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

A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R-0, the iterative eight-tetrahedron longest-edge partition (8T-LE) of R0 produces an infinity sequence of tetrahedral meshes, tau(0)={R-0}, tau(1)={Ri(1)}, tau(2)={Ri(2)}, horizontal ellipsis , tau(n)={Ri(n)}, horizontal ellipsis . 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 (eta) and any tetrahedron R-i(n )is an element of tau(n), n > 0, then eta(R-i(n ))>= 2/3 eta(R-0). The non-degeneracy constant is c=2/3 in the case of the iterative 8T-LE partition of a regular tetrahedron.