4, MR 2354878.
- ^ Lebesgue H., "Octaedres articules de Bricard", Enseign. Math., Series 2 (in French), 13 (3): 175–185, doi:10.5169/seals-41541