Memoir on the Theory of the Articulated Octahedron

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.

💡 Research Summary

This paper addresses a question posed by Mr. C. Stephanos regarding the existence of polyhedra with invariant facets that can undergo an infinite family of transformations altering only solid angles and dihedral angles. Cauchy had previously proven that no convex polyhedra exist under these transformation conditions. The author extends this result to solve Stephanos’ problem for all octahedra with triangular faces.

The core of the paper is proving the existence of deformable concave octahedra, which must necessarily be concave due to reentrant dihedral angles or intersecting facets, similar to how facets in higher-dimensional spaces behave. The author uses various techniques and proof methods to establish these deformable octahedra based on Cauchy’s theorem.

The paper provides a detailed analysis of the structure and properties of deformable octahedra, highlighting their unique characteristics and significance. This research is not only important for polyhedral theory but also contributes to understanding geometric transformations and higher-dimensional spaces. The work demonstrates that while convex polyhedra cannot be deformed under specified conditions, concave ones can, offering new insights into the nature of three-dimensional shapes and their potential variations.