Non-degeneracy of tetrahedral longest-edge trisection

Prove or disprove that iterative tetrahedral longest-edge trisection preserves non-degeneracy of tetrahedral shapes.

Background

Suarez, Abellón, Abad, and Plaza introduced a tetrahedral longest-edge trisection algorithm and tested it numerically on several classes of tetrahedra, including regular, cap, needle, sliver, and wedge-shaped elements. Their numerical experiments reported positive quality values after thirteen global refinement levels.

On the basis of those experiments, they explicitly conjectured that iterative tetrahedral longest-edge trisection is non-degenerate. The present paper constructs a degenerating trisection sequence and therefore disproves the conjecture. It is included because the conjecture is explicitly stated in the source.

References

After thirteen global refinement levels they reported positive observed quality values and, on this empirical basis, explicitly conjectured non-degeneracy of iterative tetrahedral LE trisection.

— On degeneration of tetrahedra under longest-edge n-section refinement  (2609.29153 - Korotov, 24 Sep 2026) in Section 1, subsection “Related evidence and scope”