{
  "abstract": "Let $T$ be the tetragonal disphenoid whose four congruent faces are isosceles triangles with sides $2,\\sqrt3,\\sqrt3$, so that the dihedral angles are $π/2$ along one opposite pair of edges and $π/3$ along the remaining four. By Smyth's theorem $T$ contains three embedded minimal disks meeting $\\partial T$ orthogonally; the symmetry group $D_{2d}$ of $T$ fixes one and interchanges the other two. We identify all three completely. The distinguished disk is a patch of $\\textbf{Schwarz's $\\mathrm{P}$ surface}$ and the congruent pair are patches of $\\textbf{Schwarz's $\\mathrm{D}$ surface}$, with exact areas \\[ \\operatorname{Area}(Σ^{*}_{A}) = \\frac{K\\!\\left(\\tfrac14\\right)} {K\\!\\left(\\tfrac34\\right)} = 0.781700961348056\\ldots, \\] \\[ \\operatorname{Area}(Σ^{*}_{B}) = \\frac{K\\!\\left(\\tfrac34\\right)} {2\\,K\\!\\left(\\tfrac14\\right)} = 0.639630785585503\\ldots \\] in the normalization $\\operatorname{Vol}(T)=2/3$, inradius $r_T=\\sqrt2/4$. In particular $2\\,\\operatorname{Area}(Σ^{*}_{A})\\operatorname{Area}(Σ^{*}_{B})=1$, which we show is the Bonnet relation for the adjoint pair $(\\mathrm{D},\\mathrm{P})$. The proof is a reversal of the usual strategy: rather than uniformizing the Gauss map, we identify Smyth's three Plateau contours as the $\\textit{tetrahedral four edges}$ and the $\\textit{octahedral four edges}$ --- classically solved by $\\mathrm{D}$ and $\\mathrm{P}$ respectively --- and then conjugate. Both areas agree with Surface Evolver computations of Ken Brakke to all $15$ reported digits. We also show that all twelve corners of the three disks are midpoints of edges of $T$, each of the six midpoints being used twice, and that $T$ is precisely the alcove of the affine Weyl group $\\widetilde{A}_{3}$, confirming a remark of Gandy, Cvijović, et al. (1999).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22013",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Finch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6ag96mx0gynx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22013",
  "fulltext_url": null,
  "id": "c6ag96mx0gynx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22013",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222619693",
  "pdf_url": "https://arxiv.org/pdf/2610.22013",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6ag96mx0gynx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22013",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22013"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Schwarz $\\mathrm{P}$ and Schwarz $\\mathrm{D}$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas",
  "title_text": "Schwarz P and Schwarz D as free-boundary minimal disks in a tetragonal disphenoid: exact areas",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:14Z",
  "url": "https://research.draconitic.com/paper/c6ag96mx0gynx",
  "venue": "arXiv",
  "versions": []
}