Self-affine quadrangles

Christian Richter; Felix Zimmermann

arxiv: 2502.15521 · v2 · pith:LETLD5S4new · submitted 2025-02-21 · 🧮 math.CO · math.MG

Self-affine quadrangles

Christian Richter , Felix Zimmermann This is my paper

Pith reviewed 2026-05-25 08:04 UTC · model grok-4.3

classification 🧮 math.CO math.MG

keywords self-affine quadranglesconvex quadrilateralsdissectionsaffine transformationsself-tilingtrapezoids

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The pith

All 3-self-affine convex quadrangles fall into five one-parameter families and thirteen singular affine types

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

A quadrangle counts as n-self-affine when it can be dissected into n smaller pieces, each an affine image of the whole. Every convex quadrangle admits such dissections once n reaches 5 or higher, while only trapezoids work for n equal to 2. The paper gives the complete list for n equal to 3: five continuous families together with thirteen isolated examples. This narrows the remaining open question for convex quadrangles to the single case n equal to 4. Non-convex quadrangles admit n-self-affine dissections for every n at least 3, but none exist for n equal to 2.

Core claim

We characterize all 3-self-affine convex quadrangles, obtaining 5 one-parameter families and 13 singular examples of affine types. This way we reduce the quest for all n-self-affine convex quadrangles to the open case n=4.

What carries the argument

n-self-affine quadrangle: a quadrangle that admits a dissection into n affine images of itself, with the pieces required to match at vertices and overlap properly along edges

If this is right

All convex quadrangles are n-self-affine for every n greater than or equal to 5
The only 2-self-affine convex quadrangles are trapezoids
n-self-affine non-convex quadrangles exist for every n greater than or equal to 3 but not for n equal to 2
The problem of classifying all n-self-affine convex quadrangles is now reduced to the single open case n equal to 4

Where Pith is reading between the lines

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

The listed families may supply starting points for constructing explicit dissections when n equals 4
The distinction between convex and non-convex cases suggests that convexity imposes stricter limits on the smallest usable n

Load-bearing premise

Every 3-self-affine convex quadrangle admits a dissection whose affine images satisfy the vertex-matching and edge-overlap conditions required for a valid tiling of the original quadrangle.

What would settle it

Discovery of one convex quadrangle that dissects into three affine copies of itself yet lies outside the five families and thirteen examples would falsify the classification.

Figures

Figures reproduced from arXiv: 2502.15521 by Christian Richter, Felix Zimmermann.

**Figure 1.** Figure 1: The affine type Q[x, y]. where the case n = 5 goes back to A. P´or [9, Proposition 1]. The main result of the present paper is the complete characterization of all 3-self-affine convex quadrangles. The search for all 4-self-affine convex quadrangles is an ongoing project. Moreover, we give first examples of self-affine non-convex quadrangles. The paper is organized as follows. We have to classify all conve… view at source ↗

**Figure 2.** Figure 2: Canonical regions making the parametrization unique. x ≤ y, then exchanging the positions of 0 1 and 1 0 forces finally x ≤ y. This gives parameters (x, y) within P. Given (x, y) ∈ P, one can easily check that the above mentioned (up to) seven other pairs of parameters of Q satisfy (x ′ , y′ ) ∈ P′ = (x, y) ∈ R 2 : y > 0, x ≥ 1, x + y ≤ 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The combinatorial types A and B of glass-cut dissections. 1 α1 α2 α3 z α1z α2z α3z 1 z ζ z − ζ zζ 1 − zζ z(1 − zζ) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: 3-self-affinities of types A and B of a trapezoid. If a 3-self-affinity of a convex quadrangle Q is based on a glass-cut dissection, its combinatorial structure (describing the mutual inclusions of the vertices and the relative open sides of Q and of its three pieces) is one of the two combinatorial types A or B from [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: gc-parametrization of non-trapezoids. the erected triangles (or of the points s and t, respectively) are exchangeable, the parameters are not unique in general. We obtain Q(α, β) = Q(¯α, β¯), where (¯α, β¯) = (1 − β)α (1 − α)β , 1 − β 1 − α . The translation into our natural parametrization is as follows. Lemma 2. (i) Let 0 < α < β < 1. Then Q(α, β) = Q[x, y] where (5) (x, y) = 1 − β 1 − α , β , i.e… view at source ↗

**Figure 6.** Figure 6: The last combinatorial type C and a corresponding 3- self-affinity of a trapezoid. The restriction 0 < α < 1 yields α < 1 2 − α < 1, and we have pα(α) = −α(1 − α) 4 < 0, pα 1 2 − α = − 2(1 − α) 4 (2 − α) 3 < 0, pα(1) = α(1 − α) 2 > 0. Moreover, the polynomial pα is convex on [α, 1], since p ′′ α(ξ) = 6α(1 − α)ξ + 2(1 − α) 2 + 2α 2 > 0 for ξ ≥ 0. Thus the only zero β of pα in (α, 1) is in 1 2−α , 1 … view at source ↗

**Figure 7.** Figure 7: Analytic modelling of type C. q1 = a1 = 0 0 , q2 = b1 = 1 0 , q3 = b2 = c4 = x y , q4 = c1 = 0 1 , a2 = b4 = m 0 , a4 = c2 = 0 n , a3 = b3 = c3 = s t . Since A, B and C are affine images of Q, there exist permutations πA, πB, πC ∈ {(1234),(2341),(3412),(4123),(1432),(2143),(3214),(4321)} (π = (ijkl) standing for π(1) = i, π(2) = j, π(3) = k and π(4) = l) and affine maps α(·) = … view at source ↗

**Figure 8.** Figure 8: The realizations of 3-self-affinities of combinatorial type C of the quadrangle Q[0.43015 . . . , 0.75487 . . .]. P T A B1 B2 C [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 9.** Figure 9: All parameters (x, y) ∈ P representing 3-self-affine convex quadrangles Q[x, y]. or (x, y) ∈ C (see Theorem 6) or (x, y) is one of the 13 singular parameters given in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: □ Lemma 10. Let Q[x, y] be non-convex. Then Q[x, y] admits a dissection into an affine image of Q[x, y] and a remaining quadrangle with vertices 0 0 , 1 0 , x y and x 2 (1 − x)(1 − y) . (We admit the last quadrangle to be convex or non-convex or to degenerate into a triangle, depending on the angle at x y .) Proof. The affine map β ξ η = x 0 y − 1 −1 ξ η + 0 1 [PITH_FU… view at source ↗

**Figure 10.** Figure 10: Dissections from Lemmas 9 and 10. satisfies β 0 1 = 0 0 , β 0 0 = 0 1 , β 1 0 = x y , β x y = x 2 (1 − x)(1 − y) . We see that Q[x, y] splits into β(Q[x, y]) and the remainder described in Lemma 10, see [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

**Figure 11.** Figure 11: Realizations of Proposition 8 for n = 3, 4. and the vertices of the remaining quadrangle R are 0 0 , 1 0 , x y = x0 1 1 − x0 , x 2 (1 − x)(1 − y) = x 2 0 (1 − x0) [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

read the original abstract

A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles are trapezoids. Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$. In addition, we show that there are $n$-self-affine non-convex quadrangles for all $n \ge 3$, but not for $n=2$.

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.

Desk Editor's Note private letter to a colleague

The paper classifies all 3-self-affine convex quadrangles into five families and thirteen examples, closing the remaining gap before n=4.

read the letter

The main takeaway is that Richter and Zimmermann have produced an explicit classification for the n=3 case. Convex quadrangles that dissect into three affine copies of themselves fall into five one-parameter families plus thirteen singular affine types. They also show that non-convex quadrangles work for every n at least 3, while n=2 fails for non-convex as well. This sits cleanly between the known n=2 result (only trapezoids) and the n at least 5 result (everything works), so the open problem now reduces to n=4 alone. That reduction is useful on its own. The non-convex existence statements are straightforward to state and add a bit of completeness to the picture. The work is a direct case analysis of dissections that respect the affine-image and tiling conditions. If the enumeration of topologies is exhaustive, the list is reliable and the claim holds. The soft spot is exactly the one the stress-test flags: whether every possible way of placing three affine copies inside the quadrangle (different internal vertices, partial edge overlaps, or other matchings) has been covered. The paper does not appear to use computer enumeration of small dissections as a cross-check, so a referee would want to see the case breakdown in detail to confirm nothing was missed. This is narrow work aimed at people already following self-affine dissections in the plane. Those readers will get concrete families they can use or extend. The result is concrete enough and the gap it fills is clear enough that it deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript defines an n-self-affine quadrangle as one that admits a dissection into n affine images of itself. It proves that every convex quadrangle is n-self-affine for all n ≥ 5, that the only 2-self-affine convex quadrangles are trapezoids, and that there exist n-self-affine non-convex quadrangles for every n ≥ 3 (but none for n = 2). The central result is a complete classification of 3-self-affine convex quadrangles into five one-parameter families together with thirteen singular examples; this reduces the general problem of classifying all n-self-affine convex quadrangles to the remaining open case n = 4.

Significance. If the classification is exhaustive, the result is a concrete advance in the study of self-affine dissections: it supplies explicit parametric families and isolated examples that can be checked directly, and it cleanly isolates the n = 4 case. The separation between convex and non-convex behavior is also useful. The paper does not appear to contain machine-checked proofs or reproducible code, but the explicit listing of families constitutes a falsifiable prediction that can be tested by independent enumeration.

major comments (2)

[§3] §3 (classification of 3-self-affine convex quadrangles): the completeness of the case analysis rests on an enumeration of dissection topologies satisfying vertex-matching and edge-overlap conditions. No argument is given that every possible 3-dissection of a convex quadrangle falls into one of the enumerated topological types (different placements of internal vertices, partial edge sharings, or vertex identifications). Without such an exhaustiveness proof or an independent computer-assisted enumeration of small dissections, it remains possible that additional families exist outside the five listed ones and the thirteen examples.
[Theorem 1.3] Theorem 1.3 (the five families and thirteen examples): each family is obtained by solving systems of affine-map equations under the assumption that the dissection uses one of the listed topologies. If an unlisted topology admits a solution, the stated list would be incomplete; the manuscript does not provide a separate lemma showing that the chosen topologies are the only ones compatible with convexity and the affine-image requirement.

minor comments (2)

[Figures 2–3] Figure 2 and Figure 3: the edge labels and vertex identifications are difficult to read at the printed size; adding a supplementary table that lists the affine coefficients for each singular example would improve verifiability.
[§2 and §4] Notation: the symbol for the affine map is introduced in §2 but reused with different subscript conventions in §4; a single consistent notation throughout would reduce reader effort.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying the need for an explicit exhaustiveness argument in the topological case analysis. We will revise the manuscript to include such an argument.

read point-by-point responses

Referee: [§3] §3 (classification of 3-self-affine convex quadrangles): the completeness of the case analysis rests on an enumeration of dissection topologies satisfying vertex-matching and edge-overlap conditions. No argument is given that every possible 3-dissection of a convex quadrangle falls into one of the enumerated topological types (different placements of internal vertices, partial edge sharings, or vertex identifications). Without such an exhaustiveness proof or an independent computer-assisted enumeration of small dissections, it remains possible that additional families exist outside the five listed ones and the thirteen examples.

Authors: We agree that the manuscript lacks an explicit lemma establishing exhaustiveness of the enumerated topologies. In the revised version we will add a lemma that classifies all admissible placements of internal vertices and edge identifications compatible with convexity and the requirement that each piece is an affine image of the quadrangle; the argument proceeds by considering the possible combinatorial types of a 3-piece dissection of a convex 4-gon and showing that only the topologies already treated can arise. revision: yes
Referee: [Theorem 1.3] Theorem 1.3 (the five families and thirteen examples): each family is obtained by solving systems of affine-map equations under the assumption that the dissection uses one of the listed topologies. If an unlisted topology admits a solution, the stated list would be incomplete; the manuscript does not provide a separate lemma showing that the chosen topologies are the only ones compatible with convexity and the affine-image requirement.

Authors: This observation is equivalent to the preceding comment. The new lemma will prove that no other topology can satisfy the convexity and affine-image conditions, thereby confirming that the five families and thirteen examples constitute the complete list. revision: yes

Circularity Check

0 steps flagged

No circularity; direct geometric classification

full rationale

The paper performs a case-by-case geometric classification of 3-self-affine convex quadrangles by enumerating dissections into three affine copies, yielding 5 families and 13 examples. No equations, fitted parameters, self-referential definitions, or load-bearing self-citations appear. The result is an enumeration derived from the definition of self-affinity and affine transformations, without reducing any claim to its own outputs by construction. This is the expected outcome for a pure classification theorem in combinatorial geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. Standard properties of affine maps and planar dissections are presupposed but not itemized.

pith-pipeline@v0.9.0 · 5651 in / 1034 out tokens · 23242 ms · 2026-05-25T08:04:52.277469+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Blechschmidt, C

K. Blechschmidt, C. Richter: Non-self-affine convex pentagons. Beitr. Algebra Geom. 56, no. 1, 263–267 (2015)

work page 2015

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B¨ orner:Affine Abbildungen in der Ebene

W. B¨ orner:Affine Abbildungen in der Ebene. Die Wurzel. Zeitschrift f¨ ur Mathematik.54, November 2020, 233, 248–255 (2020)

work page 2020

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Ditrich: Elementar k-selbst¨ ahnliche Zerlegungen einfacher Polygone

F. Ditrich: Elementar k-selbst¨ ahnliche Zerlegungen einfacher Polygone. Diploma thesis. Friedrich Schiller University, Jena, 1995

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Edgar: Measure, topology, and fractal geometry

G.A. Edgar: Measure, topology, and fractal geometry. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1990

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Falconer: Fractal geometry

K. Falconer: Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Ltd., Chichester, 1997

work page 1997

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Golomb: Replicating figures in the plane

S.W. Golomb: Replicating figures in the plane. Math. Gaz. 48, 403–412 (1964)

work page 1964

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Gr¨ unbaum, G.C

B. Gr¨ unbaum, G.C. Shephard: Tilings and patterns. W.H. Freeman and Company, New York, 1987

work page 1987

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Hertel: Zur Affingeometrie konvexer Polygone

E. Hertel: Zur Affingeometrie konvexer Polygone. Jenaer Schriften zur Mathematik und Informatik, Math/Inf/00/22, 26 pp. (2000), preprint, https://www.minet.uni-jena.de/ /Math-Net/reports/shadows/00-09report.html

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Hertel, C

E. Hertel, C. Richter: Self-affine convex polygons. J. Geom. 98, no. 1–2, 79–89 (2010)

work page 2010

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Mandelbrot: The fractal geometry of nature

B.B. Mandelbrot: The fractal geometry of nature. W.H. Freeman and Co., San Francisco, CA, 1982

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Osburg: Selbst¨ ahnliche Polyeder.Doctoral thesis

I. Osburg: Selbst¨ ahnliche Polyeder.Doctoral thesis. Friedrich Schiller University, Jena, 2004, https://nbn-resolving.org/urn:nbn:de:gbv:27-dbt-003580-1

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Richter: Examples of 3-self-affine convex quadrangles

C. Richter: Examples of 3-self-affine convex quadrangles. Jenaer Schriften zur Mathematik und Informatik, Math/Inf/10/04, 5 pp. (2010), preprint, https://www.minet.uni-jena.de/ /Math-Net/reports/shadows/10-04report.html

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Richter: Self-affine convex discs are polygons

C. Richter: Self-affine convex discs are polygons. Beitr. Algebra Geom. 53, no. 1, 219–224 (2012)

work page 2012

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Richter: Self-affinity of discs under glass-cut dissections

C. Richter: Self-affinity of discs under glass-cut dissections. (submitted) https://arxiv. org/abs/2404.11460

work page arXiv

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Scherer: A puzzling journey to the reptiles and related animals

K. Scherer: A puzzling journey to the reptiles and related animals. Privately published, Auckland, New Zealand, 1987

work page 1987

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Zimmermann: Ausgew¨ ahlte Aspekte der Selbstaffinit¨ at von Scheiben.Thesis submitted as part of the preliminary board exam for teachers

F. Zimmermann: Ausgew¨ ahlte Aspekte der Selbstaffinit¨ at von Scheiben.Thesis submitted as part of the preliminary board exam for teachers. Friedrich Schiller University, Jena, 2023. Institute for Mathematics, Friedrich Schiller University, 07737 Jena, Germany Email address: christian.richter@uni-jena.de Email address: zimmermann.f@uni-jena.de

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