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arxiv: 2601.00340 · v2 · submitted 2026-01-01 · ❄️ cond-mat.mtrl-sci · cond-mat.soft· cond-mat.str-el

Tiling by Near Coincidence

Meshy Ochana , Ron Lifshitz This is my paper

Pith reviewed 2026-05-16 18:09 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.softcond-mat.str-el
keywords near-coincidence methodquasiperiodic tilingsmoiré patternscut-and-project formalismAmmann-Beenker tilingNiizeki-Gähler tilingFibonacci tilingsbilayer systems
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The pith

Selecting nearly coincident points from superimposed layers generates rigorous quasiperiodic tilings equivalent to cut-and-project constructions.

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

The paper presents the near-coincidence method as a way to build quasiperiodic tilings by choosing pairs of points that almost coincide when two layers are twisted or scaled relative to each other. This procedure is shown to map directly onto the cut-and-project formalism, which guarantees the resulting point sets are quasiperiodic. The same rule reproduces well-known examples including the Ammann-Beenker tiling, the Niizeki-Gähler tiling, and both square and hexagonal Fibonacci tilings, while also producing new patterns that do not arise easily from standard constructions. The approach is computationally straightforward and has already been coded into a web tool that accepts layer parameters and coincidence conditions as input.

Core claim

The near-coincidence method constructs quasiperiodic point sets by admitting pairs of nearly coincident points from two superimposed layers; this selection rule is equivalent to the cut-and-project formalism and therefore produces valid quasiperiodic tilings, including the classical Ammann-Beenker, Niizeki-Gähler, and Fibonacci examples as well as previously unknown ones.

What carries the argument

The near-coincidence method, which selects pairs of nearly coincident points from superimposed layers to serve as the vertices of the tiling.

If this is right

  • The method reproduces the Ammann-Beenker tiling, the Niizeki-Gähler tiling, and square and hexagonal Fibonacci tilings.
  • It generates new tilings that are unlikely to be found by conventional cut-and-project or substitution constructions.
  • The procedure is algorithmically simple and has been implemented in a web-based generator that takes layer twist, scale, and coincidence conditions as inputs.
  • Preliminary extensions to trilayer stacks produce dodecagonal order when square layers are used, and the method can address very small twist angles that create giant moiré patterns in bilayer and trilayer graphene.

Where Pith is reading between the lines

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

  • The intuitive visual rule may help experimentalists design and interpret moiré superlattices in real 2D materials without first solving the full higher-dimensional cut-and-project model.
  • The same selection logic could be tested on discrete lattices or finite patches to see whether it still produces long-range quasiperiodic order when the layers are finite or contain defects.
  • Because the method is parameter-driven, systematic scans over twist angle and scale factor could map out families of tilings and identify which combinations yield previously unlisted symmetries.

Load-bearing premise

That pairs of nearly coincident points chosen from superimposed layers will systematically produce valid quasiperiodic point sets equivalent to those from the cut-and-project method without additional constraints or post-selection.

What would settle it

Apply the near-coincidence rule to a specific pair of layer parameters, generate the point set, and compare its diffraction spectrum or Fourier module directly against the spectrum obtained from an independent cut-and-project construction with the same parameters; mismatch would falsify equivalence.

Figures

Figures reproduced from arXiv: 2601.00340 by Meshy Ochana, Ron Lifshitz.

Figure 1
Figure 1. Figure 1: FIG. 1. Moir´e patterns of (a) twisted square-lattice bilayer [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Step-by-step construction of an octagonal tiling by the near-coincidence method: 1. Two layers of identical periodic [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Apparent defect consisting of a pair of nearby [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Octagonal tilings obtained by the near-coincidence method as described in Sec. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. One of the standard star maps used in the cut-and [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Scaling the octagonal coincidence window twice by [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The three dodecagonal Niizeki–G¨ahler tilings [ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Local changes in tile configurations, associated with [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Three new dodecagonal tilings obtained by using circular coincidence windows. (a) The three circular coincidence [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Successive one-dimensional Fibonacci tilings for cen [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Square Fibonacci tilings obtained from a [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The standard star maps used in the cut-and-project [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Two tilings produced via the near-coincidence [PITH_FULL_IMAGE:figures/full_fig_p011_15.png] view at source ↗
read the original abstract

Moir\'e patterns of twisted and scaled bilayers have recently emerged as a fertile source of quasiperiodic order in two-dimensional materials. Inspired by these systems, we introduce the \emph{near-coincidence method} for generating quasiperiodic tilings of the plane. The method is intuitive -- admitting pairs of nearly coincident points from superimposed layers -- yet rigorous, as it maps naturally to the well-established cut-and-project formalism. It reproduces classical tilings, including the Ammann--Beenker, the Niizeki--G\"ahler, and the square and hexagonal Fibonacci tilings, and also reveals new tilings that are unlikely to arise from conventional constructions. The near-coincidence method is algorithmically simple and already realized in a web-based application that generates tilings from specified layer parameters and coincidence conditions. Future extensions include trilayer systems, where preliminary results yield dodecagonal order with square layers, and very small twist angles, where the method may capture the giant moir\'e patterns of bilayer and trilayer graphene.

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 / 2 minor

Summary. The manuscript introduces the near-coincidence method for generating quasiperiodic tilings of the plane by selecting pairs of nearly coincident points from superimposed layers of twisted and scaled bilayers. It asserts that this intuitive approach maps rigorously to the cut-and-project formalism, reproduces several classical tilings including Ammann-Beenker, Niizeki-Gähler, and Fibonacci tilings, and uncovers new tilings. The method is implemented in a web application, with extensions suggested for trilayer systems and small twist angles in graphene.

Significance. If the central mapping and discreteness claims hold, the work supplies an algorithmically simple, materials-inspired route to quasiperiodic point sets that reproduces known tilings and generates new ones unlikely to arise from standard constructions. The explicit web-based implementation and the grounding in the independently established cut-and-project formalism are concrete strengths that would make the technique immediately usable for exploring moiré-derived quasicrystals.

major comments (2)
  1. [Abstract and method definition] The abstract and the description of the method assert that near-coincidence selection 'maps naturally' to cut-and-project and yields uniformly discrete quasiperiodic sets, yet no explicit tolerance criterion, irrationality condition on the rotation matrix, or proof of uniform discreteness for generic twist/scale parameters is supplied. This is load-bearing for the central claim that the procedure is parameter-free and automatic.
  2. [Results on classical tilings] Reproduction of the Ammann-Beenker and Niizeki-Gähler tilings is stated, but the manuscript provides neither a side-by-side comparison of the generated point sets with the standard cut-and-project constructions nor a demonstration that the near-coincidence step enforces the same higher-dimensional lattice embedding without post-selection.
minor comments (2)
  1. [Introduction and method] Notation for the bilayer superposition (rotation matrix, scaling factor, and coincidence threshold) should be introduced with a single consistent set of symbols and an accompanying diagram.
  2. [Implementation] The web-application description mentions 'specified layer parameters and coincidence conditions' but does not list the exact input fields or output formats, which would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and have made revisions to strengthen the presentation of the method and its validation.

read point-by-point responses

  1. Referee: [Abstract and method definition] The abstract and the description of the method assert that near-coincidence selection 'maps naturally' to cut-and-project and yields uniformly discrete quasiperiodic sets, yet no explicit tolerance criterion, irrationality condition on the rotation matrix, or proof of uniform discreteness for generic twist/scale parameters is supplied. This is load-bearing for the central claim that the procedure is parameter-free and automatic.

    Authors: We agree that additional explicit details are necessary to fully substantiate the mapping and the discreteness claim. In the revised version, we have expanded the Methods section to include: (i) a precise definition of the tolerance criterion, where a point pair is admitted if their distance is less than a parameter ε chosen based on the minimal lattice spacing and the twist/scale factors; (ii) the condition that the rotation angle must yield an irrational rotation matrix with respect to the lattice vectors to guarantee quasiperiodicity; and (iii) a brief argument for uniform discreteness, showing that the selected points correspond to a bounded window in the perpendicular space of the cut-and-project scheme, ensuring no accumulation points. These additions make the procedure's rigor explicit without altering its intuitive nature. revision: yes

  2. Referee: [Results on classical tilings] Reproduction of the Ammann-Beenker and Niizeki-Gähler tilings is stated, but the manuscript provides neither a side-by-side comparison of the generated point sets with the standard cut-and-project constructions nor a demonstration that the near-coincidence step enforces the same higher-dimensional lattice embedding without post-selection.

    Authors: We concur that visual and explicit verification would enhance the results section. We have added a new figure comparing the point sets obtained via near-coincidence with those from the standard cut-and-project method for both the Ammann-Beenker and Niizeki-Gähler tilings, confirming they match exactly. Furthermore, we have included a paragraph demonstrating that the near-coincidence selection inherently selects points whose perpendicular coordinates fall within the acceptance window of the higher-dimensional lattice, thus enforcing the embedding without any post-selection step. revision: yes

Circularity Check

0 steps flagged

No circularity: method explicitly anchored in independent cut-and-project formalism with external reproduction of known tilings

full rationale

The paper defines the near-coincidence method by selecting pairs of nearly coincident points from superimposed layers and asserts that this maps naturally to the established cut-and-project formalism while reproducing classical tilings (Ammann-Beenker, Niizeki-Gähler, Fibonacci). No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the validity claim rests on the external, independently established cut-and-project method rather than internal equivalence. The abstract and description provide no equations or definitions where a prediction is forced by the input selection itself. This is the common honest case of a self-contained construction grounded outside the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the near-coincidence selection rule is equivalent to the cut-and-project construction; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Quasiperiodic tilings of the plane can be obtained via the cut-and-project method from higher-dimensional periodic lattices
    The paper states that the near-coincidence method maps naturally to this established formalism.

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Reference graph

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