HEXAGONbeta

Regular Morley tetrahedra from pp-secting planes

A mix of human-written and AI-generated textHuman understanding: some partsmath.MG — Metric Geometry

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

Morley's theorem says that the adjacent angle trisectors of a triangle meet in the vertices of an equilateral triangle. Tran carried the construction to a tetrahedron by trisecting its six dihedral angles and, more generally, by pp-secting planes, with one parameter e,f,g>1e,f,g>1 for each pair of opposite edges; the resulting tetrahedron is denoted Me,f,g(ABCD)\mathcal M_{e,f,g}(ABCD). Tran showed that a regular Morley tetrahedron does not force the original tetrahedron to be regular, observed that in his examples some pair of opposite edges has four equal cross edges, and conjectured that this happens whenever Me,f,g(ABCD)\mathcal M_{e,f,g}(ABCD) is regular. We show that the conjecture is false. We place an isosceles tetrahedron in the box [−a,a]×[−b,b]×[−c,c][-a,a]\times[-b,b]\times[-c,c] and prove that Me,f,g(ABCD)\mathcal M_{e,f,g}(ABCD) is regular exactly when its vertex A′A' differs from the centre of the box and lies on one of the four lines through the centre in the directions (1,1,1)(1,1,1), (1,−1,−1)(1,-1,-1), (−1,1,−1)(-1,1,-1) and (−1,−1,1)(-1,-1,1). Every interior point of these lines other than the centre occurs for exactly one triple of parameters. When aa, bb and cc are pairwise different, no pair of opposite edges has four equal cross edges, so each of these tetrahedra is a counterexample. Two examples are given with the parameters in closed form. Finally, for quadrisecting planes, e=f=g=4e=f=g=4, an isosceles tetrahedron has a regular Morley tetrahedron only if it is regular.

Provenance statement

Deep Bhattacharjee chose the problem and directed the work. Claude (Anthropic), through Claude Code, carried out the searches and computations, wrote the checking programs and the Lean 4 proofs and drafted the paper. Refutes Tran's Conjecture 8.6. The proofs are written out in the paper; the programs and their outputs are in the release. The author approved the final version. Code and data: release v1.0.0, https://github.com/creelie/efieldsMedal/releases/tag/v1.0.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 1 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

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  2. J. Leech, Some properties of the isosceles tetrahedron , Math. Gaz. 34 (1950), 269--271. 10.2307/3611029DOI
  3. F. Morley, On the metric geometry of the plane nn-line , Trans. Amer. Math. Soc. 1 (1900), 97--115. 10.1090/S0002-9947-1900-1500528-8DOI
  4. C. O. Oakley and J. C. Baker, The Morley trisector theorem , Amer. Math. Monthly 85 (1978), 737--745. 10.1080/00029890.1978.11994688DOI
  5. Q. H. Tran, Morley's trisector theorem for isosceles tetrahedron , Acta Math. Hungar. 165 (2021), 308--315. 10.1007/s10474-021-01184-0DOI
  6. Q. H. Tran, The converse problem for the Morley tetrahedron: counterexamples, conjectures, and partial results , preprint, 2026. arXiv:2609.06553v3 (https://arxiv.org/abs/2609.06553v3)arXiv

Version history

  1. v1 · 10 pagesSubmitted by Deep BhattacharjeeInitial depositCurrentOct 23, 2026