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arxiv: 2512.22696 · v4 · submitted 2025-12-27 · 🧮 math.CO

Tiling Triangles with 2π/3 Angles

Yan X Zhang This is my paper

Pith reviewed 2026-05-16 19:10 UTC · model grok-4.3

classification 🧮 math.CO
keywords tilingtrianglesdissectioncongruent120 degreeErdőstiling numberssporadic
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The pith

Six sporadic triangles can be tiled by congruent 120-degree triangles for infinite families of N, conjecturally the only ones.

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

Researchers have long studied when one triangle can be cut into N smaller congruent triangles. For the case where the small triangles have a 120-degree angle, most situations are understood, but six unusual large triangles remain. The work provides concrete ways to make these tilings for many different N in each case. It also proposes that no other N are possible beyond those families.

Core claim

The central discovery is a set of constructions for each of the six sporadic triangles that produce tilings with any number of copies N from a particular family, together with the conjecture that these families exhaust all possibilities.

What carries the argument

The mechanism is a series of geometric constructions that repeatedly subdivide or arrange the small 120-degree triangles to fill the large one while matching angles at the vertices.

Load-bearing premise

The explicit constructions remain valid for arbitrarily large N in the families and no tilings exist outside the families for any of the six triangles.

What would settle it

Discovering a tiling using a number of copies outside the described families for any one of the six sporadic triangles.

Figures

Figures reproduced from arXiv: 2512.22696 by Yan X Zhang.

Figure 1
Figure 1. Figure 1: A List of the possible incommensurable-angles cases, where [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The basic ideal trapezoid. The marked angles are equal to [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: How to tile different parallelograms. The bigger parallelograms [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tiling more complex ideal trapezoids. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Tiling the equilateral triangle into 3 ideal trapezoids. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The basic ideal trapezoid, now tiled by 3 similar triangles with angles [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Triangles with angles 2β, 2α, α + β. The marked angles are α. The top figure has lengths c/b times that of the bottom. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Combining two tilings to make another tiling with bottom side [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Isosceles triangle with sides (mac, mac, ma(b + 2a)). potential incommensurable-angles and non-reptile cases that tile into (α, β, γ = 2π/3) triangles. Recall that these have angles: • isosceles (α, α, π − 2α); • (α, α + β, α + 2β); • (α, 2β, β + 2α); • (α, 2α, 3β); (for the purpose of this list, each item includes the variation where α and β are swapped; for example, the (α, α, π − 2α) case also includes … view at source ↗
Figure 10
Figure 10. Figure 10: Triangle with angles (α, α + β, α + 2β). 2. we can tile (mab, mac, ma(a + b)), which has angles (β, α + β, β + 2α), into m2a(a + b) copies of (a, b, c). Proof. We do the first case (the second case is symmetric). As in [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Triangle with angles (α, 2β, 2α + β) [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Triangle with angles (α, 2α, 3β). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
read the original abstract

Motivated by a question of Erd\"{o}s and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$) and the \emph{commensurable-angles} cases (where all angles of $R$ are rational multiples of $\pi$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2\pi/3$ and when $T$ is one of $6$ ``sporadic'' specific triangles, of which only $2$ were known to have constructions. For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles.

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 studies the problem of tiling a triangle T into N congruent copies of a smaller triangle R containing a 2π/3 angle, restricting attention to six sporadic triangles T. For each such T the authors supply explicit geometric constructions that realize infinite families of admissible N and conjecture that these families contain every possible N.

Significance. The explicit constructions extend the previously known cases from two to all six sporadic triangles and furnish concrete, geometrically described families; if the constructions are valid for all claimed N and the completeness conjectures hold, the work would furnish a near-complete classification for these exceptional cases, complementing the settled reptile and commensurable-angle regimes.

major comments (2)
  1. [Abstract] Abstract and concluding section: the claim that the constructed families exhaust all possible N for each of the six triangles is presented as a conjecture without any supporting argument, exhaustive enumeration for small N, or invariant/obstruction that rules out other values of N.
  2. [Construction sections] Construction sections (diagrams and coordinate descriptions): while the families are given explicitly, the text supplies no general verification that the dissections remain non-overlapping and gap-free for every N in the infinite families (e.g., no inductive argument or area/angle check that scales with N).
minor comments (2)
  1. [Introduction] Add a short table summarizing the six sporadic triangles (angles and side ratios) in the introduction for quick reference.
  2. [Construction sections] Ensure all coordinate descriptions are accompanied by explicit angle-sum or vector-closure checks that confirm closure for the general term of each family.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and helpful suggestions. We address the two major comments below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and concluding section: the claim that the constructed families exhaust all possible N for each of the six triangles is presented as a conjecture without any supporting argument, exhaustive enumeration for small N, or invariant/obstruction that rules out other values of N.

    Authors: We agree that the conjecture would be better supported by additional evidence in the text. In the revised manuscript we will add a new subsection reporting exhaustive computational enumerations for small N (up to several hundred) across the six triangles; these checks confirm that no admissible N fall outside the constructed families. We will also include a brief discussion of simple area and angle-sum obstructions that rule out certain residue classes of N, while noting that a complete proof of the conjecture remains open. revision: yes

  2. Referee: [Construction sections] Construction sections (diagrams and coordinate descriptions): while the families are given explicitly, the text supplies no general verification that the dissections remain non-overlapping and gap-free for every N in the infinite families (e.g., no inductive argument or area/angle check that scales with N).

    Authors: The constructions are given parametrically via explicit coordinates that are designed to tile by construction. We will add a general inductive argument in each construction section showing that the base tiling for the smallest N in the family extends to all larger members without overlaps or gaps; the induction step relies on the coordinate matching of edges and the preservation of the 2π/3 angle condition at each iteration. Area equality is immediate from the scaling factor, and we will include a short lemma verifying the inductive step for all six triangles. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit constructions plus open conjecture

full rationale

The manuscript supplies explicit geometric constructions (diagrams and coordinates) for families of N and states a conjecture that these families are exhaustive for the six sporadic triangles. No load-bearing step reduces by definition or self-citation to its own inputs; the completeness claim is explicitly labeled a conjecture rather than a derived necessity. No fitted parameters, self-definitional equations, or uniqueness theorems imported from the same authors appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard Euclidean geometry for plane tilings and congruence; no free parameters, ad-hoc axioms, or invented entities are introduced beyond the geometric setting.

axioms (1)
  • standard math Euclidean plane geometry with standard notions of congruence and tiling by dissection
    Invoked throughout to define valid tilings and angle conditions.

pith-pipeline@v0.9.0 · 5445 in / 1054 out tokens · 41632 ms · 2026-05-16T19:10:32.368392+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Solution of Erd\H{o}s Problem 633

    math.CO 2026-04 unverdicted novelty 9.0

    Triangles that can be tiled by congruent copies only in square numbers are fully classified.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

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    M. Beeson. No triangle can be cut into seven congruent triangles.http:// www.michaelbeeson.com/research/papers/NoSevenTiling.pdf, 2018

    work page 2018

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    M. Beeson. Tiling an equilateral triangle.https://www.michaelbeeson. com/research/papers/TriangleTilingEquilateral.pdf, 2019

    work page 2019

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    M. Beeson. Tilings of an isosceles triangle.http://www.michaelbeeson. com/research/papers/IsoscelesTilings.pdf, 2019

    work page 2019

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    M. Beeson. Triangle tiling: the case 3α+ 2β=π.http://www. michaelbeeson.com/research/papers/TriangleTiling3.pdf, 2019

    work page 2019

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    S. W. Golomb. Replicating figures in the plane.The Mathematical Gazette, 48(366):403–412, 1964

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    Laczkovich

    M. Laczkovich. Tilings of triangles.Discrete Mathematics, 140(1-3):79–94, 1995

    work page 1995

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    Laczkovich

    M. Laczkovich. Tilings of convex polygons with congruent triangles.Dis- crete & Computational Geometry, 48(2):330–372, 2012

    work page 2012

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    Laczkovich

    M. Laczkovich. Rational points of some elliptic curves related to the tilings of the equilateral triangle.Discrete & Computational Geometry, 64(3):985– 994, 2020

    work page 2020

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    Soifer.Is There Anything Beyond the Solution?, pages 47–50

    A. Soifer.Is There Anything Beyond the Solution?, pages 47–50. Springer New York, New York, NY, 2009

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    J. J. Sylvester. On subvariants, i.e. semi-invariants to binary quantics of an unlimited order.American Journal of Mathematics, 5(1):79–136, 1882. 16

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