A Family of Non-Periodic Tilings, Describable Using Elementary Tools and Exhibiting a New Kind of Structural Regularity

Miki Imura

REVIEW 4 major objections 6 minor 1 cited by

A Family of Non-Periodic Tilings, Describable Using Elementary Tools and Exhibiting a New Kind of Structural Regularity

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs non-periodic tilings from modular progression direction sequences and proves the resulting wedges fill the plane exactly.

desk verdict Explicit construction with a real proof gap: the wedge tiling is asserted, not proved, so Theorem 6.11 rests on an unverified local condition. read the letter →

arxiv 2506.07638 v1 pith:FPUTV27I submitted 2025-06-09 math.CO

Miki Imura This is my paper

classification math.CO MSC 05B4552C2052C23

keywords non-periodictilingmodularprogressionprototilewedgeshiftedstructuralregularityrotationalsymmetryplane

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

This paper proposes a family of plane tilings, called Modulo Krinkle tilings, that are non-periodic yet highly organized. The construction uses only modular arithmetic and unit vectors at fixed angles, starting from a single polygonal prototile whose two sides are direction sequences based on multiples of $m$ modulo $k$. Copies of the prototile are placed recursively in wedge-shaped regions, and wedges are rotated around a center until their boundaries match; the matching is proved by a recurrence on shifted modular progressions. The paper's central claim is that these wedges fill the whole plane exactly (Theorem 6.11), and that the resulting patterns exhibit a generalized modulo-staggered rotational symmetry that ordinary Euclidean symmetry definitions do not capture. The significance is a self-contained, explicit family of non-periodic tilings with a clearly articulated new form of structural regularity.

What carries the argument

The carrying object is the shifted modular progression with its characterization condition $C(i)$. For each $i$, the sequence starts as multiples of $m$ modulo $k$ and is transformed by replacing each occurrence of $i$ with $i+k$; condition $C(i)$ requires that all terms lie in $[i,i+k)$, that the first $k$ terms permute those integers, that the sequence repeat with period $k$, and that each next term be $a_j+m$ reduced into that interval. This condition is what makes the lower boundary of each rotated wedge align exactly with the current front: both the front sequence and the wedge lower boundary satisfy the same $C(i)$, so they agree from the matching edge onward. A second ingredient is the explicit placement rule for the wedge, with row-shift vector $\mathbf{d}_0$ equal to the sum of all lower-path unit vectors and in-row-shift vector $\mathbf{d}_1=\mathbf{v}_k-\mathbf{v}_0$, placing tiles $T_{r,c}$ in a triangular array. Together these yield the boundary-matching identities $f^{k-1}_j=b_j+k$ and $f^{n/2-1}_j=b_{j+1}+n/2$, which are the basis for Theorem 6.11.

What would settle it

For a concrete parameter triple such as $(m,k,n)=(3,7,14)$, compute the union of all tile polygons placed in the wedge up to several rows and test whether every point of the intended wedge region is covered exactly once; any uncovered or doubly covered point would refute Theorem 6.11. The same pointwise check can be applied to the rotated copies of the final sector to test the full plane tiling.

Watch

Extended reading notes

Core claim

The central discovery is that an elementary direction-sequence construction gives infinitely many non-periodic tilings whose boundary edges follow a shifted modular progression. The prototile is formed by two paths: the lower path follows the modular progression of multiples of $m$ modulo $k$ with $k$ appended, and the upper path swaps the first and last terms; both paths share start and end points and close into a simple polygon. Translating copies of this prototile by a row-shift and an in-row-shift vector produces a wedge, and rotating copies of that wedge about a center produces the plane tiling. The proof of completeness works by showing that after each wedge is placed, the front boundary sequence is exactly the next shifted modular progression, a sequence satisfying a three-part recurrence condition $C(i)$. Because both the front and the new wedge's lower boundary satisfy the same $C(i)$, they align exactly; at the final wedge the front equals the base rotated by $2\pi/t$ (without offset) or by $\pi$ about the midpoint of the first base edge (with offset), so the rotated copies close the plane without gaps or overlaps. The paper also defines $n$-fold $(m,k)$-modulo-staggered rotational symmetry as the property of being divided into $n$ congruent wedge regions rotated by $2\pi/n$, each with identical internal structure.

Load-bearing premise

The proof of completeness in Section 6 only aligns boundary direction sequences; the claim that the wedge is fully tiled with no gaps or overlaps in its interior is asserted from the recursive placement of congruent tiles, without an explicit point-to-tile map.

Editorial extensions

If this is right

For any coprime $m<k$ and any valid $n$, the construction produces an explicit non-periodic tiling of the plane whose edges are fully described by modular progressions.
Without offset, the completed tiling has $t$-fold rotational symmetry, because $k$ wedges form one sector and $t$ rotated sectors close the plane.
With offset, the completed tiling has a half-turn symmetry about the midpoint of the first base edge, and the front after $n/2$ wedges matches the truncated base rotated by $\pi$.
Every $(m,k,n)$-Modulo Krinkle tiling is composed of $n$ congruent wedge-shaped regions rotated in steps of $2\pi/n$, so each tiling is an instance of the proposed $n$-fold $(m,k)$-modulo-staggered rotational symmetry.
The boundary sequences are periodic with period $k$, so the entire structure is governed by the finite arithmetic data $(m,k,n)$.

Reading between the lines

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

The alignment argument likely extends to other final shift targets: whenever the front and the wedge lower boundary satisfy the same condition $C(i)$, any shift that makes the final front equal a rotated or translated copy of the base should give another closed tiling, not only the $k$ and $n/2$ cases treated here.
Because the construction uses only fixed unit vectors and modular arithmetic, it can be implemented as an exact symbolic tiling, making it a convenient classroom example of a non-periodic tiling that still admits a short completeness proof.
Measuring how closely the wedge boundaries in the offset cases approximate true logarithmic spirals would clarify whether modulo-staggered tilings are better classified as spiral-like or as a separate modular phenomenon; the paper does not undertake that quantitative comparison.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper constructs a family of plane tilings from modular progressions and gives them a new name, modulo-staggered rotational symmetry. The concrete construction is new as far as I know, and the boundary-sequence induction in Section 6 is coherent. But the central claim, Theorem 6.11, is not established: the wedge-shaped region of Section 4 is assumed to be tiled exactly, and the paper itself says no point-to-tile map is given. Section 6 proves only that the direction sequences of the front and the incoming wedge align. That is a property of the boundary curve, not of the region; it says nothing about whether tile interiors overlap or leave gaps. Since the prototile is non-convex with a repeated interior chain, an overlap with positive area is possible even when all boundary directions match. The non-periodicity claim in the abstract and Section 1 is also never proved; the paper only notes the prototile admits some periodic tiling.

What the paper does well is make the construction fully explicit: the tile placements are given by linear formulas, the shifted modular progression is clearly defined, and the induction in Propositions 6.8–6.10 correctly derives the front sequence evolution assuming the wedge tiles. That is a real technical contribution, but it is conditional on the missing local coverage proof.

The new symmetry notion is definitional: a figure has modulo-staggered rotational symmetry if it is divided into wedges according to a Modulo Krinkle tiling. So the conceptual novelty is currently a label, not a derived invariant. The references to spiral tilings are acknowledged in passing but not engaged with seriously.

The paper is honest about its own gap, which is a good sign. A revised version with a rigorous proof that the wedge placement covers without overlap, plus a periodicity argument, could become a solid contribution. As written, the main theorem rests on an unproved local condition, so I do not think it should be accepted.

Recommendation: reject, but send it to a referee who can assess whether the gap is fixable. It deserves a serious reviewer; the construction is explicit enough to be checked, and if the wedge tiling works, the paper would be a nice addition to tiling theory.

Referee Report

4 major / 6 minor

Summary. This paper constructs a family of tilings from a polygonal prototile whose two boundary paths are generated by a modular progression (m,k). Congruent copies are arranged by translation vectors d0 and d1 into wedge-shaped regions, and the wedges are then rotated and assembled to cover the plane. The paper claims that these tilings are non-periodic and exhibit a new 'modulo-staggered rotational symmetry.' Section 6 gives an induction on direction sequences intended to prove that the assembly is complete.

Significance. The construction is explicit and self-contained, and the parameter family is clearly specified with numerous figures. If the missing coverage, disjointness, and aperiodicity arguments were supplied, the paper could offer an instructive elementary family of non-periodic tilings. However, as it stands the main existence theorem depends on an unproved assertion in Section 4, the induction in Section 6 proves only boundary-sequence alignment, and the paper's defining notion of modulo-staggered rotational symmetry is introduced circularly. The concrete strengths are the explicit translation rules T_{r,c}=r d0 + c d1 and the verification that front direction sequences and incoming wedge lower boundaries match; the load-bearing gaps are the absence of any proof of disjoint interiors and coverage, and the absence of a proof of non-periodicity.

major comments (4)

[Section 4] The central step of the construction, that the tiles T_{r,c} with 0 ≤ c ≤ r exactly tile a wedge-shaped region without gaps or overlaps, is asserted rather than proved. The text itself states that no formal point-to-tile map is given. Since every subsequent claim, including Theorem 6.11, uses the wedge as the fundamental building block, this is a load-bearing gap. A rigorous proof must establish that the interiors of the placed tiles are pairwise disjoint and that their union is the claimed wedge region.
[Section 6 (Propositions 6.8–6.10)] Proposition 6.8 proves that the direction sequence of the incoming wedge's lower boundary matches the current front, and Propositions 6.9–6.10 derive symmetry of the final front direction sequence. Direction-sequence equality is a property of boundary curves, not of filled regions. It does not show that the new wedge lies on the exterior side of the front, that the front is a simple curve, or that neighboring wedges do not overlap with positive area. Because the prototile is non-convex and contains translated copies of the same interior chain, overlaps or gaps are possible even when all boundary directions agree. Theorem 6.11 is then stated as 'Immediate' from these propositions, so the completeness claim inherits the gap.
[Abstract and Section 7] The abstract claims 'non-periodic tilings', but nowhere in the paper is it proved that the constructed tilings have no nonzero translational symmetry. Section 7 even states that the prototile admits a periodic tiling of the plane; while this is not formally contradictory, the paper must either prove that the specific wedge-based construction is non-periodic or qualify the claim. Without such a proof, the term 'non-periodic' in the abstract and title is unsupported.
[Section 7] The definition of n-fold (m,k)-modulo-staggered rotational symmetry is circular: a figure 'exhibits' this symmetry if it is divided into n wedge-shaped regions according to a (m,k,n)-Modulo Krinkle tiling, with each region having the same internal structure. Since the property is defined in terms of the very tiling it is meant to describe, the claim that these tilings exhibit a new kind of structural regularity is tautological. An intrinsic, construction-independent definition of the symmetry would be needed for the claim to be meaningful.

minor comments (6)

[Section 3] The assertion that the interior segments of the lower and upper paths 'run parallel and never intersect' is stated without proof; since this is used to conclude that the prototile is a simple polygon, a short argument should be supplied.
[Section 4] The sentence describing which edge of the new tile shares which opposite edge of the origin tile appears to swap 'base' and 'neighbor' relative to the formal translation vectors; please reconcile the wording with the actual placement.
[Section 2] The phrase 'We use a mod b to denote' has a typographical spacing issue and should read 'We use a mod b to denote the remainder...'.
[Section 7] The statement that the prototile 'admits a periodic tiling' may confuse readers because it follows the abstract's claim of non-periodic tilings; add a clarifying sentence explaining that a prototile can admit both periodic and non-periodic tilings.
[References] Reference [5] is incomplete: 'in kongruente' should be followed by a noun, e.g., 'in kongruente Teile'.
[Figure 4] The caption should state that the figure is a schematic diagram rather than an output of the actual construction, to avoid implying a computational verification.

Circularity Check

1 steps flagged · score 5.0 of 10

One self-definitional step: the headline 'modulo-staggered rotational symmetry' is defined in terms of the Modulo Krinkle tiling itself, so the claim that these tilings exhibit it is true by definition; the tiling construction itself is not circular but has a rigor gap.

self definitional [Section 7, modulo-staggered rotational symmetry; abstract]
"We say that a figure exhibits n-fold (m, k)-modulo-staggered rotational symmetry if it is divided into n wedge-shaped regions according to a (m, k, n)-Modulo Krinkle tiling, with each region exhibiting the same internal structure."

The abstract announces that the tilings 'exhibit a distinct type of structural regularity, which we term modulo-staggered rotational symmetry.' The property is defined in Section 7 by reference to being divided into n wedge-shaped regions according to a (m, k, n)-Modulo Krinkle tiling, i.e., by the very construction whose patterns it is supposed to explain. Thus the assertion that the tilings exhibit this symmetry is an instance of the definition rather than an independently derived regularity. The paper is transparent that it is proposing a term, so this is a definitional tautology for the conceptual claim, while the wedge-decomposition facts themselves are independent.

full rationale

Aside from the Section 7 definitional step, I find no circularity in the tiling construction. Theorem 6.11 rests on Propositions 6.8 through 6.10, which compare direction sequences via the shifted modular progression; the recurrence proofs are self-contained and do not import fitted parameters or prior results. The completeness proof has a genuine rigor gap, because Section 4 concedes that no point-to-tile map is given and the propositions show boundary direction sequences coincide without proving that tile interiors are disjoint. A proof gap is not circularity. There are no load-bearing self-citations and no imported uniqueness theorems. The non-periodicity claim is asserted without proof, but that too is an unsupported claim rather than a circular reduction. The only circular step is the naming of the wedge decomposition as a new symmetry, which is definitional and explicitly flagged in Section 7.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central construction has no fitted constants. Its main hidden assumptions are geometric: the prototile is simple, the wedge fills without overlap, boundary alignment transfers to edge coincidence, and the final tilings have no translational symmetry. The new symmetry concept is an invented definition, not an independently evidenced entity.

free parameters (1)

m, k, t (and derived n)
Positive integers defining the family; not fitted to data, but the completeness proof and symmetry definition depend on the specified relationships (m < k, gcd(m,k) = 1, t >= 2).

assumptions (4)

ad hoc to paper The two polygonal paths from Section 3 form a simple polygon and do not intersect in their interiors.
Stated in Section 3 from a monotonicity remark, but used as a geometric fact underlying every tile; no detailed proof is given.
ad hoc to paper The wedge-shaped region of Section 4 is tiled without gaps or overlaps by the T_{r,c} placements.
Section 4 explicitly says no formal point-to-tile map is supplied; this is the main missing rigor.
domain assumption Alignment of boundary direction sequences implies geometric coincidence of the corresponding tile edges.
Section 6 proves sequence-level alignment but assumes this transfers to actual edge coincidence in the plane.
ad hoc to paper The constructed plane tilings have no nonzero translational symmetry.
Non-periodicity is claimed in the title and abstract but never proven; Section 7 only discusses rotational symmetry and concedes the prototile admits a periodic tiling.

invented entities (2)

(m,k,n)-Modulo Krinkle tiling
purpose: Named family of tilings constructed from modular progressions and unit vectors.
A named construction whose valid existence depends on the unproved wedge-tiling assumption.
n-fold (m,k)-modulo-staggered rotational symmetry
purpose: Proposed new symmetry concept to describe the tilings.
Defined in Section 7 in terms of being divided according to a Modulo Krinkle tiling, so the claim that the tilings exhibit it is definitional.

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Cite this review

Pith. "Pith review of A Family of Non-Periodic Tilings, Describable Using Elementary Tools and Exhibiting a New Kind of Structural Regularity." pith.science (2026). https://pith.science/paper/FPUTV27I

@misc{pith2026250607638,
  author       = {Pith},
  title        = {Pith review of: A Family of Non-Periodic Tilings, Describable Using Elementary Tools and Exhibiting a New Kind of Structural Regularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FPUTV27I}},
  note         = {Machine review of arXiv:2506.07638}
}

read the original abstract

We present a construction of a family of non-periodic tilings using elementary tools such as modular arithmetic and vector geometry. These tilings exhibit a distinct type of structural regularity, which we term modulo-staggered rotational symmetry. The construction is self-contained and does not rely on previous tiling theories or systems.

Figures

Figures reproduced from arXiv: 2506.07638 by the authors.

**Figure 2.** Two examples of the prototile constructed for (m, k) = (3, 7), with different values of n. Each edge is annotated with a direction label for clarity. When n = 2k, the interior angles at the starting point and ending point are exactly 180◦ , causing the first and last edge pairs to appear as single straight segments of doubled length. However, we still treat them as two separate edges in our construction. To facili… view at source ↗

**Figure 3.** A conceptual diagram of the prototile, with the starting point P, ending point Q, base edge B, and neighbor edge N labeled for reference. 4. Building a Wedge-Shaped Region We next describe how to construct a wedge-shaped region—a key component in the full tiling—by placing congruent copies of the prototile in the same orientation. While we do not provide a fully rigorous proof in the formal sense—such as defining … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A conceptual diagram of an initial wedge. Each tile is placed such that its base edge or neighbor edge aligns with the opposite edge of a previously placed tile. The resulting wedge is bounded by two polygonal paths extending from the origin: the lower boundary and the…

**Figure 5.** Figure 5: A variety of (m, k, n)-Modulo Krinkle tilings [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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Forward citations

Cited by 1 Pith paper

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Role of (periodic as well as aperiodic) tessellations in contemporary composition. The cases of Tesselles sonores and Le Chapeau \`a douze cornes by Marisa Acu\~na
math.HO 2026-01 unverdicted novelty 5.0 of 10

Aperiodic monotiles such as the Hat are translated into microtonal intervals in the compositions Tesselles sonores and Le Chapeau à douze cornes.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 1 Pith paper

[1]

Branko Grünbaum and G. C. Shephard.Tilings and Patterns . Dover, second edition, 2016

work page 2016
[2]

Pentaplexity: A class of non-periodic tilings of the plane.The Mathe- matical Intelligencer, 2:32–37, 1979

Roger Penrose. Pentaplexity: A class of non-periodic tilings of the plane.The Mathe- matical Intelligencer, 2:32–37, 1979

work page 1979
[3]

Kaplan, and Chaim Goodman-Strauss

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. An aperiodic monotile.Combinatorial Theory, 4(1):1–91, 2024

work page 2024
[4]

Kaplan, and Chaim Goodman-Strauss

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. A chiral aperiodic monotile.Combinatorial Theory, 4(2):1–25, 2024

work page 2024
[5]

Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente

Heinz Voderberg. Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente. Jahresbericht der Deutschen Mathematiker-Vereinigung , 46:229–231, 1936

work page 1936
[6]

Spiral tilings

Paul Gailiunas. Spiral tilings. In Reza Sarhangi, editor,Bridges: Mathematical Con- nections in Art, Music, and Science , pages 133–140. 2000. Appendix A. Gallery of Generated Patterns The following figures illustrate diverse instances of the Modulo Krinkle tiling. These patterns are generated with varying parameters and highlight the visual diversity and...

work page 2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Branko Grünbaum and G. C. Shephard.Tilings and Patterns . Dover, second edition, 2016

work page 2016

[2] [2]

Pentaplexity: A class of non-periodic tilings of the plane.The Mathe- matical Intelligencer, 2:32–37, 1979

Roger Penrose. Pentaplexity: A class of non-periodic tilings of the plane.The Mathe- matical Intelligencer, 2:32–37, 1979

work page 1979

[3] [3]

Kaplan, and Chaim Goodman-Strauss

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. An aperiodic monotile.Combinatorial Theory, 4(1):1–91, 2024

work page 2024

[4] [4]

Kaplan, and Chaim Goodman-Strauss

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. A chiral aperiodic monotile.Combinatorial Theory, 4(2):1–25, 2024

work page 2024

[5] [5]

Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente

Heinz Voderberg. Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente. Jahresbericht der Deutschen Mathematiker-Vereinigung , 46:229–231, 1936

work page 1936

[6] [6]

Spiral tilings

Paul Gailiunas. Spiral tilings. In Reza Sarhangi, editor,Bridges: Mathematical Con- nections in Art, Music, and Science , pages 133–140. 2000. Appendix A. Gallery of Generated Patterns The following figures illustrate diverse instances of the Modulo Krinkle tiling. These patterns are generated with varying parameters and highlight the visual diversity and...

work page 2000