Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples

Abstract

Trisecting the dihedral angles of an n-simplex defines its Morley simplex. We study the original simplices for which this simplex is regular. A criterion in terms of the Gram matrix of the facet normals reduces the problem to a matrix equation. The derivative of the Morley map at the regular simplex has two explicit eigenvalues, both nonzero for n≥3; thus the regular simplex is an isolated solution. We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices. We prove that this conclusion fails in every dimension 6≤ n≤200 and in every dimension n=binom k2−1 with k≥9: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for 8≤ n≤200 have dihedral symmetry of order 2(n+1), while the infinite family is based on the Johnson scheme. In dimension four we give exact constructions of two nonregular examples, defined by irreducible polynomials of degrees $18$ and $8$ with Galois groups S₁₈ and S₈. Neither example is expressible by radicals. The computer assisted existence proofs use exact rational arithmetic and intervals with outward rounding.

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BibTeX RIS

BibTeX

@misc{tran2026morley,
  title = {{Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples}},
  author = {Quang Hung Tran},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01216},
  eprint = {2610.01216},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01216},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples
AU  - Quang Hung Tran
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01216
UR  - https://arxiv.org/abs/2610.01216
AB  - Trisecting the dihedral angles of an n-simplex defines its Morley simplex. We study the original simplices for which this simplex is regular. A criterion in terms of the Gram matrix of the facet normals reduces the problem to a matrix equation. The derivative of the Morley map at the regular simplex has two explicit eigenvalues, both nonzero for n≥3; thus the regular simplex is an isolated solution. We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices. We prove that this conclusion fails in every dimension 6≤ n≤200 and in every dimension n=binom k2−1 with k≥9: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for 8≤ n≤200 have dihedral symmetry of order 2(n+1), while the infinite family is based on the Johnson scheme. In dimension four we give exact constructions of two nonregular examples, defined by irreducible polynomials of degrees $18$ and $8$ with Galois groups S_18 and S_8. Neither example is expressible by radicals. The computer assisted existence proofs use exact rational arithmetic and intervals with outward rounding.
ER  -

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