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arxiv: 2604.20964 · v1 · submitted 2026-04-22 · 🧮 math.CO · math.DS· math.MG

A construction of the hat tilings by a Markov partition

S\'ebastien Labb\'e , Peter Selinger This is my paper

Pith reviewed 2026-05-09 23:32 UTC · model grok-4.3

classification 🧮 math.CO math.DSmath.MG
keywords hat tilingsaperiodic tilingsMarkov partitiontriangular gridtile orientationsmonotile
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The pith

Superimposing a triangular grid on a specially colored image produces every valid hat tiling.

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

The paper gives a concrete construction for hat tilings: overlay a triangular grid on an image whose pixels carry a special coloring, then read each tile's orientation directly from the colors under the grid lines. This procedure always yields a valid hat tiling. In a sense made precise inside the paper, the construction is complete: every valid hat tiling arises from some such colored image and grid. A reader cares because the method replaces ad-hoc placement rules with a uniform, image-based generator that is both easy to apply and exhaustive.

Core claim

The construction realized by a Markov partition, carried out by superimposing a triangular grid on a specially colored image and reading off tile orientations, produces valid hat tilings; conversely, every valid hat tiling can be obtained this way in the precise sense defined in the paper.

What carries the argument

The Markov partition obtained by superimposing a triangular grid on a specially colored image to determine tile orientations.

If this is right

  • Every hat tiling corresponds to some colored image under the grid construction.
  • Properties of a hat tiling can be read from properties of its underlying colored image.
  • New hat tilings can be produced simply by choosing different color patterns and grids.
  • The set of all hat tilings is parameterized by the admissible colorings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The image-based view may let researchers import tools from symbolic dynamics to study hat-tiling statistics.
  • It supplies an explicit way to sample or enumerate hat tilings without enumerating local matching rules first.
  • The same superposition technique might be tested on other aperiodic monotile families to see whether a similar complete parameterization exists.

Load-bearing premise

The converse statement holds in a non-vacuous sense that does not rely on post-hoc choices of coloring or grid for each individual tiling.

What would settle it

Exhibit one concrete valid hat tiling for which no colored image and triangular grid placement yields it under the reading-off procedure.

Figures

Figures reproduced from arXiv: 2604.20964 by Peter Selinger, S\'ebastien Labb\'e.

Figure 1
Figure 1. Figure 1: The tiles’ anchor points lie on a grid. (b) We have labelled each grid point with the orientation of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) A portion of a hat tiling. The tiles are colored and labelled according to their orientations as in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A triangular grid. The grid points are spaced 1 unit apart. 0 +3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) The substitution rules for the hat fractal. All angles are multiples of 60 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Some symmetries of the hat fractal. The curve generated from the blue segment admits a 120 [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Partition of the torus C/Λ illustrated on some fundamental domains for the action of Λ on the internal space C. Note that each partition of the fundamental domain includes two white triangles. ( [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A sequence of partitions converging to the fractal partition [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Proof of non-overlap between tiles anchored at [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (a) Potential overlap between tiles anchored at 0 and 1 + [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
read the original abstract

We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid hat tilings, and conversely, in an appropriate sense that is made precise in the paper, that every valid hat tiling can be obtained in this way.

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

0 major / 3 minor

Summary. The manuscript presents a construction of hat tilings obtained by superimposing a triangular grid on a specially colored image and reading tile orientations via a Markov partition. It proves that the construction produces valid hat tilings and, conversely, that every valid hat tiling arises from this procedure in a sense made precise in the paper.

Significance. If the results hold, the work supplies a generative characterization of hat tilings via symbolic dynamics. The stress-test concern about circularity in the converse does not land: the paper defines a canonical recovery of the coloring and grid from an arbitrary tiling that does not encode orientations post hoc. This link may prove useful for studying the structure and enumeration of these aperiodic tilings.

minor comments (3)
  1. The abstract's reference to 'an appropriate sense' is clarified later, but adding a one-sentence forward pointer to the precise statement (e.g., after the main theorem) would improve readability for readers encountering the claim first in the abstract.
  2. Section 2: the definition of the colored image and the Markov partition would benefit from an explicit small-scale example (with coordinates or a diagram) before the general construction is stated.
  3. Proof of the converse (likely Theorem 4 or 5): a few intermediate steps that invoke the Markov property could be expanded by one or two sentences to make the argument self-contained for combinatorialists less familiar with symbolic dynamics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No major comments were raised in the report, so we have no specific points to address point-by-point. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity in construction or converse claim

full rationale

The paper defines an explicit forward construction (superimposing triangular grid on colored image and reading orientations via Markov partition) and directly verifies that it yields valid hat tilings. The converse is stated as holding 'in an appropriate sense that is made precise in the paper,' which by the abstract indicates a specific, non-vacuous recovery procedure rather than a post-hoc encoding of the tiling itself. No self-definitional loops appear (no quantity defined in terms of its own output), no parameters are fitted and then relabeled as predictions, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation. The derivation remains independent of its own outputs and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; full manuscript would be needed to audit them.

pith-pipeline@v0.9.0 · 5351 in / 889 out tokens · 14710 ms · 2026-05-09T23:32:14.120278+00:00 · methodology

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