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arxiv: 2604.25173 · v1 · submitted 2026-04-28 · 🧮 math.CO · math.MG

Tiling of Hyperbolic Surface by Multiple Tiles

Chunlin Li , Erxiao Wang , Wu Jie , Min Yan This is my paper

Pith reviewed 2026-05-07 16:04 UTC · model grok-4.3

classification 🧮 math.CO math.MG
keywords tilinghyperbolic surfacesEuler characteristicpolygonal tilingsenumerationedge lengthsgenus
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The pith

Tilings of negatively curved surfaces by n-gons with n at least 7 become a finite problem when the number of tiles is fixed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for surfaces with negative Euler characteristic, the possible ways to tile them using polygons with seven or more sides is limited to a finite set once the total number of tiles is specified. This finiteness allows the development of a systematic algorithm to enumerate all such tilings without infinite searching. The authors apply this to compute all two-tile tilings on surfaces of small genus and examine the variety of edge lengths required in these configurations.

Core claim

Tilings of a surface of negative Euler characteristic by n-gons with n≥7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small genus by two tiles. We also discuss the number of distinct edge lengths in multiple tile tilings.

What carries the argument

An enumeration algorithm based on the finiteness of n-gon tilings (n≥7) for fixed tile count on fixed negative Euler characteristic surfaces.

Load-bearing premise

That fixing the tile number and using n-gons with at least seven sides guarantees only finitely many distinct tilings on a given negative Euler characteristic surface.

What would settle it

An explicit example of an infinite family of distinct two-tile tilings by heptagons on a single fixed-genus surface would disprove the finiteness claim.

Figures

Figures reproduced from arXiv: 2604.25173 by Chunlin Li, Erxiao Wang, Min Yan, Wu Jie.

Figure 1
Figure 1. Figure 1: Corners and edges of a prototile heptagon. view at source ↗
Figure 2
Figure 2. Figure 2: Edge pair implies adjacent corners at vertices. view at source ↗
Figure 3
Figure 3. Figure 3: Planar diagram to vertices. The left two pictures describe the edge pairs in the same tile Tp. The right two pictures describe the edge pairs between different tiles Tp and Tq. In the upper right, the two tiles have the same counterclockwise orientation. In the lower right, the two tiles have different orientations. The adjacent corners at vertices combine to give vertices. A vertex is a circularly ordered… view at source ↗
Figure 4
Figure 4. Figure 4: One vertex of the double planar diagram (2.6). view at source ↗
Figure 5
Figure 5. Figure 5: Two versions of double planar diagrams (3.1) and (5.1). view at source ↗
Figure 6
Figure 6. Figure 6: all tilings of 2 view at source ↗
Figure 6
Figure 6. Figure 6: Tilings of 2T 2 by two congruent 7-gons, with 5 or 6 edge lengths. Thick edges of the same color have the same lengths. Normal edges have distinct lengths. 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3… view at source ↗
Figure 7
Figure 7. Figure 7: Tilings of 3P 2 by two heptagons or octagons, with 5 distinct edge lengths. 10 view at source ↗
Figure 8
Figure 8. Figure 8: Tilings of 3P 2 and 4P 2 by two congruent polygons, with maximal number of distinct edge lengths. Not included: 4P 2 , 12-gon, 8 lengths. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4… view at source ↗
Figure 9
Figure 9. Figure 9: Tilings of 3P 2 by two congruent equilateral 9-gons. 11 view at source ↗
Figure 10
Figure 10. Figure 10: Geometrical realisations for tilings in Figure 6. In the first row, the two tiles have the same view at source ↗
Figure 11
Figure 11. Figure 11: Geometrical realisations for tilings in Figure 9. view at source ↗
Figure 12
Figure 12. Figure 12: Degree 3 vertex implies some edge lengths are equal. view at source ↗
Figure 13
Figure 13. Figure 13: Two tile tilings with all distinct edge lengths. view at source ↗
read the original abstract

Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small genus by two tiles. We also discuss the number of distinct edge lengths in multiple tile tilings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript asserts that tilings of hyperbolic surfaces (negative Euler characteristic) by n-gons with n ≥ 7 form a finite set. It develops an algorithm to enumerate all such tilings for any fixed number of tiles and reports explicit computations of all two-tile tilings on surfaces of small genus; it further discusses the number of distinct edge lengths appearing in multiple-tile tilings.

Significance. If the finiteness statement and the completeness of the enumeration algorithm can be established, the explicit two-tile calculations would supply a concrete, checkable catalogue of hyperbolic tilings with minimal tile count. Such a catalogue could serve as a benchmark for computational approaches to surface tilings and might illuminate constraints on edge-length assignments in negative-curvature geometry.

major comments (2)
  1. [Abstract] Abstract: the assertion that 'tilings ... by n-gons with n≥7 is a finite problem' is stated without any argument, curvature bound, or reference establishing why the set of admissible gluings or vertex figures must be finite once the number of tiles is fixed.
  2. [Abstract] Abstract: the algorithm for enumerating tilings with a fixed tile count is described only at the level of existence; no pseudocode, search strategy, termination criterion, or verification that all admissible edge identifications and hyperbolic angle conditions are generated is supplied, rendering the reported two-tile calculations unverifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which will help improve the clarity of our manuscript. Below we respond to each major comment.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that 'tilings ... by n-gons with n≥7 is a finite problem' is stated without any argument, curvature bound, or reference establishing why the set of admissible gluings or vertex figures must be finite once the number of tiles is fixed.

    Authors: We acknowledge that an explicit argument for finiteness was not provided in the abstract. With a fixed number of tiles, the combinatorial structures are finite because there are finitely many ways to identify the edges of the tiles to form a closed surface of given Euler characteristic. The hyperbolic condition is ensured by n ≥ 7, which allows for angle deficits. We will add a concise explanation of this finiteness in the introduction of the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract: the algorithm for enumerating tilings with a fixed tile count is described only at the level of existence; no pseudocode, search strategy, termination criterion, or verification that all admissible edge identifications and hyperbolic angle conditions are generated is supplied, rendering the reported two-tile calculations unverifiable.

    Authors: The referee is correct that the abstract does not provide sufficient detail on the algorithm. Although the manuscript develops the algorithm, we will revise it to include pseudocode, a description of the search strategy, termination criteria, and verification methods to ensure all possible configurations are covered. This will make the two-tile tiling calculations verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic enumeration and explicit two-tile computations are self-contained.

full rationale

The paper states that tilings by n-gons (n≥7) on negative-Euler-characteristic surfaces form a finite set and then describes an algorithm for fixed tile count together with concrete calculations for two tiles on small-genus surfaces. No equations, parameters, or uniqueness theorems are introduced that reduce by construction to the inputs, fitted data, or prior self-citations. The finiteness claim functions as a combinatorial premise enabling the enumeration procedure rather than a derived result that loops back to itself. This matches the expected profile of a self-contained algorithmic/computational contribution in combinatorial geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that tilings by n-gons (n≥7) on negative-Euler-characteristic surfaces form a finite set when tile count is fixed; this is invoked to justify algorithmic completeness but is treated as background rather than derived here.

axioms (1)
  • domain assumption Tilings of a surface of negative Euler characteristic by n-gons with n≥7 form a finite set when the number of tiles is fixed.
    Directly stated in the abstract as the foundation for developing the enumeration algorithm.

pith-pipeline@v0.9.0 · 5340 in / 1295 out tokens · 47421 ms · 2026-05-07T16:04:14.257879+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 2 canonical work pages

  1. [1]

    H. M. Cheung, H. P. Luk, M. Yan. Tilings of the sphere by congruent pentagons IV: edge combination a4b.preprint, arXiv:2307.11453, 2023

    work page arXiv 2023

  2. [2]

    H. H. Gao, N. Shi, M. Yan. Spherical tiling by 12 congruent pentagons.J. Combin. Theory Ser. A, 120(4):744–776, 2013. 13 kp+1 kp+2 kp+1 −1 kp+1kp kp+1 kp+2 kp+1 −1 kp+1kp Figure 13: Two tile tilings with all distinct edge lengths

    2013

  3. [3]

    C. L. Li, E. X. Wang, J. Wu, M. Yan. Tiling of hyperbolic surface by a single tile, preprint, arXiv.2601.19083, 2026

    work page arXiv 2026

  4. [4]

    C. L. Li, E. X. Wang, J. Wu, M. Yan. Tiling of hyperbolic surface by maximal number of tiles, preprint, 2026

    2026

  5. [5]

    E. X. Wang, M. Yan. Tilings of the sphere by congruent pentagons I: edge combinationsa 2b2cand a3bc.Adv. in Math., 394:107866, 2022

    2022

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    Isohedral tilings by 8-, 10-and 12-gons for hyperbolic translation group of genus two

    Zamorzaeva-Orleanschi, Elizaveta. Isohedral tilings by 8-, 10-and 12-gons for hyperbolic translation group of genus two. Buletinul Academiei de S ¸tiint ¸e a Republicii Moldova. Matematica 87, no. 2 (2018): 74-84

    2018

  7. [7]

    Isohedral tilings by 14-, 16-and 18-gons for hyperbolic translation group of genus two

    Zamorzaeva-Orleanschi, Elizaveta. Isohedral tilings by 14-, 16-and 18-gons for hyperbolic translation group of genus two. Buletinul Academiei de S ¸tiint ¸e a Republicii Moldova. Matematica 89, no. 1 (2019): 91-102. 14

    2019