Frieze patterns and aperiodic tilings of the plane

Dirk Frettl\"oh; Jan Maz\'{a}\v{c}

REVIEW 2 major objections 4 minor 22 references

Frieze patterns and aperiodic tilings of the plane

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper establishes that frieze patterns—positive-integer vertex labellings obeying the diamond rule bc−ad=1 on every rhombus—can be placed on two famous aperiodic tilings: the Penrose rhombic tiling uses only four distinct values and the

desk verdict Penrose construction is new and sound; the GLB proof has a real gap and needs referee scrutiny. read the letter →

arxiv 2607.21348 v1 pith:XQ3DAC6V submitted 2026-07-23 math.CO math.MG

Dirk Frettlöh , Jan Mazáč This is my paper · ORCID

classification math.COmath.MG MSC 05B4552C23

keywords friezepatternsdiamondrulePenroserhombictilingGodrèche–Lançon–BillardaperiodictilingsConway–Coxeterfriezessupertiledecompositionsubstitution

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 tries to establish that frieze patterns, arrays of positive integers satisfying the diamond rule bc−ad=1 on every elementary rhombus, can live on aperiodic tilings rather than only on periodic rhombic grids. Its main theorems are explicit: every Penrose rhombic tiling carries a vertex labelling with only four values, and every Godrèche–Lançon–Billard tiling carries one with only three values. The Penrose construction comes from a dual-lattice projection and a residue modulo 5 that never vanishes, forcing each tile into one of two label decorations; the GLB construction comes from the tiling's unique decomposition into supertiles, with labels 1, 2, and 3 assigned to supertile corners, edge-neighbours, and interiors. In both cases, the identity (n+1)²−n(n+2)=1 yields infinitely many variants. A sympathetic reader would care because this opens frieze patterns, long tied to polygon triangulations and periodic geometry, as a new invariant of aperiodic order.

What carries the argument

The load-bearing identity is the diamond rule itself, bc−ad=1, together with the algebraic fact (n+1)²−n(n+2)=1, which makes any rhombus decorated with values n, n+1, n+1, n+2 satisfy the rule. For the Penrose tiling, the mechanism is a residue function r(v)=Σnᵢ mod 5 on the dual lattice A₄*, with vertices never having residue 0 and every tile containing exactly one shallow hole; this forces each thick or thin rhombus into one of two allowed label patterns. For the GLB tiling, the mechanism is the unique decomposition into supertiles, which yields a global 1/2/3 classification of vertices with no ambiguity across supertile boundaries. In both cases the argument is local: once the finite list

What would settle it

Enumerate all rhombi in a large patch of a GLB tiling and check the local rule: every rhombus must carry labels of the form {n, n+1, n+1, n+2} with the smallest and largest labels opposite each other; a single rhombus with the smallest and largest labels adjacent would violate the diamond rule. For the Penrose tiling, compute the residues mod 5 from pentagrid coordinates of every vertex in a finite patch; finding a vertex with residue 0, or a tile with zero or two shallow holes, would disprove the construction.

Watch

Extended reading notes

Core claim

On the Penrose rhombic tiling, every vertex can be expressed as an integer combination of four dual-lattice generators, and reducing the sum of coefficients modulo 5 gives a value r(v) that is never zero, hence takes only the values 1, 2, 3, 4. Each rhombus has exactly one shallow-hole vertex, and when a tile is oriented so that this vertex plays the role of a in the diamond rule, the four labels are always of the form {n, n+1, n+2} with the shallow hole at n, for n=1 or 2; the two possible decorations per rhombus type are checked directly and satisfy bc−ad=1. On the GLB tiling, the unique supertile decomposition, guaranteed by the primitive substitution rule, gives a globally consistent lab

Load-bearing premise

For the GLB half, the load-bearing premise is the cited unique-decomposition property: if a GLB tiling could be grouped into supertiles in two different ways, or if one vertex were both one edge from a supertile corner and an interior point of another, the 1/2/3 labelling could conflict and the diamond rule could fail.

Editorial extensions

If this is right

Frieze patterns are not tied to periodic geometry: aperiodic rhomb tilings can carry global positive-integer labellings satisfying the diamond rule on every tile.
Every one of the uncountably many locally indistinguishable Penrose tilings carries the same four-value decoration, so the frieze is an invariant of the local isomorphism class.
Every GLB tiling carries a three-value frieze despite having singular continuous diffraction and no known higher-dimensional projection description, so frieze existence does not require pure-point diffraction.
Shifting all labels by an integer gives infinitely many frieze patterns on each tiling; for the Penrose tiling, even real shifts preserve the diamond rule.
The concluding questions—whether every aperiodic rhomb tiling admits a frieze, and whether the Ammann–Beenker tiling can be labelled—are now concrete open problems with a clear local-check condition to test.

Reading between the lines

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

Since two distinct positive integers a<b cannot satisfy b²−a²=1, three values is the theoretical minimum for any positive-integer frieze on any rhombus tiling; the GLB construction therefore sits at the absolute lower bound.
The Penrose construction suggests a general recipe for cut-and-project tilings: a residue function modulo N that never vanishes and separates shallow from deep vertices. A natural test is the Ammann–Beenker tiling, where a suitable modulus or shallow-hole definition might work even though a direct analogue of the present attempt did not.
Because the labels depend only on vertices' positions relative to the supertile structure, the GLB frieze is compatible with the substitution rule and may lift to a continuous function on the tiling space, potentially encoding a cohomological invariant analogous to height functions in other substitution tilings.
Combining the two constructions suggests a classification question: aperiodic rhomb tilings admitting a finite-label frieze may be exactly those with a hierarchical vertex classification of bounded radius—mod-5 residues for Penrose, one-edge distance from supervertices for GLB.

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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

Dear colleague,

Here's the short version: the Penrose construction in this note is real and new; the GLB construction is not proved as written. The stress-test note is right to worry.

The genuinely good part is Theorem 2.1. Using the de Bruijn index r(v) mod 5 from the A4 dualisation, the authors get a global label on every Penrose vertex, r never vanishes, every tile has exactly one shallow hole, and the local decorations are forced into the shapes 1,2,2,3 and 2,3,3,4 (or rotated). Each satisfies bc−ad=1. The "one can easily check" is a bit terse, but the check is small and the claim is credible. The infinite family via scaling is trivial but harmless.

The GLB part is the soft spot. The labeling rule is: supertile vertices get 1, their edge-neighbors get 2, everyone else gets 3. The proof that the diamond rule holds says that since each rhombus lies in one supertile, the rule is immediate. That is not immediate. A rhombus in the interior of a supertile can have all four vertices at distance ≥2 from every supertile vertex; all four would be labeled 3, and the diamond rule would give 9−9=0, not 1. The 'three simple steps' only discuss supervertices, midpoints of superedges, and the consistency of label 3 for vertices contained in the interior of a supertile. They never verify that an interior rhombus has a vertex labeled 1 or 2, let alone the right arrangement. Unique composition ([21]) gives a unique grouping, but no bound on interior distances to supertile corners. So Theorem 3.1 is currently unsupported. It might be true and fixable, but the argument as it stands doesn't carry it.

The reader's 7/10 soundness score is a bit generous for the GLB section; the Penrose section alone would deserve it. Significance and novelty scores are fair. Citation pattern is fine: de Bruijn and Solomyak are the right sources, and the new connection to [13] is a real lead.

Who is this for? Someone working in frieze combinatorics or aperiodic tilings will care about the Penrose example. The GLB gap makes the paper a discussable preprint rather than a finished one.

Recommendation: send it to peer review, but make the referee focus on the GLB labeling. The referee should demand a full check of the substitution atlas or a corrected labeling. If that gets fixed, it's a nice short contribution.

Referee Report

2 major / 4 minor

Summary. The paper introduces frieze patterns supported on aperiodic rhombic tilings, i.e. positive-integer decorations of the vertices so that every rhombic tile satisfies the diamond rule bc−ad=1. It gives two constructions. For every Penrose rhombic tiling, a function r defined from A*_4 coordinates labels vertices by residues mod 5; each tile is claimed to have exactly one shallow hole, and the resulting local patterns are {1,2,2,3} or {2,3,3,4}, yielding a frieze pattern with four values (Theorem 2.1). For the Godrèche–Lançon–Billard tiling, the unique decomposition into supertiles is used: supervertices are labelled 1, vertices one edge away from a supervertex are labelled 2, and all remaining vertices are labelled 3; the paper claims this gives a frieze pattern with three values (Theorem 3.1). Corollaries assert infinitely many friezes by shifting the numerical labels.

Significance. If established, these are the first frieze patterns on aperiodic rhombic tilings, connecting Conway–Coxeter friezes with aperiodic order. The Penrose construction is explicit and parameter-free, and the GLB construction attempts to use the hierarchical decomposition in a genuinely new way. The paper is short and readable, and the Penrose part is plausible. However, the GLB proof is incomplete at a load-bearing point: the argument does not verify the diamond rule for rhombi in the interior of a supertile. Because the central construction is likely correct and the gap is repairable by a finite local check, the appropriate outcome is a major revision rather than acceptance as written.

major comments (2)

[§3, paragraph 'Since the GLB tiling can be grouped into supertiles...'] The proof of Theorem 3.1 does not rule out rhombi whose four vertices all receive label 3. With the stated rule (supervertices = 1, vertices one edge from a supervertex = 2, remaining = 3), a rhombus in the interior of a supertile can have all its vertices at graph distance at least 2 from every supervertex. For such a rhombus the diamond rule gives 3·3−3·3 = 0, not 1. The unique decomposition property cited from [21] guarantees a unique grouping into supertiles but imposes no lower bound on the distance from interior vertices to supertile corners. The 'three simple steps' only discuss supervertices and the midpoint of a super-edge; they say nothing about interior rhombi or about rhombi elsewhere along a super-edge. The authors need to supply a finite verification, for example from the GLB substitution atlas, that every rhombus inside a supertile contains a supervertex or a vertex adjace
[§2, 'One can easily check...' and Figure 4] The proof of Theorem 2.1 rests on the assertion that every tile has exactly one shallow hole and on the enumeration of possible decorations for thick and thin rhombuses. This is a finite check, but the paper only says it is easy. Because this local verification is the entire content of the Penrose theorem, the authors should list the possible residue patterns around each tile type, or give a precise citation that includes this statement. In addition, the definition of r(v)=Σ n_i mod 5 assumes that the representation v=Σ n_iπ(a_i) gives a well-defined residue; since the π(a_i) are linearly dependent in R^2, a sentence explaining uniqueness of the lift (or a reference to [18]) is needed.

minor comments (4)

[Corollaries 2.2 and 3.2] The phrase 'n may as well be an arbitrary real number' conflicts with the definition of a frieze pattern as positive integers. Either restrict n to positive integers or explicitly extend the definition to real labels.
[Figure 7 and surrounding text] The text says 'This labelling is indicated in Figure 7', but the figure caption describes only black points and arrows. Please provide a labelled version or a table showing the label pattern for each tile type within a supertile.
[§1, definition of frieze patterns on tilings] The notion of a frieze pattern on a rhombic tiling is clear from context, but it would help to state formally: a vertex labelling such that every tile's opposite vertex pairs (a,d) and (b,c) satisfy bc−ad=1. The diamond rule orientation should be specified once in full generality.
[§2, notation] The paragraph on the A*_4 construction is terse. The statement 'Since V∈A*_4\A_4, we always have Σn_i not≡0 mod 5' is a known fact, but the exact page reference [18] could be expanded to a theorem statement so the reader does not have to consult the source.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: both frieze patterns are explicit constructions; no fitted parameter is renamed as a prediction, and the cited external facts do not assume the target results.

full rationale

The paper's two constructions are explicit rather than fitted. For the Penrose tiling, the labelling is the fixed function r(v) = sum n_i mod 5 derived from the A4* dualisation construction, and the diamond rule is checked tile-by-tile from the two possible decorations of thick and thin rhombuses; no parameter is fitted to make the rule hold. The citation to [18] (one of the authors' earlier papers) is used only for a standard lattice fact about A4* and does not assume the existence of frieze patterns. For the GLB tiling, the labelling is defined by supertile-distance classes (1 for supervertices, 2 for vertices one edge away, 3 for all others) and the argument invokes Solomyak's unique decomposition property [21], an external result that neither assumes nor constructs frieze patterns. The self-citations in the paper are contextual or supply standard background facts; they are not load-bearing in a circular sense. The skeptic's concern about GLB interior rhombi possibly receiving only label 3 is a possible correctness gap in the three-step consistency argument, but it is not circularity: the labelling is not defined in terms of the diamond rule, and the missing verification is an omitted local check rather than a reduction of the theorem to its own assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. It relies on standard constructions of Penrose and GLB tilings from the literature and on two local/global consistency checks that are asserted rather than formalized.

assumptions (4)

domain assumption Penrose tilings are obtainable by dualisation of A*_4 and every vertex V∈A*_4∖A4 has Σn_i not ≡0 mod 5.
Invoked in §2 to define r(v)=Σn_i mod5 with values 1–4; supported by [3,18], not proved in the note.
domain assumption On every thick/thin Penrose rhombus the four r-values are exactly {n,n+1,n+1,n+2} with the shallow hole placed consistently so that the diamond rule holds.
Central finite check in §2; only two prototype decorations are shown, so the reader must trust the enumeration of ten orientations/two classes.
domain assumption The GLB substitution is primitive and hence every GLB tiling has the unique decomposition property [21].
Used in §3 to group any GLB tiling into supertiles; imported from [21].
ad hoc to paper The GLB vertex classification (1 at supervertices, 2 at distance-one vertices, 3 otherwise) is globally unambiguous; in particular no vertex is simultaneously of two classes and every tile sees the pattern {1,2,2,3}.
Argued in three informal steps in §3; the proof does not formalize the distance-1 graph or rule out ambiguous vertices.

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Cite this review

Pith. "Pith review of Frieze patterns and aperiodic tilings of the plane." pith.science (2026). https://pith.science/paper/XQ3DAC6V

@misc{pith2026260721348,
  author       = {Pith},
  title        = {Pith review of: Frieze patterns and aperiodic tilings of the plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQ3DAC6V}},
  note         = {Machine review of arXiv:2607.21348}
}

read the original abstract

This short note provides two examples of aperiodic frieze patterns of the plane, supported on the rhombic Penrose tiling and the Godr\`eche--Lan\c{c}on--Billard tiling. That is, we provide a decoration of their vertices with positive integers which satisfy the diamond rule, in analogy to the usual (in)finite frieze patterns as defined by Conway and Coxeter.

Figures

Figures reproduced from arXiv: 2607.21348 by the authors.

**Figure 1.** Periodic pattern on the left and a patch of Penrose rhombic tiling. this grid as a periodic rhomb tiling of the plane. Of course, a finite frieze pattern is only a tiling of some strip in such a rhombic tiling. But if we allow the labels to be arbitrary integers, rather than positive integers, we can extend the frieze pattern throughout the entire tiling, still obeying the diamond rule: on top of the top row of zero… view at source ↗

**Figure 2.** Projections of the generators ai into the tiling plane. It follows that every vertex v of a Penrose tiling can be obtained as a projection of a point V ∈ A∗ 4\A4 and V = X 4 i=1 niai , ni ∈ Z, thus giving v = P4 i=1 niπ(ai ). Since V ∈ A∗ 4\A4 , we always have P i ni ̸≡ 0 mod 5, see [18, page 5]. We can then assign to each vertex v = P4 i=1 niπ(ai ) the evaluation r(v) = P i ni mod 5. We note that the function r cor… view at source ↗

**Figure 3.** Penrose rhombuses with indicated shallow hole vertex, and the corresponding naming of the vertices which allows to define the diamond rule bc − ad = 1. With this rule, the function r provides a valid frieze pattern of any Penrose rhombic tiling. For that, observe that the thick rhombus appears in ten different orientations. They split into two classes, which are related by a 2π 10 -rotation. Since r cannot be zero o… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The two possible vertex decorations for the thick rhombuses. Each of the tiles appears in five different orientations. The double circle indicates the shallow hole. The same holds for the thin rhombuses ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: The two possible vertex decorations for the thin rhombuses. Each of the tiles appears in five different orientations. The double circle indicates the shallow hole [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: A frieze pattern fulfilling the diamond rule superimposed on a Penrose rhomb tiling. Image from [2] with approval from the authors [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: The action of the original GLB substitution rule on thick and thin rhombuses. The black points and arrows serve to mark the orientation of each tile. The tiling is invariant under the fourth iteration of the substitution rule. Since the substitution rule is primitive …

**Figure 8.** Figure 8: A patch of GLB tiling (grey) with its level-1 supertile structure (black). label 2. The remaining vertices all get label 3. This labelling is indicated in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: A frieze pattern fulfilling the diamond rule with labels 1, 2, 3 superimposed on a GLB tiling. 4. Concluding remarks Here we introduced the first two examples of frieze patterns on aperiodic rhomb tilings. This raises several questions: Are there frieze patterns on eve…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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