Old and new geometric polyhedra with few vertices

TL;DR

This paper classifies and geometrically realizes all triangulations of the 2-torus and projective plane with specific vertex-labeled graphs, revealing pairs with identical 1-skeletons but no shared faces, using geometric models.

Contribution

It provides explicit geometric realizations of all such triangulations, including pairs with identical 1-skeletons but no common faces, in low-dimensional spaces.

Findings

All 12 triangulations of the 2-torus with $O_4$ are realized in 3-space.

All 12 triangulations of the projective plane with $K_6$ are realized in 4-space.

Identified pairs of triangulations sharing the same 1-skeleton but no common face.

Abstract

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.