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Tilings of Closed Surfaces by Congruent Most Curvilinear Polygons

  • Haofang Sun
  • Min Yan
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Abstract

We study tilings of closed surfaces by congruent most curvilinear polygons. By relating such tilings to branched coverings determined by planar diagrams and vertex types, we establish necessary and sufficient conditions for a given closed topological surface to admit a metric of constant curvature supporting a tiling with prescribed combinatorial data and number of tiles. We also characterize minimal tilings, which do not descend through nontrivial unbranched coverings to tilings with fewer tiles, in terms of common block decompositions of the associated permutations. Explicit permutation constructions yield infinite families of minimal tilings with prescribed vertex types, even when the planar diagram and vertex types are fixed. In the hyperbolic setting, these results also give finite-index subgroups of the universal tiling symmetry groups that are maximal among torsion-free subgroups, with the index equal to the number of tiles.

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Cite

BibTeXRIS
BibTeX
@misc{sun2026tilings,
  title = {{Tilings of Closed Surfaces by Congruent Most Curvilinear Polygons}},
  author = {Haofang Sun and Min Yan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03786},
  eprint = {2610.03786},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03786},
}
RIS
TY  - UNPB
TI  - Tilings of Closed Surfaces by Congruent Most Curvilinear Polygons
AU  - Haofang Sun
AU  - Min Yan
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.03786
UR  - https://arxiv.org/abs/2610.03786
AB  - We study tilings of closed surfaces by congruent most curvilinear polygons. By relating such tilings to branched coverings determined by planar diagrams and vertex types, we establish necessary and sufficient conditions for a given closed topological surface to admit a metric of constant curvature supporting a tiling with prescribed combinatorial data and number of tiles. We also characterize minimal tilings, which do not descend through nontrivial unbranched coverings to tilings with fewer tiles, in terms of common block decompositions of the associated permutations. Explicit permutation constructions yield infinite families of minimal tilings with prescribed vertex types, even when the planar diagram and vertex types are fixed. In the hyperbolic setting, these results also give finite-index subgroups of the universal tiling symmetry groups that are maximal among torsion-free subgroups, with the index equal to the number of tiles.
ER  -