  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.
