Are all convex polyhedra tetragonalizable?
A convex polyhedron in 3-space is called tetragonizable by a family of pyramids, all of them erected on the facets of that polyhedron, if the two pyramidal faces containing one of the edges of the original polyhedron are coplanar, thus forming a convex quadrangle which is then a facet of the obtained tetragonalization. E.g., the cube is a tetragonalization of the regular tetrahedron, the rhombic dodecahedron of the cube, and the rhombitriacontahedron of the regular icosahedron. Based on the nice (and unsettled) conjecture that each convex polyhedron is tetragonizable, and supporting this conjecture, the author proves that all simplicial polyhedra (having only triangular facets) are tetragonizable, and that all midscribable polyhedra (all whose edges are tangent to a sphere) have the same property.NEWLINENEWLINE Further parts of this nice paper refer to in-, mid- and circumscribable polyhedra and the conjecture that every tetragonizable polyhedron admits a continuum of tetragonalizations.
scientific article; zbMATH DE number 6638579