Non-degeneracy of tetrahedral longest-edge bisection

Prove or disprove that the solid angles of tetrahedra remain nondegenerate under repeated tetrahedral longest-edge bisection.

Background

Rivara and Levin formulated a non-degeneracy conjecture for tetrahedral longest-edge bisection based on numerical experiments. Specifically, they conjectured that repeated refinement would not cause the tetrahedral solid angles to degenerate.

The paper’s explicit bisection construction provides a counterexample, showing that the conjecture does not hold for all tetrahedra. It is included because the unresolved conjecture is stated verbatim with an explicit uncertainty marker.

References

In 1992, Rivara and Levin explicitly conjectured, for tetrahedral longest-edge bisection, that the solid angles do not degenerate as repeated refinement proceeds to infinity, and they presented empirical evidence in support of this conjecture.

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