Old and new geometric polyhedra with few vertices

This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph \(O_4\) which is isomorphic to the complete four-partite graph \(K_{2,2,2,2}\); it is known that there exist precisely twelve such triangulations. We find all the 12 triangulations in a Schlegel diagram of the hyperoctahedron and realize all of them geometrically with the same 1-skeleton in 3-space. In particular, we identify two geometric polyhedral tori (both without self-intersections) with the same 1-skeleton in 3-space, but without a single common face, or in other words their intersection (as point-sets) is only their common 1-skeleton. Similarly, all the twelve triangulations of the 2D projective plane with the vertex labeled complete graph \(K_6\) are found in a Schlegel diagram of the 5-simplex and all are realized geometrically with the same 1-skeleton in 4-space; especially we obtain a pair of triangulations of the M\"{o}bius 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. The constructed polyhedra are modeled and visualized with GeoGebra.