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PreprintCC BY 4.0arXiv

The Converse Problem for the Morley Tetrahedron: Counterexamples, Conjectures, and Partial Results

  • Quang Hung Tran
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Abstract

In a paper in Acta Mathematica Hungarica the author proved that the Morley tetrahedron of an isosceles tetrahedron, obtained by trisecting the six dihedral angles, is again isosceles, and proposed two converse conjectures. We show that both are false. There is a nonisosceles tetrahedron T1 and an isosceles, nonregular tetrahedron T2 whose Morley tetrahedra are regular, and there are nonisosceles tetrahedra, even a two-parameter family of tetrahedra without any symmetry, whose Morley tetrahedra are isosceles. In T1 and in T2 there is a pair of opposite edges such that the other four edges are equal, and we conjecture that a regular Morley tetrahedron always forces this. We prove the conjecture for every tetrahedron with a nontrivial symmetry, and we show that, up to similarity, the regular tetrahedron, T1 and T2 are the only tetrahedra with this edge pattern and a regular Morley tetrahedron. The tetrahedron T2 has AB=CD=1 and AC=AD=BC=BD=√((21+4√6)/45), while T1 is given by a root of a sextic with Galois group S6 and cannot be expressed by radicals. We also prove that a tetrahedron with a regular Morley tetrahedron is regular if it is orthocentric, if it is isodynamic, if its three sums of opposite edges are equal, if it has three equal edges at a vertex, if it has an equilateral face, or if none of its dihedral angles is larger than 95∘; the tetrahedron T2 has two dihedral angles of about 98.7∘. For isosceles Morley tetrahedra we conjecture that (AB2−CD2)(AC2−BD2)(AD2−BC2)≥0 and that AB=CD forces a second pair of equal opposite edges. We also conjecture that a tetrahedron with AC=BD whose Morley tetrahedron satisfies A'B'=B'C'=C'D'=D'A' has a nontrivial symmetry. Some proofs are computer assisted; they use exact rational arithmetic or interval arithmetic with outward rounding.

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Cite

BibTeXRIS
BibTeX
@misc{tran2026converse,
  title = {{The Converse Problem for the Morley Tetrahedron: Counterexamples, Conjectures, and Partial Results}},
  author = {Quang Hung Tran},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.06553},
  eprint = {2609.06553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06553},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Converse Problem for the Morley Tetrahedron: Counterexamples, Conjectures, and Partial Results
AU  - Quang Hung Tran
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06553
UR  - https://arxiv.org/abs/2609.06553
AB  - In a paper in Acta Mathematica Hungarica the author proved that the Morley tetrahedron of an isosceles tetrahedron, obtained by trisecting the six dihedral angles, is again isosceles, and proposed two converse conjectures. We show that both are false. There is a nonisosceles tetrahedron T_1 and an isosceles, nonregular tetrahedron T_2 whose Morley tetrahedra are regular, and there are nonisosceles tetrahedra, even a two-parameter family of tetrahedra without any symmetry, whose Morley tetrahedra are isosceles. In T_1 and in T_2 there is a pair of opposite edges such that the other four edges are equal, and we conjecture that a regular Morley tetrahedron always forces this. We prove the conjecture for every tetrahedron with a nontrivial symmetry, and we show that, up to similarity, the regular tetrahedron, T_1 and T_2 are the only tetrahedra with this edge pattern and a regular Morley tetrahedron. The tetrahedron T_2 has AB=CD=1 and AC=AD=BC=BD=√((21+4√6)/45), while T_1 is given by a root of a sextic with Galois group S_6 and cannot be expressed by radicals. We also prove that a tetrahedron with a regular Morley tetrahedron is regular if it is orthocentric, if it is isodynamic, if its three sums of opposite edges are equal, if it has three equal edges at a vertex, if it has an equilateral face, or if none of its dihedral angles is larger than 95^∘; the tetrahedron T_2 has two dihedral angles of about 98.7^∘. For isosceles Morley tetrahedra we conjecture that (AB^2−CD^2)(AC^2−BD^2)(AD^2−BC^2)≥0 and that AB=CD forces a second pair of equal opposite edges. We also conjecture that a tetrahedron with AC=BD whose Morley tetrahedron satisfies A'B'=B'C'=C'D'=D'A' has a nontrivial symmetry. Some proofs are computer assisted; they use exact rational arithmetic or interval arithmetic with outward rounding.
ER  -