{
  "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 $n\\ge3$; 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 $6\\le n\\le200$ and in every dimension $n=\\binom k2-1$ with $k\\ge9$: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for $8\\le n\\le200$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01216",
  "author_total": 1,
  "authors": [
    {
      "name": "Quang Hung Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fra3sfsdvr93a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01216",
  "fulltext_url": null,
  "id": "fra3sfsdvr93a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01216",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7216422606",
  "pdf_url": "https://arxiv.org/pdf/2610.01216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fra3sfsdvr93a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples",
  "title_text": "Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/fra3sfsdvr93a",
  "venue": "arXiv",
  "versions": []
}