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REVIEW 3 major objections 4 minor 24 references

Neoplatonic solids

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Every 6-net with up to 50 vertices admits an undented Euclidean realization built from unit equilateral triangles, and every prime 6-net in this range has a unique convex ideal hyperbolic realization.

desk verdict A genuinely new finite census with a plausible but not fully auditable computer-assisted proof; worth engaging, but the central existence theorem hangs on software the paper doesn't let you check. read the letter →

arxiv 2607.26363 v1 pith:NTNBOYH2 submitted 2026-07-29 math.MG math.CO

Peter Doyle , Matthew Ellison This is my paper · ORCID
classification math.MGmath.CO MSC 52B7051M1052C26
keywords 6-netneoplatonicsolididealhyperbolicpolyhedronequilateraltrianglesimplicialtriangulationundentedcomputer-assistedproofsphere
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 works toward a conjecture that every 6-net — a simplicial triangulation of the 2-sphere in which no vertex touches more than six triangles — can be realized as an undented Euclidean polyhedron made of unit equilateral triangles, and also as an ideal hyperbolic polyhedron with equilateral ideal-triangle faces. The authors prove the Euclidean part of this conjecture for all 10,412,340 combinatorially distinct 6-nets with at most 50 vertices, and they prove the ideal-hyperbolic part for every prime 6-net in the same range, including uniqueness of the convex ideal realization. The proof is computer-assisted: it combines complete enumeration of prime nets, numerical homotopy from ideal to Euclidean realizations, and interval-arithmetic certification that an exact solution lies near each approximate one, with a separating-triangle decomposition that reduces non-prime nets to prime factors. If the conjecture holds in general, these 'neoplatonic solids' form an infinite family that extends the tetrahedron, octahedron, and icosahedron, and the methods here demonstrate a template for rigorous computer-assisted existence proofs in polyhedral geometry.

What carries the argument

The central objects are 6-nets — simplicial triangulations of the 2-sphere with maximum vertex degree at most 6 — with a net called prime if every 3-cycle bounds a face. An undented polyhedron is one where every vertex has a local support plane, so the exterior dihedral angles around each vertex sum to at least 0. The argument rests on three mechanisms: (1) an effective inverse function theorem with interval-arithmetic error bounds, which certifies that an exact Euclidean realization lies near a numerically computed approximate one, with coplanarity conditions substituted at degree-6 vertices that flatten; (2) a bracket method of super- and subsolutions for the equations defining ideal hyper

What would settle it

Run an independent implementation of the enumeration and the interval-arithmetic certification on every 6-net with v≤50 and check that the prime/non-prime counts agree with 8,239,684 and 2,172,656 and that every net passes the embeddedness and undentedness checks; any mismatch or any net without a valid certificate would disprove the claim.

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Extended reading notes

Core claim

The central claim is an existence theorem: every 6-net with v≤50 has an undented realization in Euclidean 3-space whose faces are unit equilateral triangles, and every prime 6-net with v≤50 has a unique convex ideal realization in hyperbolic 3-space. The Euclidean proof is computer-assisted: approximate realizations are obtained by numerical homotopy from the ideal ones, and an effective inverse function theorem with interval arithmetic proves that an exact realization lies nearby; when a degree-6 vertex flattens, coplanarity conditions replace some edge-length equations. Uniqueness of the Euclidean realization is not proved. Non-prime nets are handled by cutting along separating triangles,

Load-bearing premise

The whole theorem rests on the unverified correctness of the archived computer programs that enumerate the nets and certify each existence claim; there is no formal machine-checked proof of the software, so any undetected bug in the enumeration, the interval bounds, or the homotopy could invalidate the result even if the underlying conjecture is true.

Editorial extensions

If this is right

  • Every 6-net with v≤50 has a neoplatonic realization; up to combinatorial isomorphism there are 10,412,340 such nets, of which 8,239,684 are prime and 2,172,656 are non-prime.
  • Every prime 6-net with v≤50 has a convex ideal hyperbolic realization, unique up to isometry; this provides a large certified family of ideal equilateral hyperbolic polyhedra.
  • The non-prime nets with v≤50 are completely classified into three families, all realized by gluing regular tetrahedra and octahedra; these gluings are checked exactly with rational coordinates or interval-extended angle bounds, and undentedness is proven by a spherical isoperimetric inequality.
  • The volume of the ideal icosahedron is computed to high precision, a numerical invariant that can serve as a check on future constructions.
  • The numerical homotopy from ideal to Euclidean realizations offers experimental evidence for the neoplatonic homotopy conjecture: that Euclidean neoplatonics are the zero-length limits of a one-parameter family of undented hyperbolic polyhedra.

Reading between the lines

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

  • The certification recipe — find a numerical solution, then close the gap with interval arithmetic and an effective inverse function theorem — is a template that could be applied to other equilateral polyhedral realization problems, for example triangulations with maximum degree 7 or with prescribed non-unit edge lengths, whenever a homotopy from a hyperbolic or other known solution exists.
  • If the general neoplatonic conjecture is true, the family of neoplatonic solids provides a discrete uniformization statement: every sphere triangulation of degree at most 6 has a canonical Euclidean and hyperbolic equilateral representative. Proving the paper's neoconvex rigidity hypothesis would imply uniqueness of these representatives and yield a rigidity theorem that permits local nonconvexity
  • The super- and subsolution bracketing for ideal realizations closely parallels circle-packing existence proofs; the authors highlight that the edge-length rule here is a product rather than a sum. A plausible route to the full ideal conjecture is to construct bracketing functions for all 6-nets, which the computations suggest must become extremely tight as the vertex count grows.
  • The volume-maximization conjecture for ideal neoplatonics is computationally testable within the current census: one could compute the algebraic volume of each certified ideal realization and check whether any alternative ideal geodesic 2-cycle with the same combinatorics has larger volume, which would either support or falsify the conjecture in the tested range.
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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

3 major / 4 minor

Summary. The paper defines 6-nets as simplicial triangulations of the 2-sphere with maximum degree at most 6 and studies two conjectured realizations: an 'undented' Euclidean polyhedron built from unit equilateral triangles (a neoplatonic solid) and an equilateral ideal hyperbolic polyhedron. The main finite result is Corollary 2: every 6-net with v ≤ 50 has a Euclidean neoplatonic realization; there are 8,239,684 prime and 2,172,656 non-prime nets, totaling 10,412,340. Proposition 1 asserts existence of Euclidean realizations for all prime 6-nets with v ≤ 50, and Proposition 3 asserts existence and uniqueness of convex ideal neoplatonic realizations for the same range. The proofs combine exact rational constructions for two simple non-prime classes, a Perron-type super/subsolution argument for ideal existence, and a large computer-assisted certification using archived software for the prime Euclidean case.

Significance. If the computational certificates are sound, the paper establishes a substantial finite case of the neoplatonic conjecture and provides a strong test of the broader conjectural framework. The exact rational constructions for the 145 nets in the first two non-prime classes, the independent count cross-checks against the census [7], and the use of Rivin's uniqueness theorem for ideal convex realizations are genuine strengths. However, the central Euclidean existence proof is currently not independently auditable from the text, and several load-bearing computational steps are described in only one or two sentences.

major comments (3)
  1. [2, Proposition 1] The proof of Proposition 1 is a black-box reference to [12] and [10]. It does not state the nonlinear system being solved, the dimension/degrees of freedom, the norm used in the effective inverse function theorem, the interval-arithmetic library or radii, or the certificate format. The key sentence 'In cases where the limit lies flat at one or more degree-6 vertices, some of the equations specifying edge lengths get replaced by coplanarity conditions' is not expanded into a precise mathematical statement: which equations, exactly how the substitution is performed, and why the resulting system implies existence of an actual neoplatonic realization. Since Corollary 2's prime case relies entirely on this proposition, this is a load-bearing gap in auditability.
  2. [3, third class] The class-3 argument, which handles 2,172,511 nets, is compressed into two sentences: 'After identifying combinatorial duplicates...' and 'An interval embeddedness check proves that each has an exact embedded realization.' The paper does not explain how the certified intervals from the prime case are extended across attached tetrahedra, what predicate is used for combinatorial duplicate identification, or what the interval embeddedness check verifies (e.g., no self-intersections, positive dihedral angles, local support planes). These details are essential because the class-3 count and the existence assertions for all non-prime v ≤ 50 depend on them.
  3. [6] The ideal-existence proof uses a Perron-type method, but the construction of the super- and subsolutions u_0, u_1 is described only by example and by the assertion 'we chose super- and subsolutions with all defects equal to ±1/500... ±1/4000.' The paper does not specify how u_0 and u_1 are generated for each of the 8.2 million prime nets, how the descent algorithm is proved to terminate, or what certificates verify the bracketed triangle, Delaunay, and boundary inequalities. The statement 'we can check that as long as u0 ≤ u ≤ u1 the triangle inequalities hold' is a computational claim, but no check program or certificate format is described at a level that a reader can verify. Proposition 3's existence portion therefore rests on an unstated computational protocol.
minor comments (4)
  1. [3] The three-class decomposition is clear in broad strokes, but the relation between the number of tetrahedral/octahedral factors and the number of vertices is not explicitly derived; equations such as v = 3 + 3m + t appear without supporting definitions of m and t. A short derivation would improve readability.
  2. [4] The term 'algebraic volume' for an oriented ideal cycle is not defined or referenced. Since it is used to state Conjecture (Volume max), a definition or reference is needed.
  3. [8 and References] Reference [3] is given as 'Seek: Atlas of neoplatonic solids' with no URL or DOI; this is not a usable citation. Also, the acknowledgments mention that computations were carried out by Claude Code and ChatGPT; if the journal has a policy on AI-assisted computation, this should be stated more explicitly.
  4. [2] The introductory characterization of prime nets states 'A 6-net is prime just if it is the tetrahedron, or if it has no vertex of degree 3 and is not a stack of two or more octahedra.' This is asserted without proof; a short justification or reference would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central existence claims rest on independent constructions and external citations; main limitations are auditability, not circularity.

full rationale

I walked the derivation chain. Proposition 1 (prime 6-nets have Euclidean neoplatonic realizations) is proved by numerical approximation plus an effective inverse-function theorem with guaranteed error bounds, using the archived software [10] and the general method of Ellison [12]. The approximate seed is obtained by numerical homotopy from ideal realizations, but the exact existence conclusion does not rest on the seed: it rests on the IFT interval certificate. Proposition 3 (ideal existence and uniqueness) is proved from the independent Perron super/subsolution construction for Springborn's equations, with uniqueness from Rivin [18], neither of which is derived from the paper's own conclusions. Corollary 2 combines Proposition 1, the explicit three-class decomposition for non-prime nets (with direct rational-coordinate constructions for classes 1–2 and interval checks for class 3), and cross-checks the resulting counts against the external census [7]. No fitted parameter is renamed as a prediction, no theorem is imported from the authors' own prior work to forbid alternatives, and no definition is made in terms of the quantity to be derived. The dependence on [10], [12], and unreproduced interval-certificate details is a reproducibility/auditability concern, not a circularity concern; the paper itself flags the experimental status of the homotopy conjecture. Hence score 0.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The proof rests on external theorems (Rivin; BPS/Springborn; spherical isoperimetry) and on a substantial computational oracle. The only hand-chosen numbers are finite proof-management parameters such as defect sizes and bracket functions; no physical constants are fitted. No new physical entities are postulated.

free parameters (2)
  • Per-case super/subsolution defect magnitude δ_v = 1/500 (v=4..24), 1/1000 (v=25), ..., 1/4000 (v=50)
    §6: chosen by hand to make the Perron bracket work; the values are tightened as v grows, so the finite proof depends on these arbitrary choices.
  • Per-net bracket pair (u_0,u_1) = Not listed for each net
    §6: the existence of bracketing functions satisfying triangle, flatness, boundary-convexity, and Delaunay inequalities is asserted per prime net and verified computationally; these functions are proof inputs rather than derived values.
assumptions (8)
  • domain assumption Rivin's theorem: a net has at most one convex equilateral ideal hyperbolic realization.
    Used in Prop. 3 for uniqueness; cited as [18], not proved in the paper.
  • domain assumption Springborn–Bobenko–Pinkall–Luo characterization of ideal equilateral realizations by functions u with len(ab)=u(a)u(b), flatness, boundary convexity, and Delaunay inequalities.
    §6: external theory used to construct and certify ideal realizations.
  • domain assumption Effective inverse function theorem with interval arithmetic is correctly implemented in the archived software.
    §2 Proposition 1 proof: central Euclidean existence depends on this computational oracle.
  • domain assumption Prime 6-net characterization: prime iff tetrahedron, or no degree-3 vertex, and not a stack of two or more octahedra.
    §2: asserted without proof; load-bearing for enumeration and the separating-triangle classification.
  • standard math Spherical isoperimetric inequality bound used to show tetrahedral attachments preserve undentedness.
    §3: external theorem applied for degrees 3, 4, and 5.
  • domain assumption Cutting along all separating triangles decomposes any non-prime 6-net into prime factors, and the three stated classes exhaust the non-prime nets with v≤50.
    §3: classification is asserted with enumerative counts rather than derived from first principles in the text.
  • ad hoc to paper At flat degree-6 vertices, replacing edge-length equations by coplanarity conditions yields valid certified existence statements.
    §2 Proposition 1 proof: this limiting rule is introduced specifically for degenerate limits and is not derived in the text.
  • domain assumption Enumeration oracles (buckygen, edgebreaker/CLERS, Goedgebeur variant, and the [7] census) are correct and complete.
    §8: generation of all 6-nets relies on these programs; counts are cross-checked but the software is not formally verified.

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Cite this review

Pith. "Pith review of Neoplatonic solids." pith.science (2026). https://pith.science/paper/NTNBOYH2

@misc{pith2026260726363,
  author       = {Pith},
  title        = {Pith review of: Neoplatonic solids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NTNBOYH2}},
  note         = {Machine review of arXiv:2607.26363}
}
abstract

A \emph{6-net} is a simplicial triangulation of the $2$-sphere with maximum degree $\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these \emph{neoplatonic solids} and \emph{ideal neoplatonics}. A net is \emph{prime} if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realization yields an approximate Euclidean neoplatonic, and a computer-assisted proof shows that a true Euclidean neoplatonic lies nearby, though we do not prove uniqueness. Using the separating-triangle decomposition, we extend Euclidean existence to all $10{,}412{,}340$ $6$-nets with $v\leq50$, counted up to combinatorial isomorphism.

Figures

Figures reproduced from arXiv: 2607.26363 by the authors.

Figure 1
Figure 1. Realizing an ideal neoplatonic. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Realizing the ideal icosahedron. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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Works this paper leans on

24 extracted references · 7 canonical work pages

  1. [7]

    McKay, and Ulrike von Nathusius,Back- track search and look-ahead for the construction of planar cubic graphs with restricted face sizes, MATCH Commun

    Gunnar Brinkmann, Brendan D. McKay, and Ulrike von Nathusius,Back- track search and look-ahead for the construction of planar cubic graphs with restricted face sizes, MATCH Commun. Math. Comput. Chem. 48(2003), 163–177,https://match.pmf.kg.ac.rs/electronic_versions/ Match48/match48_163-177.pdf

    2003

  2. [12]

    Matthew Ellison,Realizing abstract simplicial complexes with specified edge lengths, 2023,https://doi.org/10.48550/arXiv.2312.05376

    work page Pith review arXiv doi:10.48550/arxiv.2312.05376 2023

  3. [10]

    Peter Doyle,Software for the computer-assisted proof of Euclidean neo- platonic realizations, Zenodo,https://doi.org/10.5281/zenodo.21609862, 2026, Software and check outputs for6-nets withv≤50. 14

    work page doi:10.5281/zenodo.21609862 2026

  4. [1]

    Ian Agol,Lower bounds on volumes of hyperbolic Haken 3-manifolds, 1999, https://doi.org/10.48550/arXiv.math/9906182

    work page Pith review arXiv doi:10.48550/arxiv.math/9906182 1999

  5. [2]

    A. D. Alexandroff,Existence of a convex polyhedron and of a convex sur- face with a given metric (English summary), Sb. Math. (1942), 61–65, Adden- dum tohttps://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid= sm&paperid=6131&option_lang=eng. [3]Seek:Atlas of neoplatonic solids

    1942

  6. [4]

    Daniel Audet,Déterminants sphérique et hyperbolique de Cayley–Menger, Bulletin AMQ51(2011), 45–52,https://www.amq.math.ca/wp-content/ uploads/bulletin/vol51/no2/chronique_note_math.mai11.pdf

    2011

  7. [5]

    Bobenko, Ulrich Pinkall, and Boris A

    Alexander I. Bobenko, Ulrich Pinkall, and Boris A. Springborn,Discrete confor- mal maps and ideal hyperbolic polyhedra, Geom. Topol.19(2015), 2155–2215, https://doi.org/10.2140/gt.2015.19.2155

    work page doi:10.2140/gt.2015.19.2155 2015

  8. [6]

    McKay,The generation of fullerenes, J

    Gunnar Brinkmann, Jan Goedgebeur, and Brendan D. McKay,The generation of fullerenes, J. Chem. Inf. Model.52(2012), 2910–2918,https://doi.org/ 10.1021/ci3003107, orhttps://arxiv.org/abs/1207.7010

    arXiv 2012

Show all 24 references
  1. [8]

    Second mémoire, Journal de l’École Polytechnique9(1813), 87–98, Reprinted in Œuvres com- plètes, Série 2, Tome 1, pp

    Augustin-Louis Cauchy,Sur les polygones et les polyèdres. Second mémoire, Journal de l’École Polytechnique9(1813), 87–98, Reprinted in Œuvres com- plètes, Série 2, Tome 1, pp. 26–38,https://books.google.com/books?id= SxczAQAAMAAJ&pg=PA87

  2. [9]

    Robert Connelly,Review of Convex Polyhedra by A. D. Alexandrov, March 2006, https://pi.math.cornell.edu/~connelly/alexandrov.pdf

    2006

  3. [11]

    Peter Doyle, Matthew Ellison, and Zili Wang,Filling a triangulation of the 2-sphere, 2024,https://doi.org/10.48550/arXiv.2303.10773

    2024 doi arXiv

  4. [13]

    Hans Freudenthal and B. L. van der Waerden,Over een bewering van Euclides, Simon Stevin25(1947), 115–121,https://zbmath.org/0030.17302

    1947

  5. [14]

    thesis, Ghent University, 2013,http://hdl.handle.net/1854/ LU-3235187

    Jan Goedgebeur,Generation algorithms for mathematical and chemical prob- lems, Ph.D. thesis, Ghent University, 2013,http://hdl.handle.net/1854/ LU-3235187

    2013

  6. [15]

    Feng Luo,Combinatorial Yamabe flow on surfaces, Comm. Contemp. Math.6 (2004), 765–780,https://doi.org/10.1142/S0219199704001501, orhttps: //arxiv.org/abs/math/0306167

    2004 arXiv

  7. [16]

    Tome III, Imprimerie Impériale, Paris, 1818,https://archive.org/details/ lesuvresdeuclide03eucl

    François Peyrard (ed.),Les Œuvres d’Euclide en Grec, en Latin et en Français. Tome III, Imprimerie Impériale, Paris, 1818,https://archive.org/details/ lesuvresdeuclide03eucl

  8. [17]

    Otto Rausenberger,Konvexe pseudoreguläre Polyeder, Zeitschrift für mathe- matischen und naturwissenschaftlichen Unterricht46(1915), 135–142,https: //books.google.com/books?id=tQMgAQAAMAAJ&pg=PA135

    1915

  9. [18]

    3, 553–580,https://doi.org/10

    Igor Rivin,Euclidean structures on simplicial surfaces and hyperbolic volume, Annals of Mathematics139(1994), no. 3, 553–580,https://doi.org/10. 2307/2118572

    1994

  10. [19]

    Jarek Rossignac,Edgebreaker: Connectivity compression for triangle meshes, IEEE Trans. Vis. Comput. Graph.5(1999), 47–61,https://doi.org/10. 1109/2945.764870, orhttps://faculty.cc.gatech.edu/~jarek/papers/ EdgeBreaker.pdf

    1999

  11. [20]

    Jarek Rossignac, Alla Safonova, and Andrzej Szymczak,3D compression made simple: Edgebreaker on a corner-table, Proceedings of Shape Modeling In- ternational, 2001,https://doi.org/10.1109/SMA.2001.923399, orhttps: //www.cs.cmu.edu/~alla/edgebreaker_simple.pdf, pp. 278–283

    2001

  12. [21]

    I. Kh. Sabitov,The volume as a metric invariant of polyhedra, Discrete Comput. Geom.20(1998), 405–425,https://doi.org/10.1007/PL00009393, 15 orhttps://www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/ Sabitov/20n4p405.pdf

    1998 doi

  13. [22]

    Sleator, Robert E

    Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston,Rotation dis- tance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc.1(1988), 647–681,https://doi.org/10.2307/1990951

    1988 doi

  14. [23]

    Geom.64(2020), 63–108,https://doi.org/10.1007/ s00454-019-00132-8, orhttps://arxiv.org/abs/1707.06848

    Boris Springborn,Ideal hyperbolic polyhedra and discrete uniformization, Discrete Comput. Geom.64(2020), 63–108,https://doi.org/10.1007/ s00454-019-00132-8, orhttps://arxiv.org/abs/1707.06848

    2020 arXiv

  15. [24]

    Thurston,Shapes of polyhedra and triangulations of the sphere, 1998, https://doi.org/10.48550/arXiv.math/9801088

    William P. Thurston,Shapes of polyhedra and triangulations of the sphere, 1998, https://doi.org/10.48550/arXiv.math/9801088

    1998 doi arXiv

  16. [25]

    E. B. Vinberg,Volumes of non-Euclidean polyhedra, Russian Math. Surveys48 (1993), 14–45,https://doi.org/10.1070/RM1993v048n02ABEH001011. 16

    1993 doi

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