On 3-triangulation of toroids

Abstract

As toroid (polyhedral torus) could not be convex, it is questionable if it is possible to 3-triangulate them (i.e. divide into tetrahedra with the original vertices). Here, we will discuss some examples of toroids to show that for each vertex number n ≥ 7, there exists a toroid for which triangulation is possible. Also we will study the necessary number of tetrahedra for the minimal triangulation.

2-toroids and their 3-triangulation

“…4. For 1-toroids in [22] was proved that existence of additional branches in its graph of connection would not change estimated value T min . Here, we may assume that if any additional branch appears in graph G of P , it belongs either to 1-toroid T 1 or to T 2 .…”

“…In [22] was proved that minimal triangulation of 1-toroid from series S with n vertices have n tetrahedra. It follows that for described 2-toroid P , T min = n 1 + n 2 = n + 4.…”

“…Interesting triangulations are described in the papers of Edelsbrunner, Preparata, West [9] and Sleator, Tarjan, Thurston [14]. Some characteristics of triangulation in a three-dimensional space are given by Chin, Fung, Wang [6], Develin [7] and Stojanović [20][21][22], and in n-dimensional space by Lee [10]. Algorithms for investigating triangulation in three-dimensional space are given in [23,24].…”

“…It is obtained as an example of polyhedron without diagonals [5,15,16]. Some properties of 3-triangulation for other 1-toroids are given in [22].…”

Minimal 3-triangulations of $p$-toroid

“…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.…”