Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

Erxiao Wang; Jinjin Liang; Wen Chen

REVIEW 3 major objections 4 minor 23 references

Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Three families exhaust all irrational-angle sphere tilings.

desk verdict A substantial classification result; the stress-test's 'missing' extended edge in Lemma 8 is actually already in Table 5, and the paper deserves a serious referee. read the letter →

arxiv 2505.16629 v1 pith:CPALNHD2 submitted 2025-05-22 math.CO

Wen Chen , Jinjin Liang , Erxiao Wang This is my paper · ORCID

classification math.CO MSC 52C2005B45

keywords sphericaltilingtrianglemonotilenon-side-to-sidehalfvertexirrationalangleextendededgeadjacentdeductionmatchinglemma

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

An old open thread in spherical tiling theory asks which congruent triangles can cover the sphere without meeting side-to-side, meaning some tile corners lie in the interior of another tile's side. This paper closes that thread for every triangle that has at least one angle irrational in degrees. It proves that such tilings are exactly: one-parameter families of triangles admitting two-layer earth map tilings with 2k ≥ 6 tiles (plus rotation modifications when k is even), a one-parameter family of triangles each admitting a unique 8-tile tiling, and a single triangle admitting a unique 16-tile tiling. All the angular and side-length data for these monotiles are listed explicitly. A reader who wants to know whether a given irrational triangle can tile the sphere without side-to-side contact can now read the answer off the three listed families.

What carries the argument

The argument is carried by a small set of combinatorial instruments adapted to half vertices. The irrational angle lemma (Lemma 7) forces the angle-count vector of every vertex to lie on an affine integer line through (1,1,1) and one known vertex vector; otherwise the angle equations would have a unique rational solution, contradicting irrationality. The matching lemma (Lemma 8) then enumerates every possible extended-edge equality—patterns such as b+ka = ka+b or c+(k+1)a = b+ka+b, plus the great-circle case—under the assumption that the largest angle α never occurs at a half vertex. Around these sit the balance lemma, the adjacent angle deduction, and the counting identities for full and half vertices, which together reduce the infinite classification problem to a finite table of possible vertex combinations that can be checked one by one.

What would settle it

Run an exhaustive search for non-side-to-side tilings of the sphere by congruent scalene triangles whose angles include an irrational number of degrees, for tile counts up to the bounds permitted by the counting identities; any output not congruent to a two-layer earth map tiling, the 8-tile tiling, or the 16-tile tiling, or any extended edge not listed in Table 5, would disprove the theorem.

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Extended reading notes

Core claim

The paper's main theorem asserts a complete classification. In a non-side-to-side tiling of the sphere by congruent triangles with any irrational angle, the tile and the tiling must be one of the three listed types: the two-layer earth map family with 2k ≥ 6 triangles and its even-k rotation modifications, the one-parameter 8-tile family, or the sporadic 16-tile tiling. The phrase 'non-side-to-side' means that not every triangle corner is a corner of the tiling; those exceptional points are half vertices. Since degenerate and isosceles cases had already been classified, the proof assumes a convex scalene triangle and derives the full list of possible vertex combinations, eliminating every combination that cannot be completed. The result is stated up to rotation and global flip of the sphere.

Load-bearing premise

The load-bearing premise is that Lemma 8 lists every possible extended-edge pattern when the largest angle never sits on a half vertex; if that enumeration misses a pattern, a tiling outside the three families could survive the case analysis.

Editorial extensions

If this is right

If the theorem is correct, the irrational-angle branch of the classification is closed; any future example must lie in one of the three families of Table 3.
The explicit formulas in Table 3 turn the existence question into a check of angular data: a triangle with an irrational angle can tile non-side-to-side exactly when it fits one of the listed parameter ranges.
The 8-tile and 16-tile tilings are the only sporadic irrational tilings, so no further isolated examples can appear in this branch.
The matching-lemma and vertex-statistics scheme gives a finite-case template that the paper's final section proposes to extend to the remaining rational-angle case.

Reading between the lines

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

If the matching lemma is indeed exhaustive, a mechanical search over all length-word matchings along a line with a > b > c under the no-α-half-vertex rule would reproduce exactly the patterns of Table 5; finding a new pattern would mean the theorem needs revisiting.
The theorem's rigidity suggests that non-side-to-side tiling with an irrational angle forces high symmetry, which is why the only families are layered earth maps plus two small exceptional tilings.
A natural next test is to apply the same vertex-combination enumeration to rational-angle triangles at small denominators; the paper's f=36 example already indicates that new sporadic tilings will appear there, so the rational classification is expected to be richer.

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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 headline is: this paper is a real advance, and the specific worry in the stress-test note about Lemma 8 does not survive contact with the text. The alleged missing extended edge c+(k+1)a=b+ka+b is the same equality as entry 4 in Table 5, b+ka+b=(k+1)a+c, just written in reverse order. So the Matching Lemma's table is not missing that case. That removes the only concrete incompleteness charge I saw.

What is genuinely new: the Irrational Angle Lemma and the Matching Lemma, and the complete classification of non-side-to-side spherical tilings by congruent triangles when one angle is irrational. The three families in the main theorem—two-layer earth map tilings, the 8-tile family, and the 16-tile sporadic—are concrete, and the geometric data are given explicitly. The proof incorporates earlier isosceles and right-triangle classifications rather than hand-waving them. The AAD technique from the pentagon papers is imported and extended sensibly.

Soft spots: Lemma 7 is overclaimed. It says 'all vertices n' satisfy the line condition, but the proof works only for full vertices (the right-hand side in the linear system is 2, not 1, for half vertices). The applications only use the full-vertex version, so this is an exposition bug, not a hole in the classification. The Matching Lemma's proof is terse in places; the exclusions in Figures 13–15 are plausible but a referee will want to check that no pattern is waved off too fast. No code or data accompanies the case analysis, so independent reimplementation is a nontrivial project. That is typical for this literature.

Overall: the central argument holds up as far as I can see, and the classification is a major result for discrete geometry. The paper deserves a serious referee, ideally someone who knows the Dawson–Doyle examples and the Ueno–Agaoka edge-to-edge classification. I would bring it to a tilings reading group, and I would cite it if I work on spherical tilings. Send it to peer review.

Referee Report

3 major / 4 minor

Summary. The paper develops tools for classifying non-side-to-side tilings of the sphere by congruent triangles, specifically the irrational angle lemma (Lemma 7) and the matching lemma (Lemma 8) enumerating possible extended edges. The main theorem asserts that, for a triangle with any angle irrational in degrees, all non-side-to-side tilings are: (1) two-layer earth map tilings with 2k≥6 tiles and their rotational modifications for even k; (2) a one-parameter family of triangles each admitting a unique 8-tile tiling; and (3) a unique triangle admitting a unique 16-tile tiling. The proof derives possible anglewise vertex combinations (AVC) from low-degree full or half vertices, then eliminates or constructs tilings case by case, with exact geometric data in Table 3.

Significance. If correct, this completes a natural and previously open case in the classification of monohedral spherical triangle tilings, extending the edge-to-edge classification of Ueno-Agaoka and the isosceles/right-triangle results of Dawson and Doyle. The paper's strengths include explicit geometric data, detailed local diagram arguments, and a clear reduction to finitely many AVC cases using the authors' adjacent-angle-deduction method. The main theorem is specific and falsifiable, and no fitted parameters appear. However, the completeness of the classification depends on Lemma 8's exhaustive list of extended edges, and the omission identified below means the central claim is not yet established as written; the human case analysis also needs careful verification at the points where Lemma 8 is invoked.

major comments (3)

[§2, Lemma 8 and Table 5] In the third case of the proof of Lemma 8, after analyzing the case x=a, the text explicitly concludes three types of extended edges: c+ka+b=b+ka+c, c+(k+1)a+c=b+ka+b, and c+(k+1)a=b+ka+b. Table 5 lists only the first two of these (entries 3 and 7); the third type is missing. Thus the lemma's assertion that Table 5 contains all possible extended edges is internally inconsistent with its own proof. This is load-bearing: Section 3 repeatedly invokes Lemma 8 to conclude specific side relations, e.g. in Case α^2β ('by Lemma 8, we have a=2c'), in Case α^2βγ (where b=2c is concluded), and in Cases α^4 and β^4 (where a=2c or b=2c is derived). If the missing pattern c+(k+1)a=b+ka+b (for k=0, c+a=2b) is geometrically realizable, the list of possible extended edges is incomplete and the subsequent AVC-based exclusions could miss tilings outside the three claimed families. The authors should add the missing entry and either derive a contradiction from it or show that it leads only to already-listed tilings, and then re-verify every invocation of Lemma 8.
[§2, Lemma 7] Lemma 7 is stated for 'all vertices' (both full and half), but its proof uses the equation n·(α,β,γ)=2 for every vertex n, which holds only for full vertices; half vertices have angle sum 1 (π), not 2. The statement is in fact false for half vertices: in Case α^3 with f=6, the half vertex βγ=(0,1,1) does not lie on the line through m=(3,0,0) and (2f/(f+4))u=(6/5,6/5,6/5). The lemma should be restated for full vertices only, with a separate argument (e.g., doubling a half vertex to a full vertex) for the half-vertex columns AVC1 in Tables 6–8. As written, the derivation of the AVC1 columns, which are used throughout Section 3, is not justified by the stated lemma.
[Theorem statement and §2] The abstract and the theorem state a classification of all non-side-to-side tilings by congruent triangles with any irrational angle, but the proof explicitly restricts to scalene triangles: Section 2 says 'Dawson has completely classified ... isosceles triangles. Therefore, we only need to discuss convex and scalene triangle.' The theorem should either be restricted to scalene triangles or should explicitly incorporate the isosceles classification from [7,8]; otherwise the statement is broader than what is proven. This is not a mere wording issue, because the listed three families do not include Dawson's isosceles families, and the paper does not state whether Dawson's classification contains any irrational-angle isosceles tilings outside these families.

minor comments (4)

[§2, Lemma 8] Lemma 8 refers to 'Table 6' but the displayed table of extended edges is numbered 'Table 5'; the cross-reference should be corrected.
[Figure 15] The references to 'the third picture' and 'the fourth picture' in the proof of Lemma 8 are difficult to follow because the subfigures in Figure 15 are not individually labeled; please add labels or explicit descriptions.
[§3, Tables 6–8] The sentence 'All other AVC in Table 6, 7, 8 are derived similarly' is very terse, especially for the half-vertex columns AVC1; after Lemma 7 is corrected, the derivation of these columns should be spelled out or a clear reference to the doubling argument should be given.
[§3.2] The sentence 'The first six cases admit no half vertex' refers to Table 7 but does not list which six cases are meant; please make this explicit.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorem is derived from local matching/counting lemmas and prior independent classifications; the only self-citation (AAD) is not load-bearing in a circular way.

full rationale

The derivation chain is not circular. The main theorem is obtained by applying Lemma 7 (Irrational Angle Lemma) and Lemma 8 (Matching Lemma) to the finite list of low-degree vertices from Lemma 6/Table 4, then eliminating configurations via adjacent angle deduction and AVC comparisons. None of these steps assumes the conclusion: Lemma 7 is proved from Lemma 1 and the irrationality hypothesis; Lemma 8 is a local enumeration of possible extended edges; Lemma 6 is a counting argument from Euler's formula. The known classifications of Dawson and Dawson-Doyle are used only to identify already-known examples, not to force the three claimed families. The AAD tool is cited from the authors' own prior work [22, Lemma 10], but it is a general parameter-free lemma about angle arrangements and does not contain the spherical-triangle classification being proved; it is not fitted to the target data and no prediction is renamed from a fit. There is a separate internal-exhaustiveness concern: in the third case of Lemma 8 the proof derives an extended edge c+(k+1)a = b+ka+b that does not appear in Table 5. That is a potential correctness gap in the matching lemma, but it is not circularity, because the lemma is not defined in terms of the main theorem and the theorem is not assumed in its proof. No circular step can be exhibited, so the score reflects only the minor, non-load-bearing self-citation of the AAD technique.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard spherical geometry, Euler's formula, and the combinatorial definitions of tiling, plus the assumption that prior classifications of isosceles and right-triangle tilings are correct. No ad hoc fitted parameters are introduced; the parameter families in the theorem are the output of the classification, not inputs.

assumptions (5)

standard math Euler's formula v - e + f = 2 for spherical tilings
Used in equations (2.1)-(2.4) to derive bounds on vertex counts.
standard math Spherical triangle angle sum: α+β+γ = π + area, giving α+β+γ = 1 + 4/f in units of π
Lemma 1, the basis of the Irrational Angle Lemma.
domain assumption The tiling is non-side-to-side with straight sides
Definition of the classification problem.
domain assumption The triangle is convex and scalene with α > β > γ
Lemma 4 and Lemma 5 reduce the problem to this setting; isosceles and right triangles are covered by prior classifications.
domain assumption Every half vertex has degree ≥ 2 and every full vertex has degree ≥ 3
Follows from convexity and non-degeneracy; used in Table 4.

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Cite this review

Pith. "Pith review of Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle." pith.science (2026). https://pith.science/paper/CPALNHD2

@misc{pith2026250516629,
  author       = {Pith},
  title        = {Pith review of: Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPALNHD2}},
  note         = {Machine review of arXiv:2505.16629}
}

abstract

We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with $2n$($n\geq3$) tiles, together with rotational modifications for even $n$; a 1-parameter family of triangles each admitting a unique tiling with $8$ tiles; and a sporadic triangle admitting a unique tiling with $16$ tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.

Figures

Figures reproduced from arXiv: 2505.16629 by the authors.

**Figure 1.** A geodesic triangle on the sphere. ∗Corresponding author (wang.eric@zjnu.edu.cn). Research was supported by National Natural Science Foundation of China NSFC-RGC 12361161603 and Key Projects of Zhejiang Natural Science Foundation LZ22A010003. 1 arXiv:2505.16629v1 [math.CO] 22 May 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** One-layer and two-layer earth map tilings and their modifications. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Triangular subdivisions of Platonic solids and their rotation modifications. [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

Figures from the paper (22 more)

**Figure 4.** Figure 4: Non-side-to-side tilings by congruent isosceles triangles. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

**Figure 5.** Figure 5: Non-side-to-side tilings by congruent right triangles. [PITH_FULL_IMAGE:figures/full_fig_p003_5.png]

**Figure 6.** Figure 6: The subdivision of a special quadrilateral monotile. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: Non-side-to-side triangular tilings by subdividing quadrilateral tilings. [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: T(4α 2γ; 4β 2γ) and T(8α 2βγ; 4β 2γ 2 ). angles (α, β, γ) sides (α, π − α, 4π f ) a = arccos cos α(1−cos 4π f ) sin 4π f sin α b = π − a c = arccos cos 4π f −cos2 (α) sin2(α) (π − γ 2 , π 2 − γ 2 , γ), γ ∈ ( π 4 , π 2 ) a = arccos tan γ 2 cos γ−1 sin γ isosce…

**Figure 9.** Figure 9: Full vertex A and half vertex B. The degree of a vertex is the number of corners at the vertex. For example, the full vertex A in [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

**Figure 10.** Figure 10: Mismatched tiles, an extended edge, and two stop vertices. [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 11.** Figure 11: Adjacent Angle Deduction of β α α γ . 7 [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]

**Figure 12.** Figure 12: All possible non-side-to-side tilings by congruent degenerate triangles. [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]

**Figure 15.** Figure 15: Matching of b + · · · and c + · · · along an extended line 10 [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]

**Figure 13.** Figure 13: Matching of a + · · · and c + · · · along an extended line α b γ β c α a no α b γ γ b α β a γ β c α α γ β γ γ α b a a b [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

**Figure 16.** Figure 16: A possible half vertex when γ is not in any half vertex. 3 Proof of the Main Theorem In a non-side-to-side abc-tiling, a full vertex of degree 3,4,5 or a half vertex of degree 2 in [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]

**Figure 17.** Figure 17: The possibilities for α 3 . Case αβ2 From [PITH_FULL_IMAGE:figures/full_fig_p014_17.png]

**Figure 18.** Figure 18: The possibilities for α 2β = γβ α βα γ βα γ . Therefore, α 2β = γβ α βα γ γα β . Then T4 in [PITH_FULL_IMAGE:figures/full_fig_p014_18.png]

**Figure 19.** Figure 19: Two possibilities for γβ α βα γ γα β . Therefore, γ2γ3 · · · = βγ α αγ β γβ α . Then we have f = 8 and AVC = {α 2β, β2γ 4 ; βγ2}. Therefore, T4 and T5 are determined by the AVC in [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]

**Figure 20.** Figure 20: Two possibilities of α 2β. Case α 2γ We get a unique tiling T(4α 2γ; 4β 2γ) in the same way as T(4α 2β; 4βγ2 ) in Case α 2β, since β > γ has nothing to do with the deduction and we may switch β and γ. Then we have α = 1 − γ 2 , β = 1 2 − γ 2 and γ ∈ ( 1 4 , 1 3 ). Two…

**Figure 21.** Figure 21: Two possibilities for α 2βγ. αβ γ 1 βγ α 2 · · · = αβ γ 1 βγ α 2 αγ β 12 γβ α 13 determine T12, T13. Similarly, we determine T14, T15 and T16. Then we get a unique tiling T(8α 2βγ; 4β 2γ 2 ), the 3D picture for tiling is shown in the third of [PITH_FULL_IMAGE:figures…

**Figure 22.** Figure 22: The tiling for α 2βγ = αγ β γβ α βα γ βα γ . By calculation, we get α = 3 4 , β = arctan √ 2 π ≈ 0.3041, γ = 1 2 − arctan √ 2 π ≈ 0.1959, a = 2π 3 , b = π 2 , c = π 4 . Case α 4 From [PITH_FULL_IMAGE:figures/full_fig_p017_22.png]

**Figure 23.** Figure 23: The possibilities for a half vertex α α. For f = 16, we have AVC = {α 4 , αβ2γ 2 ; α 2}. If a = 2c, in the adjacent tiles of αβ2γ 2 , there is a half vertex at β or γ. If b = 2c, then we get α = 1 2 , β = 1 2 , γ = 1 4 by calculation. Dawson has discussed in [7]. For …

**Figure 24.** Figure 24: We get β1 · · · is a half vertex by Lemma 8, a contradiction. If α 2β 2γ = βγ α 1 γα β 2 βα γ 3 γβ α 4 αβ γ 5 , this determine T1, T2, T3, T4 and T5 in the second of [PITH_FULL_IMAGE:figures/full_fig_p019_24.png]

**Figure 24.** Figure 24: Three possibilities for α 2β 2γ. Case α 2βγ2 From [PITH_FULL_IMAGE:figures/full_fig_p020_24.png]

**Figure 25.** Figure 25: f = 36, T(6α 4 , 2β 6 , 6βγ6 ; 6α 2 , 6β 3 ). More details will given in [13]. References [1] C. Adams. The Tiling Book: An Introduction to the Mathematical Theory of Tilings. American Mathematical Society, Providence, RI, 2022. [2] C. Adams, C. Edgar, P. Hollander, L…

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Reference graph

Works this paper leans on

23 extracted references · 17 canonical work pages

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Pith tools

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

[1] [1]

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

2022

[2] [2]

Adams, C

C. Adams, C. Edgar, P. Hollander, L. Jacoby. The non-edge-to-edge tilings of the sphere by regular polygons. Discrete Comput. Geom. 72 (2024) 1029-1085

work page 2024

[3] [3]

Akama, E

Y. Akama, E. Wang, M. Yan. Tilings of sphere by congruent pentagons III: edge combi- nationa 5. Adv. Math. (2022) 0394:107881

work page 2022

[4] [4]

H. M. Cheung, H. P. Luk, M. Yan. Tilings of the sphere by congruent quadrilaterals or triangles. Preprint, arXiv:2204.02736v3

arXiv

[5] [5]

H. M. Cheung, H. P. Luk, M. Yan. Tilings of the sphere by congruent pentagons IV: edge combinationa 4b. Preprint, arXiv:2307.11453

arXiv

[6] [6]

H.L. Davies. Packings of spherical triangles and tetrahedra. Proc. Colloquium on Convex- ity, Copenhagen (1967) 42-51

work page 1967

[7] [7]

R.J.M. Dawson. Tilings of the sphere with isosceles triangles. Discrete Comput. Geom. 30 (2003) 467-487. 21

work page 2003

[8] [8]

R.J.M. Dawson. An isosceles triangle that tiles the sphere in exactly three ways. Discrete Comput. Geom. 30 (2003) 459-466

work page 2003