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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.

We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised triangles endowed with the Poincare hyperbolic metric are studied. The self-improvement of this transformation is easily proven, as well as the minimum angle condition. We give a lower bound for the maximum of the smallest angles of the triangles produced by the iterative SE duplication α=π6. This bound does not depend on the shape of the initial triangle.