A $\sigma$-morphic convex protoset

Aleksa D\v{z}uklevski

REVIEW 4 major objections 4 minor 29 references

A $\sigma$-morphic convex protoset

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A protoset of six convex polygons is constructed that tiles the plane in exactly countably many noncongruent ways.

desk verdict A promising construction of a convex sigma-morphic protoset, but the main theorem's proof leaves the crucial vertex-forcing argument deferred and the counting step unjustified, so the result is not established as written. read the letter →

arxiv 2507.20867 v1 pith:TOKL73WV submitted 2025-07-28 math.CO

Aleksa Džuklevski This is my paper · ORCID

classification math.CO MSC 52C2005B4552A37

keywords tilingtessellationσ-morphictilepolymorphicconvexprotosetrecompositionprototiles

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 note establishes that there is a protoset $P$ consisting only of convex polygons — four hexagons and two pentagons — which is $\sigma$-morphic: it tiles the Euclidean plane in exactly $\aleph_0$ many noncongruent ways up to isometry. The significance is that every earlier construction of a $\sigma$-morphic protoset used bumps and nicks, which force matching by non-convex edge shapes; the paper shows the same countably-infinite tiling behaviour can be achieved inside the class of convex prototiles. The construction is carried out by recomposing a known two-tile non-convex $\sigma$-morphic protoset: each bump or nick is replaced by two straight edges forming a unique angle $\alpha$, and the subdivision points are chosen so that every vertex has exactly one forced surrounding. A secondary result is a new three-tile non-convex protoset (Theorem 3.1) whose $\aleph_0$ tilings are obtained by inserting an arbitrary finite number of red rows.

What carries the argument

The load-bearing object is the angular replacement of a bump or nick. In the base protoset, matching is forced by a protrusion entering an indentation; here each such edge is replaced by two straight edges meeting at an angle $\alpha$, one pair protruding outward and one pair receding inward, so that only an $\alpha$-corner of the opposite sign can fit it. The paper adds two layers of control on the subdivided polygons: the angle sums available at any vertex are restricted (the only way to reach $360^\circ$ with $\alpha,\beta,\gamma$ is $\alpha+\beta+\gamma$, and no sum produces $\theta$ or $180^\circ-\theta$), and the lengths of consecutive inner edges are all distinct, so a 3-valent vertex can be surrounded in only one order and orientation. Recomposition — marking the new polygons so that the original tile edges reappear — transfers the forced tilings of the base protoset to the convex protoset $P$.

What would settle it

Take the six polygons with a proposed choice of $\alpha$, $Q$, $R$, and $Z$; compute all sums of tile angles that equal $360^\circ$ and all sums that equal $180^\circ-\theta$ for an inner angle $\theta$. If any unintended $360^\circ$ sum appears, or any two consecutive inner edges have equal length, the vertex surrounding is not unique and the protoset admits an extra local configuration, opening the way to uncountably many tilings.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.2: there exists a protoset $P$ of convex prototiles that is $\sigma$-morphic. In the proof, a known two-tile non-convex $\sigma$-morphic protoset (the red outlines in Figure 7) is divided by inner segments into four convex hexagons and two convex pentagons. The subdivision is chosen so that (i) the angle $\alpha$ of each new angular bump or nick, together with the angles $\beta,\gamma$, gives the only triple of tile angles summing to $360^\circ$; (ii) no two consecutive inner edges of any tile have equal length; and (iii) no sum of tile angles equals $\theta$ or $180^\circ-\theta$ for any inner angle $\theta$. These conditions make each vertex of every tile admit exactly one possible cyclic arrangement of the tiles around it, and the edge-length inequalities force the order and orientation of the tiles. Consequently the tilings by $P$ are in bijective correspondence with the tilings by the base protoset, and since the latter has exactly $\aleph_0$ noncongruent tilings, so does $P$.

Load-bearing premise

The construction rests on the assumption that one can choose the angle $\alpha$ and the positions of the points $Q$, $R$, and $Z$ so that all required inequalities hold simultaneously — every vertex has a unique forced surrounding, no two consecutive inner edges are equal, no forbidden angle sum appears, and every polygon stays convex — with no explicit choice actually exhibited.

Editorial extensions

If this is right

If the proof is correct, this is a $\sigma$-morphic protoset whose prototiles are all convex polygons, so countably-many-tiling behaviour is compatible with convexity at the level of protosets.
The protoset $P$ inherits the tilings of the base protoset, including a periodic tiling; the paper recalls that any $\sigma$-morphic protoset must admit a periodic tiling.
The new three-tile protoset of Theorem 3.1 gives a second mechanism for $\aleph_0$ tilings: a discrete parameter counts the finitely many red rows, with all but two tilings obtained this way.
By the classification facts surveyed in the paper, no single convex tile can be $\sigma$-morphic: the possible convex monohedral tilers yield either finitely many or continuum many tilings.

Reading between the lines

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

The proof's genericity step suggests a general recipe: any finite non-convex protoset whose forcing is purely local and vertex-based could be recomposed into convex polygons by replacing each bump or nick with an angular chain and choosing the chain angle outside the finitely many forbidden sums; testing this on other base constructions would show how far the method extends.
The paper does not give explicit coordinates for $\alpha, Q, R, Z$; a concrete instantiation would let a computer check every vertex angle sum and edge-length inequality, turning the existence theorem into a verified explicit family of convex polygons.
Because the construction uses six prototiles, a natural next test of the method is to search for a smaller convex protoset; the paper itself asks whether three or even two convex prototiles could suffice.

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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

The honest take: the paper contains a genuinely new idea and a promising construction, but its central theorem is not established as written. The author does two things. First, a new 3-tile sigma-morphic protoset with bumps and nicks (Theorem 3.1) whose proof is largely complete and is a nice addition in its own right. Second, the headline: a protoset of six convex polygons claimed to be sigma-morphic, obtained by 'convexifying' Schmitt's two-tile protoset. That would be the first convex example and a real answer to a natural question. The construction is new, and the survey of existing c-morphic and sigma-morphic machinery is useful and well-cited.

The soft spot is the proof of Theorem 3.2. The key step--forcing the correct tiles to appear around every vertex--is explicitly deferred: 'We first have to make sure that the correct tiles appear around the vertex ... but we'll leave that for later.' The later argument never appears. Everything after that, including the edge-length forcing around Q and R, assumes the surrounding tile set is already known, so the local picture is not actually proved. The genericity claim for choosing alpha and the points Q,R is plausible: the forbidden configurations form a finite union of algebraic conditions, so an open dense set of good parameters should exist. But that existence alone does not supply the missing forcing proof. Second, the step from 'P-tilings mark out Schmitt tilings' to 'P has exactly countably many tilings' is unjustified. Recomposition gives a map from P-tilings to Schmitt tilings, but one also needs to show the map is countable-to-one and that no continuum of P-tilings live over a single Schmitt tiling. That is not done. The omitted proof for the Figure 8 protoset is a minor sin in comparison.

There is no apparent fatal flaw in the construction itself; it looks repairable. But a referee should demand a complete argument before accepting the theorem. This paper deserves to be sent out for review--the problem is natural, the approach is new, and a careful referee could either fill the gaps or find a counterexample. I would not cite it as a proven result yet, but I would mention it as a construction claimed to be sigma-morphic with a proof sketch. Bring it to reading group if you want a case study in how much detail counts as proof in tiling arguments.

Referee Report

4 major / 4 minor

Summary. The paper studies σ-morphic protosets (protosets that tile the plane in exactly countably many noncongruent ways). It surveys known constructions, presents a new non-convex σ-morphic protoset of three tiles (Theorem 3.1, Figure 6), and states the main result Theorem 3.2: a protoset P consisting of convex prototiles (shown in Figure 7) that is σ-morphic, obtained by recomposing Schmitt's σ-morphic protoset into convex polygons. The proof of Theorem 3.2 is a sketch: it proposes to choose angles and edge lengths generically so that every vertex of every tile is forced to be surrounded exactly as in Figure 7, then argues that the local forcing around vertices Q and R propagates to make P behave like Schmitt's protoset. A second convex protoset (Figure 8) is also announced with its proof omitted.

Significance. If the proof of Theorem 3.2 can be completed, the result would be the first example of a σ-morphic protoset consisting solely of convex prototiles, answering a natural variant of the open problem of Grünbaum and Shephard and showing that convexity is not an obstruction to countability of tiling classes. The new non-convex protoset of Theorem 3.1 and the geometric ideas for subdividing bump-and-nick tiles into convex pieces would also be of independent interest. The paper is clearly written and includes a useful survey; the constructions are explicitly tied to Schmitt's prior work, and the central claim does not depend on the conclusion it sets out to prove, so there is no circularity. However, as it stands the proof of the main theorem contains substantial gaps, as detailed below.

major comments (4)

[§3.2, proof of Theorem 3.2] The proof explicitly defers the key vertex-forcing step: 'We first have to make sure that the correct tiles appear around the vertex... by a bit more complicated argument, but we'll leave that for later.' This step never appears. The subsequent order-forcing arguments for vertices Q and R assume that the correct set of tiles around each vertex has already been established. Since the theorem's conclusion requires that every vertex in every tiling by P have exactly the local configuration of Figure 7, the missing argument is load-bearing: without it, extra tilings, possibly continuum many, are not excluded.
[§3.2, genericity of the parameters α and QR] The existence of a suitable angle α and of subdivision points Q, R, Z is justified by 'almost inevitable' genericity, stated as 'the set of forbidden angles is finite' and 'continuum many placements... taking one that will fulfill the requirements on the angles is then almost inevitable'. The forbidden conditions, however, include both angle equalities and edge-length equalities (for instance, distinctness of consecutive inner edges such as |QS| ≠ |QT|), which are algebraic equations and inequalities in the parameters (α and the position of QR). The bad parameter set is a finite union of proper real-algebraic subsets which is typically infinite, not a finite set. A rigorous existence proof (for example by a dimension/measure argument or an explicit construction) for a parameter choice satisfying all angle and length constraints simultaneously is needed.
[§3.2, final paragraph] The conclusion 'As now the protoset P behaves in the same way that the pair of prototiles discovered by Schmitt... we have shown that P is indeed σ-morphic' does not follow from the recomposition definition used in the paper. The definition of recomposition only requires that the marks of any P-tiling form some Schmitt tiling; it does not state that the induced map from P-tilings to Schmitt tilings is countable-to-one, nor that every Schmitt tiling admits at least one P-tiling. Even if every P-tiling marks a Schmitt tiling, there could be continuum many P-tilings marking the same Schmitt tiling. The proof must show that the fibers of this map are countable (or alternatively count P-tilings directly).
[§3.2, Figure 8] The second convex protoset shown in Figure 8 is announced with the proof omitted ('We will, however, omit the proof as it uses similar ideas as the previous one'). Since the section presents this as an additional result, the paper either needs to provide the proof or explicitly downgrade the statement to a conjecture. As written, the claim that the protoset is σ-morphic is unproved.

minor comments (4)

[§3.1, Theorem 3.1] There is a typo in the theorem statement: 'The protest shown in Figure 6 (a) is σ-morphic' should read 'protoset'.
[§3.2, recomposition definition] In the definition of recomposition, 'any tiling T1 admitted by T1' should read 'any tiling admitted by T1', and similarly for 'marks on the tiles are the vertices and edges of some tiling admitted by T2' the grammar needs fixing.
[§3.2, paragraph after Theorem 3.2] The claim that 'each triangle is c-morphic while quadrangles can either be monomorphic or c-morphic' is stated without proof or reference; a citation or a brief justification would be helpful.
[§3.2, proof of Theorem 3.2, tiles 7 and 8] The geometric argument concerning the edge ZW and the points Z′, Z′′, Z′′′, W′ relies on a configuration that is not fully described in the text; adding a sub-figure or a precise coordinate description would improve readability and verifiability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 3.2 rests on external inputs (Schmitt's protoset, the recomposition definition, and convex-tiling classifications); the proof's self-asserted omissions and genericity leaps are correctness gaps, not circular reductions.

full rationale

The derivation chain for the central claim (Theorem 3.2) contains no step that reduces to its own input. The load-bearing premises are external: Schmitt's sigma-morphic protoset is cited from Schmitt's own work [24]; the recomposition notion is quoted from Grunbaum-Shephard [6]; and the convex-polygon tilability facts (pentagon classes, hexagon families, n >= 7 impossibility) come from standard sources [1, 6, 10, 13, 17, 18]. Nothing is fitted and then renamed a prediction, and no uniqueness theorem is imported from the author's own prior work. The only self-citations are to [2] (Basic-Dzuklevski-Slivkova) in Sections 2.1 and 4, used for context, and they are not load-bearing for Theorem 3.2. Under the reviewing rule, the manuscript's own assertions of missing support are flagged explicitly: Section 3.2 defers the key vertex-forcing argument ("We first have to make sure that the correct tiles appear around the vertex... but we'll leave that for later") and never returns to it; the closing inference ("As now the protoset P behaves in the same way that the pair of prototiles discovered by Schmitt from Figure 5 we have shown that P is indeed sigma-morphic") relies on that missing forcing claim, since recomposition alone only maps each P-tiling to some Schmitt tiling and does not by itself give countable fibers or surjectivity; the generic choice of alpha and of the segment QR is asserted ("almost inevitable since the set of forbidden angles is finite") without proof; and the Figure 8 claim is left with "We will, however, omit the proof". Each of these is a completeness or correctness risk, not a circularity: the paper nowhere defines P's sigma-morphicity in terms of, or fits parameters to, the very conclusion it draws.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The construction relies on Schmitt's sigma-morphic protoset, the classification of convex tilable polygons, and a genericity assumption about the parameter choices. No fitted numeric constants or ad hoc entities are introduced beyond the constructed tiles.

free parameters (2)

Angle alpha of the convex bumps and nicks = Not specified; chosen generically
The construction requires alpha such that alpha plus beta plus gamma equals 360 degrees is the only angle sum involving alpha, beta, or gamma, and such that the resulting tiles are convex. Existence is argued by genericity but no explicit value is given.
Positions of subdivision vertices Q, R, Z and resulting edge lengths = Not specified; chosen in an open set
The proof requires that no two consecutive inner edges of any tile are equal and that certain angles are unique. The paper argues such choices exist by a continuity argument, but does not pin them down.

assumptions (3)

domain assumption Schmitt's protoset from Figure 5 is sigma-morphic.
The new convex protoset is constructed by recomposing Schmitt's tiles, and the proof assumes the known tiling behavior of that protoset to conclude sigma-morphism of P.
domain assumption The classification of convex polygons that tile the plane is complete.
Used to assert that no single convex tile is sigma-morphic, motivating the protoset formulation. The pentagon classification relies on a non-peer-reviewed computer-assisted proof, as the paper itself notes.
ad hoc to paper A generic choice of subdivision parameters satisfies the angle-sum and edge-length constraints.
The proof's forcing argument depends on the existence of such a choice, which is only justified informally by a genericity argument.

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Pith. "Pith review of A $\sigma$-morphic convex protoset." pith.science (2026). https://pith.science/paper/TOKL73WV

@misc{pith2026250720867,
  author       = {Pith},
  title        = {Pith review of: A $\sigma$-morphic convex protoset},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TOKL73WV}},
  note         = {Machine review of arXiv:2507.20867}
}

abstract

We say that a tile is $\sigma$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $\sigma$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $\sigma$-morphic. The result is interesting since all the constructions of $\sigma$-morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cleverly dividing the tiles of the set of tiles discovered by Schmitt into convex tiles so that they behave in the same manner.

Figures

Figures reproduced from arXiv: 2507.20867 by the authors.

**Figure 1.** A c-morphic tile. Note that the blue outline is the same as the yellow one, allowing the orange tiles to be oriented in a different direction, without requiring any change in the rest of the tiling. This freedom of choice at every layer gives rise to c many tilings. obtain a tiling of the plane is by extending this triangle into ever larger and larger triangles, with which we associate an infinite sequence of number… view at source ↗

**Figure 2.** Two tiles that exhibit a hierarchical structure to [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** An "equilateral triangle" of "third order". To obt [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: is forced, with the only difference that the vertices at which the apexes of 3 tiles meet are a different distance apart. This distance can in turn only be a natural number, due to the symmetric protrusions on the edges of the big tile, which are "discretizing" the bou…

**Figure 5.** Figure 5: (a) The σ-morphic protoset discovered by Schmitt [24]. (b) A patch from one of the tilings admitted by this protoset. All but a finite amount of "anomalous" tilings are obtained in a similar fashion, by taking hexagons like this one which are comprised of triangles con…

**Figure 6.** Figure 6: (a) A protoset of 3 prototiles that is σ-morphic. (b) One of the tilings admitted by this protoset. All but two tilings are obtained by varying the number of rows of the burgundy tiles, which can take a value of any non-negative integer. upper side of the purple tile f…

**Figure 7.** Figure 7: For instance, the angles of the lowermost three til [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 7.** Figure 7: Prototiles of the protoset P. There are 4 distinct convex hexagons in P and 2 distinct convex pentagons. The original protoset of Schmitt that we are subdividing to obtain P is shown in red (without the bumps and nicks) and is referred to as the original protoset in th…

**Figure 8.** Figure 8: The convex protoset obtained by recomposition of p [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Adams, The Tiling Book: An Introduction to the Mathematical Theory of Tilings, American Mathematical Society (2022)

C. Adams, The Tiling Book: An Introduction to the Mathematical Theory of Tilings, American Mathematical Society (2022)

work page 2022

[2] [2]

Bašić, A

B. Bašić, A. Džuklevski, and A. Slivková, Solutions to Se ven and a Half Prob- lems on Tilings. Electron. J. Comb., 30, 2023

work page 2023

[3] [3]

Dolbilin, The countability of a tiling family and the p eriodicity of a tiling

N. Dolbilin, The countability of a tiling family and the p eriodicity of a tiling. Discrete Comput Geom 13, 405–414 (1995)

work page 1995

[4] [4]

Danzer, and N

L. Danzer, and N. Dolbilin, Delone Graphs; Some Species a nd Local Rules (1997)

work page 1997

[5] [5]

Grünbaum and G

B. Grünbaum and G. C. Shephard, Patch-determined tiling s, Math. Gaz. 61 (1977), 31–38

work page 1977

[6] [6]

Grünbaum and G

B. Grünbaum and G. C. Shephard, Tilings and Patterns , W. H. Freeeman and Company, New York (1987)

work page 1987

[7] [7]

Jeandel, R

R. Jeandel, R. & M. Rao, An aperiodic set of 11 Wang tiles. A dvances in Combinatorics, #1/1–37 (2021)

work page 2021

[8] [8]

Kari, A small aperiodic set of Wang tiles

J. Kari, A small aperiodic set of Wang tiles. Discret. Mat h., 160, 259-264 (1996)

work page 1996