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arxiv: 2603.27827 · v2 · pith:SPOQAKQGnew · submitted 2026-03-29 · 🧮 math.CO

Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

Arun Maiti This is my paper

Pith reviewed 2026-05-21 10:20 UTC · model grok-4.3

classification 🧮 math.CO
keywords Heesch numberhyperbolic tilingsconvex monotileshomogeneous tilingsweakly aperiodic tilingstiling theoryhyperbolic geometry
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The pith

Convex monotiles in the hyperbolic plane can surround themselves any number of times without tiling the entire plane.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 2603.27827 by Arun Maiti.

Figure 1.1
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 b… view at source ↗
Figure 2.1
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. 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 wit… view at source ↗
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] view at source ↗
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.

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5559 in / 1261 out tokens · 49082 ms · 2026-05-21T10:20:32.264137+00:00 · methodology

discussion (0)

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Reference graph

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