Pairs of polyhedra sharing the same 1-skeleton in 3D and 4D spaces, without a single common face

Abstract

All the twelve triangulations of the (2-)torus with the vertex labeled complete four- partite graph K_{2,2,2,2} are found in a Schlegel diagram of the hyperoctahedron and are all realized geometrically with the same 1-skeleton in 3-space. In particular, we identify two geometric polyhedral tori (each without self-intersections) with the same 1-skeleton in 3-space, but without a single common face; in other words, their intersection (as point-sets) is only their common 1-skeleton. Similarly, all the twelve triangulations of the (2-dimensional) projective plane with the vertex labeled complete graph K_6 are found in a Schlegel diagram of the 5-simplex and are all realized geometrically with the same 1-skeleton in 4-space; especially we obtain a pair of triangulations of the Möbius band and a pair of triangulated projective planes with the same 1-skeleton (within each pair) in 3-space and 4-space, respectively, without a single common face.