2012/03/04 by Raoul Bricard, Bricard, Raoul
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #math.HO #math.MG
paper · pdf · doi:10.48550/arxiv.1203.1286
Translator: Evangelos A. Coutsias, March 23, 2010. Translation from the French original of Raoul Bricard's Mémoire sur la théorie de l'octaèdre articulé, J.Math.Pures Appl. 1897, 3, 113-150. (E. A. Coutsias, [email protected], Mathematics Dept., University of New Mexico). With 13 figures. Includes translations of the original problem posed by C. Stephanos and Bricard's answer
arxiv created 2012/03/04 · openalex publication_date 2012/03/04 · arxiv updated 2012/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Mr. C. Stephanos posed the following question in the Intermédiaire des Mathématiciens: "Do there exist polyhedra with invariant facets that are susceptible to an infinite family of transformations that only alter solid angles and dihedrals?" I announced, in the same Journal, a special concave octahedron possessing the required property. Cauchy, on the other hand, has proved that there do not exist convex polyhedra that are deformable under the prescribed conditions. In this Memoir I propose to extend the above mentioned result, by resolving the problem of Mr. Stephanos in general for octahedra of triangular facets. Following Cauchy's theorem, all the octahedra which I shall establish as deformable will be of necessity concave by virtue of the fact that they possess reentrant dihedrals or, in fact, facets that intercross, in the manner of facets of polyhedra in higher dimensional spaces.