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 = {R1}, τ2 = {R2}, . . ., τn = {Rn }, . . .. 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 Rn Є τn, n > 0, then η Rn ≥ 2/3 η(R0). The non-degeneracy constant is c = 2/3 in the case of the iterative 8T-LE partition of a regular tetrahedron.