{
  "abstract": "The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13757",
  "author_total": 1,
  "authors": [
    {
      "name": "Haroldo Costa Silva Filho"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2r106q5bfq7mh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13757",
  "fulltext_url": null,
  "id": "2r106q5bfq7mh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13757",
  "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.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213359465",
  "pdf_url": "https://arxiv.org/pdf/2609.13757",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2r106q5bfq7mh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13757",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13757"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields",
  "title_text": "The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/2r106q5bfq7mh",
  "venue": "arXiv",
  "versions": []
}