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Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

T0 review · 1 major / 1 minor · reviewed 2026-05-21 · grok-4.3

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Pith's one-line read Convex monotiles in the hyperbolic plane can surround themselves any number of times without tiling the entire plane.

desk verdict The paper claims to resolve the Heesch problem for homogeneous tilings in the hyperbolic plane with a corollary for convex monotiles and adds first examples of weakly aperiodic convex monotiles from duals. read the letter →

arxiv 2603.27827 v2 pith:SPOQAKQG submitted 2026-03-29 math.CO

Arun Maiti This is my paper · ORCID
classification math.CO
keywords Heeschnumberhyperbolictilingsconvexmonotileshomogeneousweaklyaperiodictilingtheorygeometry
checked against alexander_duality_circle_linkingFoundation.DimensionForcing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that the Heesch number for convex monotiles in hyperbolic geometry has no upper bound. It first resolves the question for homogeneous tilings by constructing families where surround counts grow without limit, then invokes a corollary to transfer the result to convex monotiles. A sympathetic reader would care because this means there is no fixed maximum number of concentric layers a single tile can form before it must either tile the plane or become impossible to surround further. The work additionally produces the first known weakly aperiodic convex monotiles as duals of these homogeneous tilings.

What carries the argument

Homogeneous (semi-regular) tilings of the hyperbolic plane, together with their duals, which are used to build families of convex monotiles with increasing finite surround counts.

What would settle it

An explicit convex monotile in the hyperbolic plane together with a proof that no arrangement of congruent copies can surround it more than a fixed finite number of times would falsify the unboundedness claim.

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Extended reading notes

Core claim

We construct homogeneous tilings in the hyperbolic plane in which a tile admits arbitrarily large finite numbers of concentric surrounds by congruent copies without admitting a full tiling of the plane. This shows the Heesch number is unbounded for homogeneous tilings. The same unboundedness then follows for convex monotiles by the stated corollary. Duals of the homogeneous tilings also yield the first examples of weakly aperiodic convex monotiles.

Load-bearing premise

The constructions and properties shown for homogeneous tilings carry over directly to convex monotiles by the corollary without needing separate checks that hyperbolic geometry or convexity imposes a uniform bound on surround counts.

Editorial extensions

If this is right

  • For any positive integer N there exist convex monotiles in the hyperbolic plane that admit at least N concentric surrounds without tiling the plane.
  • The Heesch problem is settled negatively for the class of homogeneous tilings and, by corollary, for convex monotiles.
  • Duals of homogeneous tilings supply the first known weakly aperiodic convex monotiles.
  • Any search for convex monotiles with bounded Heesch number in hyperbolic geometry must exclude or modify the homogeneous case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar constructions might be adapted to produce convex monotiles with controlled but large Heesch numbers in other non-Euclidean settings.
  • The existence of weakly aperiodic convex monotiles in hyperbolic geometry raises the question of whether stronger aperiodicity results are possible with the same dual technique.
  • The result suggests that bounded-Heesch-number problems in hyperbolic geometry are likely to require tiles whose vertex figures or edge lengths vary in ways forbidden by homogeneity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper claims to resolve the Heesch problem for homogeneous (semi-regular) tilings in the hyperbolic plane by constructing families with arbitrarily large Heesch numbers, asserts this as a corollary for convex monotiles, and additionally exhibits the first known weakly aperiodic convex monotiles obtained as duals of homogeneous tilings.

Significance. If the constructions are valid, the result would resolve an open question on the Heesch number in hyperbolic geometry for these tile classes and supply new examples of weakly aperiodic monotiles; the explicit constructions for the homogeneous case constitute a concrete strength.

major comments (1)
  1. [Corollary after §3] Corollary following §3: the transfer of unbounded Heesch numbers from homogeneous tilings to convex monotiles is stated directly but supplies no separate verification that convexity and the hyperbolic metric do not introduce angle-deficit or curvature constraints capable of bounding the surround count; a concrete check (e.g., angle-sum calculation or explicit layer construction under convexity) is required for the corollary to be load-bearing.
minor comments (1)
  1. [final section] Notation for the dual construction in the final section could be clarified by adding a short diagram labeling the correspondence between homogeneous tiles and their dual monotiles.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough reading and for identifying a point where the presentation of the corollary could be strengthened. We address the major comment below and will incorporate the requested verification in the revised manuscript.

read point-by-point responses

  1. Referee: [Corollary after §3] Corollary following §3: the transfer of unbounded Heesch numbers from homogeneous tilings to convex monotiles is stated directly but supplies no separate verification that convexity and the hyperbolic metric do not introduce angle-deficit or curvature constraints capable of bounding the surround count; a concrete check (e.g., angle-sum calculation or explicit layer construction under convexity) is required for the corollary to be load-bearing.

    Authors: The homogeneous tilings in our constructions are formed by regular convex polygons meeting at vertices in the hyperbolic plane, so the prototiles are already convex. The explicit families we construct demonstrate that the angle deficits at vertices allow the number of surrounding layers to grow without bound while maintaining convexity and avoiding gaps or overlaps. To address the referee's concern directly, we will add a dedicated paragraph (or short subsection) after the corollary that performs an explicit angle-sum verification for one representative family: we compute the hyperbolic angle sum for successive layers around a central tile and show that the deficit permits arbitrarily large finite Heesch numbers without curvature imposing an upper bound. This will include a concrete numerical example confirming that convexity does not cap the surround count. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation chain self-contained; no circular reductions identified

full rationale

The paper claims a resolution of the Heesch problem via new constructions for homogeneous tilings, followed by a corollary for convex monotiles. No quoted steps reduce a claimed prediction or first-principles result to its own inputs by definition, fitted parameters, or load-bearing self-citations. The abstract and described structure rest on original constructions rather than renaming known results or smuggling ansatzes. This is the normal case of an independent derivation against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The abstract invokes standard facts from hyperbolic geometry and tiling theory but introduces no explicit free parameters, new entities, or ad-hoc axioms beyond those already established in the field.

assumptions (1)
  • standard math Standard axioms and properties of hyperbolic plane geometry and monohedral tilings
    The resolution and corollary rely on background results about tilings in the hyperbolic plane that are assumed known.

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Cite this review

Pith. "Pith review of Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles." pith.science (2026). https://pith.science/paper/SPOQAKQG

@misc{pith2026260327827,
  author       = {Pith},
  title        = {Pith review of: Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPOQAKQG}},
  note         = {Machine review of arXiv:2603.27827}
}
read the original abstract

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.

Figures

Figures reproduced from arXiv: 2603.27827 by the authors.

Figure 1.1
Figure 1.1. Schematic of the layer-by-layer growth of a [4, 5, 4, 5] tiling centered around X0 For a d-tuple k = [k1, k2, · · · , kd], a fan of type k around a vertex is a configuration of d faces in cyclic order around the vertex with the number of sides k1, k2, · · · , kd such that all edges incident to v are shared by two faces. A partial fan around v of type k is a configuration of faces around v such that all but two edges… view at source ↗
Figure 2.1
Figure 2.1. Neighborhoods of type F2 around odd faces 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Completing a partial neighborhood of a 10-gon It is easy to see that the procedure does not introduce any new partial neighborhoods of the odd faces in the (i + 1)-th layer. Hence the induction hypothesis holds for i + 1-layer. Thus we can construct a partial tiling of n layers of type k. Next, we will show that kn does not admit n + 1 layers. We claim that there is a 2i + 5-gon in the i + 1-th layer with a partial … view at source ↗
Figures from the paper (1 more)
Figure 2.3
Figure 2.3. Figure 2.3: Sequence of enforced neighborhoods The process results in neighborhoods of type F1 around a sequence of adjacent odd faces as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_2_3.png]

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Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    138, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2019, pp

    Nathalie Aubrun, Sebastián Barbieri, and Etienne Moutot,The Domino Problem is Unde- cidable on Surface Groups, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) (Dagstuhl, Germany), Leibniz International Proceedings in Informatics (LIPIcs), vol. 138, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2019, pp. 46:1–46:14

    work page 2019

  2. [2]

    Bojan Bašić,The Heesch number for multiple prototiles is unbounded, C. R. Math. Acad. Sci. Paris353(2015), no. 8, 665–669. MR 3367631

    work page 2015

  3. [3]

    thesis, Harvard University, 1964

    Robert Berger,The undecidability of the domino problem, Ph.d. thesis, Harvard University, 1964

    work page 1964

  4. [4]

    Robert Berger,The undecidability of the domino problem, Mem. Amer. Math. Soc. No.66 (1966), 72

    work page 1966

  5. [5]

    1, 138–187

    Eduardo Brandani da Silva and Giovana Higinio de Souza,Uniform tilings, border automata and orbifolds of the hyperbolic plane, International Journal of Geometry11(2022), no. 1, 138–187

    work page 2022

  6. [6]

    Geom.65(2021), no

    Basudeb Datta and Subhojoy Gupta,Semi-regular tilings of the hyperbolic plane, Discrete Comput. Geom.65(2021), no. 2, 531–553. MR 4212977

    work page 2021

  7. [7]

    Math.159(2005), no

    Chaim Goodman Strauss,A strongly aperiodic set of tiles in the hyperbolic plane, Invent. Math.159(2005), no. 1, 119–132. 9

    work page 2005

  8. [8]

    Branko Grünbaum and G. C. Shephard,Incidence symbols and their applications, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978), Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 199–244

    work page 1978

Show all 22 references
  1. [9]

    ,Tilings and patterns, W. H. Freeman and Company, New York, 1987

    1987

  2. [10]

    44, 4010–4016

    Emmanuel Jeandel,The periodic domino problem revisited, Theoretical Computer Science 411(2010), no. 44, 4010–4016

    2010

  3. [11]

    44, 4010–4016

    ,The periodic domino problem revisited, Theoretical Computer Science411(2010), no. 44, 4010–4016

    2010

  4. [12]

    Kari,The tiling problem revisited, Lect

    J. Kari,The tiling problem revisited, Lect. Notes Comput. Sci.4664(2007), 72–79

    2007

  5. [13]

    7, 111911, 7

    Arun Maiti,Quasi-vertex-transitive maps on the plane, Discrete Math.343(2020), no. 7, 111911, 7. MR 4079360

    2020

  6. [14]

    Arun Maiti,Pseudo-homogeneous tiling of the hyperbolic plane, arXiv e-prints (2023), arXiv:2302.05661

    2023

  7. [15]

    1, 29–84

    Maurice Margenstern,The domino problem of the hyperbolic plane is undecidable, Theoret- ical Computer Science407(2008), no. 1, 29–84

    2008

  8. [16]

    Maurice Margenstern,The periodic domino problem is undecidable in the hyperbolic plane, Reachability Problems (Berlin, Heidelberg) (Olivier Bournez and Igor Potapov, eds.), Springer Berlin Heidelberg, 2009, pp. 154–165

    2009

  9. [17]

    G. A. Margulis and S. Mozes,Aperiodic tilings of the hyperbolic plane by convex polygons, Israel J. Math.107(1998), 319–325

    1998

  10. [18]

    Michael Rao,Exhaustive search of convex pentagons which tile the plane, arXiv e-prints (2017), arXiv:1708.00274

    2017 arXiv

  11. [19]

    Renault,The uniform locally finite tilings of the plane, Journal of Combinatorial Theory, Series B98(2008), no

    D. Renault,The uniform locally finite tilings of the plane, Journal of Combinatorial Theory, Series B98(2008), no. 4, 651–671

    2008

  12. [20]

    Tarasov,On the heesch number for the hyperbolic plane, Math Notes88(2010), 97–102

    A.S. Tarasov,On the heesch number for the hyperbolic plane, Math Notes88(2010), 97–102

    2010

  13. [21]

    com, 2021

    Marek ˘Ctrnáct,More about hybrid hyperbolic tilings,https://www.reddit.com/r/math/ comments/pbvj5l/more_about_hybrid_hyperbolic_tilings/?utm_source=chatgpt. com, 2021

    2021

  14. [22]

    HaoWang,Proving theorems by pattern recognition — ii, TheBellSystemTechnicalJournal 40(1961), no. 1, 1–41. 10

    1961

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