{
  "abstract": "For each fixed $n$, we prove a positive lower bound for the diameter of a compact simple geodesic $n$-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of $n$ for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer $q\\ge2$, a geodesic construction with $2q+3$ sides and diameter less than $3/q$, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05438",
  "author_total": 2,
  "authors": [
    {
      "name": "Yixi Liao"
    },
    {
      "name": "Erxiao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/44664jzwyr9yf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05438",
  "fulltext_url": null,
  "id": "44664jzwyr9yf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05438",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220705376",
  "pdf_url": "https://arxiv.org/pdf/2610.05438",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/44664jzwyr9yf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05438",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05438"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Area and diameter gaps for hyperbolic monotiles",
  "title_text": "Area and diameter gaps for hyperbolic monotiles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/44664jzwyr9yf",
  "venue": "arXiv",
  "versions": []
}