Similarity Classes in the Eight-Tetrahedron Longest-Edge Partition of a Regular Tetrahedron

Abstract

A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R0, the iterative eight-tetrahedron longest-edge partition (8T-LE) of R0 produces an infinity sequence of tetrahedral meshes, τ0={R0}, τ1={Ri1}, τ2={Ri2},…, τn={Rin},…. In this paper, it is proven that, in the iterative process just mentioned, only two distinct similarity classes are generated. Therefore, the stability and the non-degeneracy of the generated meshes, as well as the minimum and maximum angle condition straightforwardly follow. Additionally, for a standard-shape tetrahedron quality measure (η) and any tetrahedron Rin∈τn, n>0, then ηRin≥23ηR0. The non-degeneracy constant is c=23 in the case of the iterative 8T-LE partition of a regular tetrahedron.