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Tilings of the sphere by congruent pentagons IV: Edge combination $a^4b$ with general angles

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Pith's one-line read This paper proves that every edge-to-edge tiling of the sphere by congruent $a^4b$ pentagons with an irrational angle appears in a short explicit list.

desk verdict A genuinely shorter route to a known classification, with real gaps in the case-completeness argument; worth refereeing but needs the skipped branches filled. read the letter →

arxiv 2412.08492 v1 pith:34EFBH7P submitted 2024-12-11 math.CO

Junjie Shu , Yixi Liao , Erxiao Wang This is my paper · ORCID
classification math.CO MSC 52C2005B4551M20
keywords sphericaltilingcongruentpentagonsedge-to-edgea4bpentagonalmostequilateralirrationalanglepentagonalsubdivisionearthmap
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 paper sets out to settle which edge-to-edge tilings of the sphere can be made from congruent pentagons with four equal sides and one different side, the edge combination $a^4b$, under the 'general angles' assumption that at least one tile angle is irrational in degrees. Its central claim is that every non-symmetric such tiling is either a one-parameter pentagonal subdivision of the octahedron or icosahedron, or belongs to a sequence of one-parameter families built on a non-symmetric three-layer earth map tiling with $4m$ tiles, together with explicit flip modifications and extra rearrangements in the $f=8k+4$ case. The symmetric case, already reduced to quadrilateral tilings, gives the tetrahedron subdivision and a symmetric earth-map family. If the classification is right, the a priori continuous freedom in the angles collapses to a short explicitly parametrized list, and the paper, together with the announced rational-angle sequel, completes the classification of all edge-to-edge congruent pentagon tilings of the sphere for this edge combination.

What carries the argument

The engine of the proof is the Irrational Angle Lemma: given three distinct vertex types whose exponent vectors, together with the angle-sum vector $(1,1,1,1,1)$, are linearly independent, every other vertex must lie in the same three-dimensional affine subspace, because otherwise the angle system would force a rational solution for an irrational angle. That lemma turns a continuous angle problem into finite lists of candidate vertex combinations. These candidates are then pruned by the Balance Lemma, the Parity Lemma, and adjacent-angle deduction (AAD), the local rule that propagates tile labels and angles around a vertex until the whole tiling is forced; the pentagonal subdivision and the three-layer earth map construction provide the geometric templates for the tilings that survive.

What would settle it

A concrete check would be an exhaustive enumeration, for small sphere sizes such as $f=28$, $36$, and $60$, of all angle-wise vertex combinations consistent with the irrational-angle, balance, and parity lemmas and with the geometric equations (2.3)--(2.5); any resulting edge-to-edge tiling whose vertex set is not among those in Theorem 1 or Theorem 2 would be a counterexample, and the skipped 'similar' branches are the natural place to begin.

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Extended reading notes

Core claim

The paper's central discovery is Theorem 2: for a non-symmetric $a^4b$-tiling with general angles, the only possible tilings are the pentagonal subdivisions of the octahedron $T(24\alpha\delta\epsilon,8\gamma^3,6\beta^4)$ and of the icosahedron $T(60\alpha\delta\epsilon,20\gamma^3,12\beta^5)$, and a sequence of one-parameter families admitting the non-symmetric 3-layer earth map tiling $T(4m\, \alpha\delta\epsilon,2m\,\beta^2\gamma,2\gamma^m)$ for every $m\ge4$. Each odd $m=2k+1$ family also admits a standard flip modification, and when $f=8k+4$ with $\alpha=\beta=(1-4/f)\pi$ there are many further tilings formed by rearranging UFO blocks, all counted in Table 3. The proof reaches this list by showing that every alternative combination of degree-3 vertex types either violates the irrational-angle lemma, the balance lemma, or the parity lemma, or dies in an adjacent-angle deduction, and that the surviving combinations propagate uniquely to the listed tilings.

Load-bearing premise

The classification is complete only if the case trees in Sections 3–6 are exhaustive, including the branches dismissed as 'similar' without written deduction, and only if the UFO multiplicities counted visually from 3D computer models in Table 3 are correct.

Editorial extensions

If this is right

  • A fixed $a^4b$ pentagon with an irrational angle can have only the tilings listed in Theorem 2 (or the symmetric ones in Theorem 1), and the paper gives an explicit count of how many distinct tilings each prototile admits.
  • The completion of the $a^4b$ case closes the last edge combination in the decomposition of pentagonal tilings, so with the sequel all edge-to-edge tilings of the sphere by congruent pentagons will be classified.
  • The listed families come with exact angle and side-length formulas, so each tiling can be constructed and checked directly rather than only abstractly.
  • Letting one of the pentagon's angles become $\pi$ turns these tilings into new non-edge-to-edge quadrilateral tilings, extending the known quadrilateral tiling landscape.

Reading between the lines

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

  • The handwritten case trees are exactly the kind of finite enumeration that could be machine-verified; replacing the 'similar' dismissals and the visual UFO counts with a formal check would put the completeness claim on fully checkable ground.
  • The UFO-flip mechanism suggests that all the $f=8k+4$ variations are generated from one earth-map tiling by a small set of local moves, which would explain the polynomial counting formulas in Table 3 and might generalize to other tiling classifications.
  • The degenerate-pentagon quadrilateral tilings may offer test cases for extending the known edge-to-edge quadrilateral classification to non-edge-to-edge settings, since they are produced from explicit pentagon families rather than by ad hoc construction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. This paper studies edge-to-edge tilings of the sphere by congruent pentagons with edge combination a^4b, under the assumption that at least one angle is irrational in degrees. The main result, Theorem 2, states that the only non-symmetric tilings are two 1-parameter pentagonal subdivisions of the octahedron and icosahedron, and a sequence of 1-parameter families whose generic member supports a non-symmetric 3-layer earth map tiling with 4m tiles, together with flip modifications and additional rearrangements when f=8k+4. Theorem 1 records the symmetric cases from previous work. The proof uses the Irrational Angle Lemma, the parity and balance lemmas, and extensive adjacent-angle-deduction (AAD) case analysis organized in tables. Exact and numerical geometric data, tile counts, and some induced non-edge-to-edge quadrilateral tilings are also given.

Significance. If the classification is correct and complete, it provides a valuable independent verification of the a^4b case previously treated in [3], and it contributes a concise source of explicit prototile data, 3D pictures, and tiling counts for all listed families. The Irrational Angle Lemma gives a transparent way to reduce the rational-linear-algebra constraints to finite combinatorial cases. The paper is explicit about its main strengths: exact parameter ranges and geometric data are provided for every family, and the total number of tilings for a fixed prototile is stated. The main limitation is that the proof is not fully checkable as written because several exclusion branches are asserted rather than derived; those branches are exactly where completeness of the classification must be verified.

major comments (4)
  1. [Section 5, Proposition 4] The exclusion of the seven rows of Table 12 other than {γ3} and {αγ2} is not proved. The sentence 'The other cases lead to contradictions in a similar and much easier way' follows only the worked rows {γ3} and {αγ2}, but each row has its own angle system, its own Irrational Angle Lemma relation, and its own AAD chain. Since Theorem 2 asserts completeness, a surviving row would change the classification. Please either display the reductions for all rows or give a precise reduction to the displayed cases.
  2. [Section 3, Proposition 1] After the worked example for {αδ2,γϵ2,βγ2}, the proof states 'All other pairs are discussed similarly by Table 5 and 6.' The tables record input data and final contradictions but not the individual AAD deductions for the remaining ten pairs of Lemma 15. Because the AAD deductions are the actual argument that closes each branch, 'similarly' is not checkable from the manuscript. At minimum, the tables should be expanded to make each contradiction reproducible, or each pair should be referred to a specific published case.
  3. [Sections 5-6, Propositions 5-7 and 10] Several propositions defer to large tables with the phrase 'similar to Proposition 1' or say the proof is completed by the table. For example, Proposition 5 refers to Tables 14-20, Proposition 6 to Tables 21-22, Proposition 7 to Tables 23-27, and Proposition 10 to Tables 28-29. These tables state vertex types, angle values, and the irrational-angle relations, but the AAD derivation is omitted for whole blocks. This is a load-bearing completeness gap, not a presentation issue.
  4. [Section 3, Table 3] The multiplicity counts in Table 3 are support for the claim that the total number of different tilings is counted explicitly. The text says these counts were obtained 'by playing with the 3D pictures in GeoGebra,' and no independent counting rule is supplied. Since the number of tilings for a fixed prototile is one of the stated new results, please provide a reproducible enumeration or a rigorous combinatorial derivation of the counts in the last column of Table 3.
minor comments (3)
  1. [Section 4, first paragraph] The text refers to 'a special tile from Lemma 2', but the special-tile lemma is Lemma 4; please correct the reference.
  2. [Throughout] The two-dimensional tables are often hard to read because the 'Contradiction' column abbreviates a multi-step AAD argument to a phrase such as 'No β2···'. Please add a short legend describing the columns and the convention for reading angle words.
  3. [Section 1, Introduction] The statement that the sequel shows 'there are no more new prototiles, nor new tilings' is only an announcement; please label it explicitly as a claim proved in [5].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the a4b classification is derived by case analysis plus the external geometric lemma [3]; self-cited lemmas are routine prior results, not the target theorem.

full rationale

The central claim of Theorem 2 is not obtained by defining its output as its input. The exclusion branches are argued from Lemma 13 (Irrational Angle Lemma), which uses only the angle-sum Lemma 3 and linear algebra, together with adjacent-angle-deduction (AAD) reasoning, and the geometric feasibility checks use Lemma 11, explicitly imported from the external 174-page classification [3] by Cheung, Luk, and Yan. The same-series citations ([7], [9], [10], [11]) supply prior published lemmas (parity, balance, special-tile, Lemma 8, Lemma 9, Lemma 10) that are parameter-free results about spherical pentagon tilings in general; none is defined in terms of the a4b classification being proved, and none is used to assert the target theorem itself. The pentagonal subdivision families are constructed directly by subdividing Platonic faces, and their parameter ranges are given explicitly in Table 1 rather than being fitted from the tiling data. There are, however, many genuinely omitted case analyses: Proposition 1 dismisses 'All other pairs are discussed similarly by Table 5 and 6', Proposition 4 dismisses seven Table 12 rows as 'similar and much easier', Proposition 3 appeals to Table 11 'similar to Proposition 1', and Table 3's UFO counts are said to come from 'playing with the 3D pictures in GeoGebra'. These are completeness and verification gaps (correctness risk), not circularity: they do not define the conclusion into the premises, and they do not rename a known result as a new one. The self-citations are load-bearing only as routine foundational lemmas, which are published, checkable, and not equivalent to the a4b completeness claim, so the derivation is self-contained in the sense relevant to circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on a large toolkit imported from earlier papers. The authors' own series ([10], [11], [2], [9]) supplies the AAD technique, the parity lemma, the balance lemma, the special tile lemma and the basic counting identities; these are self-citations but grounded in previously published classifications of the same program. The external paper [3] supplies Lemma 11, the necessary and sufficient conditions for angles and an edge length to define an almost equilateral pentagon, used to prove geometric existence of every family. The family parameters t and α are the genuine moduli of the 1-parameter families, not fitted constants, and the excluded values t0 are exactly the equilateral degeneracies. No invented entities appear; the UFO is a named four-tile patch, not a postulated object. The most fragile postulates are the completeness of the AAD framework and the deferred rational-angle claim.

assumptions (9)
  • standard math Euler-type counting identities f = 12 + 2Σ(k-3)v_k and v_3 = 20 + Σ(3k-10)v_k for pentagonal tilings of the sphere
    Equations (2.1) and (2.2) in Section 2, cited from [10]; these are standard Euler characteristic and degree-counting consequences for spherical tilings.
  • standard math The angle sum of the tile is 3 + 4/f and no vertex contains all five angles
    Lemma 3, cited from [10, Lemma 4]; follows from the spherical Gauss-Bonnet relation for f congruent pentagons.
  • domain assumption Every tiling has a special tile with at most one vertex of degree greater than 3
    Lemma 4, cited from [10, Lemma 1]; used in Sections 4-6 as the starting point of the local case splits (Figures 25-31).
  • domain assumption Parity lemma: the number of δ and ε angles at any vertex is even
    Lemma 5, cited from [11, Lemma 10]; used throughout the AAD deductions and depends on the edge combination a^4b.
  • domain assumption Balance lemma constraints on δ and ε occurrences at vertices
    Lemma 6, cited from [9, Lemma 6]; used to restrict the possible vertex combinations in Sections 3-6.
  • domain assumption The three equations (2.3)-(2.5) are necessary and sufficient for angles plus edge a to define an almost equilateral pentagon
    Lemma 11, cited from the external paper [3, Lemma 18]; load-bearing for the geometric existence of every family in Tables 1-2.
  • domain assumption The AAD (adjacent angle deduction) technique is a complete local constraint: the displayed arrangements exhaust all possible edge and angle configurations around a given vertex
    Framework from [10, Section 2.5]; every case elimination in Sections 3-6 assumes that the listed AAD branches cover all possibilities.
  • domain assumption At least one of the five angles is irrational in degrees
    Scope condition of the paper, called general angles; used to power the Irrational Angle Lemma 13. The rational-angle case is deferred to the sequel [5].
  • ad hoc to paper The rational-angle case introduces no new prototiles or tilings
    Stated in the Introduction as 'It turns out that there are no more new prototiles, nor new tilings', with proof deferred to the unpublished sequel [5]; used to assert the completeness of the overall pentagonal classification.

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Pith. "Pith review of Tilings of the sphere by congruent pentagons IV: Edge combination $a^4b$ with general angles." pith.science (2026). https://pith.science/paper/34EFBH7P

@misc{pith2026241208492,
  author       = {Pith},
  title        = {Pith review of: Tilings of the sphere by congruent pentagons IV: Edge combination $a^4b$ with general angles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34EFBH7P}},
  note         = {Machine review of arXiv:2412.08492}
}
abstract

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with $12, 24$ and $60$ tiles; and a sequence of $1$-parameter families of pentagons admitting non-symmetric $3$-layer earth map tilings together with their various rearrangements under extra conditions. Their parameter moduli and geometric data are all computed in both exact and numerical form. The total numbers of different tilings for any fixed such pentagon are counted explicitly. As a byproduct, the degenerate pentagons produce naturally many new non-edge-to-edge quadrilateral tilings. A sequel of this paper will handle $a^4b$-pentagons with all angles being rational in degree by solving some trigonometric Diophantine equations, to complete our full classification of edge-to-edge tilings of the sphere by congruent pentagons.

Figures

Figures reproduced from arXiv: 2412.08492 by the authors.

Figure 1
Figure 1. Pentagons with the edge combinations a 4 b. We use α n1 β n2 γ n3 δ n4 ϵ n5 to mean a vertex having n1 copies of α, n2 copies of β, etc.. The anglewise vertex combination(s), abbreviatled as AVC, is the col￾lection of all vertices in a tiling. Then the notation T(16αδϵ, 8β 2γ, 2γ 4 ) means the tiling has exactly 16 vertices αδϵ, 8 vertices β 2γ, and 2 vertices γ 4 , and is uniquely determined by them. In general the… view at source ↗
Figure 2
Figure 2. A symmetric 3-layer earth map tiling and its two standard flip modi [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Pentagonal subdivision of a regular triangle face. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (34 more)
Figure 4
Figure 4. Figure 4: Pentagonal subdivision of the tetrahedron, octahedron and icosahe [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: The 1-parameter family of non-symmetric 3-layer earth map tilings [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: A generic non-symmetric 3-layer earth map tiling and its standard [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: A special flip modification when f = 16 and α = π 2 = γ [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: The only UFO for a tiling with AVC 1 and 60 tiles. [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: All modifications for the non-symmetric 3-layer earth map tiling with [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: as δ δ · · · , δ δ · · · , β δ · · · , δ δ β · · · , and denote the consecutive angle segments as δ δ, δ δ , β δ, δ δ β . δ ϵ γ α β ϵ γ α β δ ϵ γ α β δ δ ϵ γ α β δ ϵ γ α β ϵ γ α β δ ϵ γ α β α δ γ ϵ δ β α γ ϵ δ β ϵ γ α β δ ϵ γ α β α δ γ ϵ δ β [PITH_FULL_IMAGE:figures/…
Figure 11
Figure 11. Figure 11: For f = 16, T{16αδϵ, 8βγ2 , 2β 4} and T{8αδϵ, 8βδϵ, 8αγ2 , 2β 4}. If α 2βγ is a vertex, we have the unique AAD for α 2βγ = γα β δβ α δβ α ϵγ α , βα γ αβ δ αβ δ αγ ϵ , γα β δβ α ϵγ α δβ α or βα γ αβ δ αγ ϵ αβ δ . The first and sec￾ond case gives a vertex αδ · · · . By …
Figure 12
Figure 12. Figure 12: The non-symmetric 3-layer earth map tiling [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: The standard flip modification of the 3-layer earth map tiling. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: The loop of edges dividing the sphere into two identical halves. [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: αγ f+4 8 = βα γ ϵγ α · · · ϵγ α . When f = 20, [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: αγ f+4 8 = βα γ ϵγ α · · · ϵγ α αγ ϵ . • In [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: gives a tiling with three UFO: {T7, T8, T9, T10}, {T4, T9, T10, T11} and {T1, T5, T14, T15}. After flipping the UFO {T7, T8, T9, T10}, the new tiling still has three UFO and the other two UFO are: {T3, T6, T9, T10} and {T1, T5, T14, T15}. A1 A2 A3 A4 A5 A6 A7 A8 A9 A1…
Figure 18
Figure 18. Figure 18: αγ f+4 8 = βα γ αγ ϵ · · · αγ ϵ and α2γ1 · · · = α 2γ. When α2γ1 · · · = αβγ, just like in [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: αγ f+4 8 = βα γ αγ ϵ · · · αγ ϵ and α2γ1 · · · = αβγ. When f = 20, [PITH_FULL_IMAGE:figures/full_fig_p023_19.png]
Figure 20
Figure 20. Figure 20: αγ f+4 8 = βα γ αγ ϵ · · · αγ ϵ and α2γ1 · · · = αγ f+4 8 . When f = 20, [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
Figure 21
Figure 21. Figure 21: γ f−4 8 δϵ = αγ ϵ · · · αγ ϵ β δ ϵ δ ϵ γ . When f = 20, [PITH_FULL_IMAGE:figures/full_fig_p024_21.png]
Figure 22
Figure 22. Figure 22: The half sphere HS1 from γ f−4 8 δϵ = ϵγ α · · · ϵγ α β δ ϵ δ ϵ γ . A1(A2) A2(A1) A3(A12) A4(A11) A5(A10) A6(A9) A7(A8) A8(A7) A9(A6) A10(A5) A11(A4) 1 2 3 4 5 A12(A3) 6 7 8 10 9 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: The half sphere HS2 from γ f−4 8 δϵ = ϵγ α · · · ϵγ α αγ ϵ β δ ϵ δ ϵ γ . A1(A2) A2(A1) A3(A12) A4(A11) A5(A10) A6(A9) A7(A8) A8(A7) A9(A6) A10(A5) A11(A4) 1 2 3 4 5 6 A12(A3) 7 8 9 11 10 [PITH_FULL_IMAGE:figures/full_fig_p025_23.png]
Figure 24
Figure 24. Figure 24: The half sphere HS3 from γ f−4 8 δϵ = ϵγ α · · · αγ ϵ · · · αγ ϵ β δ ϵ δ ϵ γ . 25 [PITH_FULL_IMAGE:figures/full_fig_p025_24.png]
Figure 25
Figure 25. Figure 25: αδ2 , βδ2 or γδ2 is the unique a 2 b-vertex type [PITH_FULL_IMAGE:figures/full_fig_p027_25.png]
Figure 26
Figure 26. Figure 26: Special tile for the edge combination a 4 b. Proposition 4. There is no non-symmetric a 4 b-tiling with general angles, such that βδϵ is the unique a 2 b-vertex type and α 2β is a vertex. Proof. By Lemma 8, one of {αβγ, α2γ, αγ2 , β2γ, βγ2 , γ3 , αγ3 , βγ3 , γ4 , γ5} …
Figure 27
Figure 27. Figure 27: Since βδϵ is the unique a 2 b-vertex type and H = α1 · · · , the degree 3 vertex ϵ1 · · · = βδϵ determines T3, T4. The two pictures show two possible arrangements of T5. In the right-hand side of [PITH_FULL_IMAGE:figures/full_fig_p029_27.png]
Figure 27
Figure 27. Figure 27: H = α1 · · · . In the right-hand side of [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]
Figure 28
Figure 28. Figure 28: Since βδϵ is a 2 b-vertex and H = β1 · · · , we have the degree 3 vertex ϵ1 · · · = βδϵ, which determines T4, T5. The four pictures show four possible arrangements of T3. β δ α ϵ γ β α γ ϵ δ β α δ ϵ γ ϵ δ β α γ β 1 2 3 4 5 6 β δ α ϵ γ α γ α β δ ϵ β α δ ϵ γ ϵ δ β α γ β…
Figure 29
Figure 29. Figure 29: , we have the degree 3 vertex β1δ4 · · · = βδϵ, which determines T5. γ α ϵ β δ β δ ϵ γ α β α β γ ϵ δ α 1 2 3 4 5 6 γ α ϵ β δ β δ ϵ γ α β δ β ϵ γ α γ ϵ δ β α 1 2 3 4 5 6 [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]
Figure 30
Figure 30. Figure 30: H = δ1 · · · . In the first picture, we have α1β4 · · · = αβ2 , αβγ. In the second pic￾ture, we have α1γ3 · · · = α 2γ, αβγ or αγ2 . If α1γ4 · · · = α 2γ, then β1 · · · = αβ2 , αβγ, β3 , β2γ, βγ2 . In the third picture, we have α5γ1 · · · = αβγ or αγ2 , α1α4 · · · = α…
Figure 31
Figure 31. Figure 31: H = ϵ1 · · · . In the first picture, we have α3β1 · · · = αβ2 or αβγ, α1α4 · · · = α 3 or α 2γ. In the second picture, we have γ1 · · · = αβγ, αγ2 , β 2γ, βγ2 or γ 3 . In the third picture, we have α5γ1 · · · = α 2γ or αβγ. If α5γ1 · · · = αβγ, we have H = {α 2 δϵ, δ3…
Figure 32
Figure 32. Figure 32: Therefore AVC ⊂ {βδϵ, α2γ, αβγ, βγ2 , δ2 ϵ 2 , γ2 δϵ}. The AAD of βγ2 is αβ δ αγ ϵ γ , αβ δ ϵγ α αγ ϵ or αβ δ ϵγ α ϵγ α , this implies αδ · · · or αϵ · · · is a vertex, contradiction the AVC [PITH_FULL_IMAGE:figures/full_fig_p035_32.png]
Figure 33
Figure 33. Figure 33: Some quadrilaterals as degenerate pentagons. [PITH_FULL_IMAGE:figures/full_fig_p044_33.png]
Figure 34
Figure 34. Figure 34: The pentagonal subdivisions of the platonic solids with [PITH_FULL_IMAGE:figures/full_fig_p044_34.png]
Figure 35
Figure 35. Figure 35: The pentagonal subdivisions of the platonic solids with [PITH_FULL_IMAGE:figures/full_fig_p045_35.png]
Figure 36
Figure 36. Figure 36: The degenerate non-symmetric 3-layer earth map tilings with 20 tiles [PITH_FULL_IMAGE:figures/full_fig_p045_36.png]

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Forward citations

Cited by 2 Pith papers

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

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

    math.CO 2025-05 conditional novelty 8.0 of 10

    Any non-side-to-side spherical tiling by congruent triangles with an irrational angle is either a 2-layer earth map tiling, an 8-tile family, or the unique 16-tile sporadic tiling.

  2. Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles

    math.CO 2025-07 conditional novelty 5.0 of 10

    Rational-angle a^4b pentagonal sphere tilings are exactly three families: a 12-tile tetrahedral subdivision, a 4m-tile symmetric family with flips, and a 20-tile non-symmetric case.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.