2-toroids and their 3-triangulation

Abstract

Abstract. It is known that we can always 3-triangulate (i.e. divide into tetrahedra) convex polyhedra but not always non-convex polyhedra. Here we discuss possibilities and properties of 3-triangulation of 2-toroids, i.e. polyhedra topologically equivalent to sphere with 2 handles, and develop the concepts of piecewise convex polyhedra and graph of connection.

Minimal 3-triangulations of $p$-toroid

“…But, for P p 3p+4 it is questionable if it has geometric realization. In [19] double-Császár 2-toroid, which is P 2 10 from this series was introduced. It was proved that it is 3-triangulable 2-toroid with the smallest number of vertices, n = 10.…”

“…It is obtained as an example of polyhedron without diagonals [4], [11], [12]. Some other examples of 1-toroids are given in [13], [14], while in [18], [19] 3-triangulations of 1-toroids and 2-toroids are discussed. In [3], [6] some combinatorial properties of p-toroids are given.…”