Memoir on the Theory of the Articulated Octahedron

Abstract

Mr. C. Stephanos posed the following question in the Interm\'ediaire des Math\'ematiciens: "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.