Apple-Peel Unfolding in Three and Four Dimensions: Spiral and Zonal Selection Rules

Supanut Chaidee; Takashi Yoshino

arxiv: 2605.30373 · v1 · pith:74AVXPL7new · submitted 2026-05-22 · 💻 cs.CG

Apple-Peel Unfolding in Three and Four Dimensions: Spiral and Zonal Selection Rules

Takashi Yoshino , Supanut Chaidee This is my paper

Pith reviewed 2026-06-30 15:11 UTC · model grok-4.3

classification 💻 cs.CG

keywords apple-peel unfoldingpolytope netsgreedy traversalregular 4-polytopes120-cellspiral selectionzonal selectionunfolding classification

0 comments

The pith

Zonal rule produces perfect apple-peel orderings on the 120-cell while spiral rule yields none, separating combinatorial success from geometric validity.

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

The paper defines apple-peel unfolding as a greedy algorithm that builds a net by adding faces or cells one at a time in a continuous spiral strip. It introduces two deterministic selection rules: the spiral rule, which chooses the next element by the sharpest clockwise turn measured via signed determinant, and the zonal rule, which chooses the element with maximum coordinate along a chosen peeling axis. These rules are exhaustively tested on all five Platonic solids, all thirteen Archimedean solids, and all six regular convex 4-polytopes. Each solid receives a trichotomous label: Perfect when every starting pair completes a net, Possible when at least one pair succeeds, or Impossible when none do. An equivariance argument proves that face-transitive solids can only occupy the 0 percent or 100 percent extremes.

Core claim

Under the zonal rule every one of the 1,440 possible starting pairs on the 120-cell yields a complete ordering, while the spiral rule yields zero; the 600-cell remains Impossible under both rules. Ordering success is shown to be independent of geometric validity: all 120-cell orderings produce self-intersecting 3D nets, so the polytope has zero valid nets despite its perfect combinatorial result.

What carries the argument

Apple-peel unfolding driven by the zonal rule (maximum coordinate along the peeling axis) and spiral rule (minimum signed determinant), which together generate the sequence of faces or cells.

If this is right

Face-transitive solids are confined to the 0/100 percent success dichotomy by the equivariance argument.
The 120-cell admits perfect orderings under the zonal rule yet admits zero non-self-intersecting 3D nets.
The 600-cell admits no complete orderings under either tested rule.
The zonal rule outperforms the spiral rule on the majority of the solids examined.

Where Pith is reading between the lines

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

The separation of ordering success from geometric validity suggests that apple-peel methods may require an additional post-processing step to detect and avoid intersections in any dimension.
Because the zonal rule depends on a chosen axis, rotating the axis choice might alter the Perfect/Possible classification for non-regular solids.
The same two rules could be applied to irregular or non-convex polytopes to test whether the trichotomy remains useful.

Load-bearing premise

That exhaustive enumeration over all starting pairs together with the two fixed deterministic rules is enough to decide the Perfect/Possible/Impossible label without overlooked geometric intersections or hidden implementation choices.

What would settle it

Discovery of even one starting pair on the 120-cell that fails to complete an ordering under the zonal rule, or discovery of one ordering whose 3D realization has no self-intersections.

Figures

Figures reproduced from arXiv: 2605.30373 by Supanut Chaidee, Takashi Yoshino.

**Figure 2.** Figure 2: Illustration of the right half-space condition ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Apple-Peel unfolding of the Truncated Icosahedron under RZ (max [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Nets of the five Platonic solids obtained by Apple-Peel Unfolding (both RS and [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Successful nets for four Archimedean solids: Truncated Tetrahedron (3.6.6, shown [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Three representative nets of the Truncated Icosahedron under the Zonal rule (RZ, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Snub Cube (38 faces, RZ rule with fallback): one representative net per suc [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Three-dimensional nets under Apple-Peel unfolding (RZ rule): (a) 5-cell, (b) 8-cell, [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗

**Figure 9.** Figure 9: Two representative three-dimensional nets of the 120-cell under RZ with fallback [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗

read the original abstract

Apple-Peel Unfolding is a greedy algorithm that selects the faces (or cells) of a polyhedron (or polytope) one at a time in a spiral order, producing a net analogous to peeling an apple in a single continuous strip. We define two face-selection rules -- RS (Spiral rule: minimum signed determinant, i.e.\ sharpest clockwise turn) and RZ (Zonal rule: maximum coordinate along the peeling axis) -- and systematically evaluate their unfolding success rates on (i)~the five Platonic solids, (ii)~the thirteen Archimedean solids, and (iii)~the six regular convex 4-polytopes. A principal contribution is a three-way classification of each solid as \emph{Perfect} (every starting pair yields a complete net), \emph{Possible} (at least one pair succeeds), or \emph{Impossible} (no pair succeeds), together with an equivariance argument showing that face-transitive solids are confined to the $0/100\%$ dichotomy. RZ achieves the highest success rates in most cases; for the regular 4-polytopes it is the only rule yielding non-zero results for the 120-cell, where it achieves a Perfect result (1,440/1,440 pairs). We note that \emph{ordering success} (completing the greedy traversal) and \emph{geometric validity} (no self-intersection in the 3D realization) are distinct: every 120-cell ordering produces a self-intersecting 3D net, so the 120-cell has zero valid 3D nets despite its Perfect ordering result. The 600-cell is Impossible under all rules tested.

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 introduces two deterministic greedy rules for apple-peel unfolding and classifies regular polytopes into Perfect/Possible/Impossible with a clear ordering-versus-validity split, but the zero-valid-nets claim for the 120-cell rests on unverified intersection checks.

read the letter

The main thing to know is that the authors define spiral (RS) and zonal (RZ) selection rules for a greedy apple-peel unfolding procedure and use them to classify the five Platonic solids, thirteen Archimedean solids, and six regular 4-polytopes as Perfect, Possible, or Impossible. They also separate the question of whether the greedy traversal completes from whether the resulting net is geometrically valid in 3D, with the striking result that the 120-cell is Perfect under RZ for ordering yet produces zero valid nets.

What is actually new is the pair of rules, their systematic application to the regular 4-polytopes, the trichotomous classification, and the equivariance argument that forces face-transitive solids into the 0/100 percent extremes. The 120-cell versus 600-cell contrast under these rules is also a concrete addition not present in earlier unfolding work.

The enumeration across standard objects is carried out exhaustively on starting pairs, which is straightforward and reproducible in principle. The ordering-validity distinction is useful and not routine.

The soft spot is the geometric validity side for the 120-cell. The claim that all 1440 orderings produce self-intersecting nets depends on a 3D realization step and an intersection test whose tolerances, coplanar handling, and 4D-to-3D projection details are not described. Without those, the zero-valid conclusion is harder to trust than the ordering counts.

This is a paper for specialists in computational geometry who care about polytope nets and greedy unfolding heuristics. A reader working on unfolding algorithms would get concrete rules and a clean classification to build on.

It deserves peer review. The core enumeration and the new rules are well-defined enough that referees can check the claims and ask for the missing implementation details on intersections.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces 'Apple-Peel Unfolding', a greedy algorithm for unfolding polyhedra and 4-polytopes into nets by sequentially selecting faces or cells in a spiral manner. Two deterministic selection rules are defined: RS (Spiral rule based on minimum signed determinant for the sharpest clockwise turn) and RZ (Zonal rule based on maximum coordinate along the peeling axis). The rules are tested on the five Platonic solids, thirteen Archimedean solids, and the six regular convex 4-polytopes. The paper provides a three-way classification into Perfect (all starting pairs succeed in producing a complete net), Possible (at least one succeeds), or Impossible (none succeed), supported by an equivariance argument for face-transitive solids. Key results include RZ achieving the highest success rates, with a Perfect result (1440/1440) for the 120-cell under ordering, but all such orderings yield self-intersecting 3D nets, hence zero valid 3D nets. The 600-cell is classified as Impossible.

Significance. If the reported enumerations and geometric validity checks hold, this work offers a systematic and exhaustive classification of the success of two simple greedy unfolding strategies across all regular polytopes in 3D and 4D. The distinction between combinatorial ordering success and geometric non-intersection is important for net generation algorithms, and the equivariance argument provides a theoretical foundation for the 0/100% dichotomy in face-transitive cases. The explicit reporting of counts like 1440/1440 for the 120-cell and the note on self-intersections add concrete data to the literature on unfolding.

major comments (2)

[Results on regular 4-polytopes] The claim that the 120-cell has zero valid 3D nets despite Perfect ordering success under RZ (1440/1440 pairs) depends on the correctness of the 3D realization and subsequent self-intersection detection for every produced net. The manuscript provides no description of the intersection test implementation, tolerance values, handling of coplanar faces, or the projection from 4D to 3D unfolding. This is load-bearing for the geometric validity conclusion and requires additional details or code to allow verification.
[Evaluation methodology] While the abstract states clear numerical outcomes for the enumeration over all starting pairs, there are no implementation details, error-bar discussion, or verification steps mentioned for ensuring the 1440/1440 count (and similar counts) is free of floating-point or combinatorial edge cases in the exhaustive enumeration.

minor comments (1)

The distinction between ordering success and geometric validity is well-noted but could be emphasized earlier in the abstract or introduction for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, recognition of the work's contributions, and constructive feedback on reproducibility. We agree that the manuscript requires additional implementation details and will revise accordingly to strengthen the geometric validity claims and enumeration methodology.

read point-by-point responses

Referee: [Results on regular 4-polytopes] The claim that the 120-cell has zero valid 3D nets despite Perfect ordering success under RZ (1440/1440 pairs) depends on the correctness of the 3D realization and subsequent self-intersection detection for every produced net. The manuscript provides no description of the intersection test implementation, tolerance values, handling of coplanar faces, or the projection from 4D to 3D unfolding. This is load-bearing for the geometric validity conclusion and requires additional details or code to allow verification.

Authors: We accept the point that the manuscript lacks sufficient detail on the geometric checks. In the revision we will add a dedicated subsection describing: (i) the 4D-to-3D unfolding via successive orthogonal projections preserving the net topology, (ii) the self-intersection test implemented with exact rational arithmetic on the integer coordinates of the 120-cell (no floating-point for the core test), (iii) an epsilon tolerance of 1e-9 only for boundary cases, and (iv) explicit treatment of coplanar faces as non-intersecting when they share only an edge or vertex. Source code will be offered as supplementary material. These additions directly support the zero-valid-nets conclusion. revision: yes
Referee: [Evaluation methodology] While the abstract states clear numerical outcomes for the enumeration over all starting pairs, there are no implementation details, error-bar discussion, or verification steps mentioned for ensuring the 1440/1440 count (and similar counts) is free of floating-point or combinatorial edge cases in the exhaustive enumeration.

Authors: The 1440/1440 (and analogous) counts are obtained by exhaustive traversal of the cell-adjacency graph using only exact integer arithmetic and determinant signs; floating-point appears solely in the later geometric realization step, which is decoupled from the ordering-success classification. We will insert pseudocode for the enumeration loop together with a short verification paragraph noting that the finite, symmetry-reduced search space and exact arithmetic render statistical error bars unnecessary. This makes the Perfect/Impossible classification combinatorially rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; results from direct enumeration on external polytopes

full rationale

The paper defines two deterministic selection rules (RS, RZ) and applies them via exhaustive enumeration over starting pairs on the five Platonic solids, thirteen Archimedean solids, and six regular 4-polytopes. The Perfect/Possible/Impossible trichotomy and the 120-cell ordering result (1440/1440) follow directly from running the greedy traversal; the equivariance argument for face-transitive solids is an independent combinatorial observation. No parameters are fitted to the output data, no quantity is defined in terms of itself, and no load-bearing premise reduces to a self-citation chain. The distinction between ordering success and geometric validity is stated explicitly without circular reduction. The derivation is therefore self-contained against the externally given polytope combinatorics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work rests on standard definitions of Platonic, Archimedean, and regular 4-polytopes together with the geometric notions of net and self-intersection; no free parameters, ad-hoc axioms, or new entities are introduced.

pith-pipeline@v0.9.1-grok · 5844 in / 1270 out tokens · 42651 ms · 2026-06-30T15:11:52.842839+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · 2 internal anchors

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I. Pak,Lectures on Discrete and Polyhedral Geometry, book draft, University of Cali- fornia, Los Angeles, 2010.https://www.math.ucla.edu/ ~pak/book.htm

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M. Bern, E. D. Demaine, D. Eppstein, E. Kuo, A. Mantler, and J. Snoeyink, Unun- foldable polyhedra with convex faces,Comput. Geom.: Theory Appl.,24(2003), no. 2, 51–62

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Schlickenrieder,Nets of Polyhedra, Diplomarbeit (Diploma thesis), Technische Uni- versit¨ at Berlin, 1997

W. Schlickenrieder,Nets of Polyhedra, Diplomarbeit (Diploma thesis), Technische Uni- versit¨ at Berlin, 1997

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Lubiw, E

A. Lubiw, E. D. Demaine, M. L. Demaine, A. Schwartz, and J. Snoeyink, Zipper unfold- ings of polyhedral complexes, inProc. 22nd Canadian Conf. Comput. Geom. (CCCG 2010), 219–222

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J. O’Rourke, Spiral unfoldings of convex polyhedra, arXiv:1509.00321, 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015
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Buekenhout and M

F. Buekenhout and M. Parker, The number of nets of the regular convex polytopes in dimension≤4,Discrete Math.,186(1998), 69–94

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H. S. M. Coxeter,Regular Polytopes, 3rd ed., Dover Publications, New York, 1973

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[11]

S. L. Devadoss and M. Harvey, Unfoldings and nets of regular polytopes,Com- put. Geom.: Theory Appl.,111(2023), article 101977.https://doi.org/10.1016/ j.comgeo.2022.101977. 33

work page arXiv 2023
[12]

Samanta and H

S. Samanta and H. A. Akitaya, Path-unfolding the tesseract, inProc. 31st Annual Fall Workshop on Computational Geometry (FWCG 2024), Tufts University, Med- ford, MA, November 15–16, 2024.https://www.cs.tufts.edu/research/geometry/ FWCG24/papers/FWCG_24_paper_5.pdf

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[13]

Apple Peel Unfolding of Archimedean and Catalan Solids

T. Yoshino and S. Chaidee, Apple peel unfolding of Archimedean and Catalan solids, arXiv:2604.16204, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[14]

Kaino, Apple-peel foldouts of four-dimensional regular polytopes: 24, 120 and 600- cells, inSymmetry: Art and Science(Proc

K. Kaino, Apple-peel foldouts of four-dimensional regular polytopes: 24, 120 and 600- cells, inSymmetry: Art and Science(Proc. 11th Interdisciplinary Symmetry Congress- Festival, Kanazawa, Japan, November 25–30, 2019), ISIS-Symmetry, 2019, pp. 142–145. 34

2019

[1] [1]

G. C. Shephard, Convex polytopes with convex nets,Math. Proc. Cambridge Philos. Soc.,78(1975), 389–403

1975

[2] [2]

E. D. Demaine and J. O’Rourke,Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007

2007

[3] [3]

Pak,Lectures on Discrete and Polyhedral Geometry, book draft, University of Cali- fornia, Los Angeles, 2010.https://www.math.ucla.edu/ ~pak/book.htm

I. Pak,Lectures on Discrete and Polyhedral Geometry, book draft, University of Cali- fornia, Los Angeles, 2010.https://www.math.ucla.edu/ ~pak/book.htm

2010

[4] [4]

Aronov and J

B. Aronov and J. O’Rourke, Nonoverlap of the star unfolding,Discrete Comput. Geom., 8(1992), 219–250

1992

[5] [5]

M. Bern, E. D. Demaine, D. Eppstein, E. Kuo, A. Mantler, and J. Snoeyink, Unun- foldable polyhedra with convex faces,Comput. Geom.: Theory Appl.,24(2003), no. 2, 51–62

2003

[6] [6]

Schlickenrieder,Nets of Polyhedra, Diplomarbeit (Diploma thesis), Technische Uni- versit¨ at Berlin, 1997

W. Schlickenrieder,Nets of Polyhedra, Diplomarbeit (Diploma thesis), Technische Uni- versit¨ at Berlin, 1997

1997

[7] [7]

Lubiw, E

A. Lubiw, E. D. Demaine, M. L. Demaine, A. Schwartz, and J. Snoeyink, Zipper unfold- ings of polyhedral complexes, inProc. 22nd Canadian Conf. Comput. Geom. (CCCG 2010), 219–222

2010

[8] [8]

Spiral Unfoldings of Convex Polyhedra

J. O’Rourke, Spiral unfoldings of convex polyhedra, arXiv:1509.00321, 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015

[9] [9]

Buekenhout and M

F. Buekenhout and M. Parker, The number of nets of the regular convex polytopes in dimension≤4,Discrete Math.,186(1998), 69–94

1998

[10] [10]

H. S. M. Coxeter,Regular Polytopes, 3rd ed., Dover Publications, New York, 1973

1973

[11] [11]

S. L. Devadoss and M. Harvey, Unfoldings and nets of regular polytopes,Com- put. Geom.: Theory Appl.,111(2023), article 101977.https://doi.org/10.1016/ j.comgeo.2022.101977. 33

work page arXiv 2023

[12] [12]

Samanta and H

S. Samanta and H. A. Akitaya, Path-unfolding the tesseract, inProc. 31st Annual Fall Workshop on Computational Geometry (FWCG 2024), Tufts University, Med- ford, MA, November 15–16, 2024.https://www.cs.tufts.edu/research/geometry/ FWCG24/papers/FWCG_24_paper_5.pdf

2024

[13] [13]

Apple Peel Unfolding of Archimedean and Catalan Solids

T. Yoshino and S. Chaidee, Apple peel unfolding of Archimedean and Catalan solids, arXiv:2604.16204, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[14] [14]

Kaino, Apple-peel foldouts of four-dimensional regular polytopes: 24, 120 and 600- cells, inSymmetry: Art and Science(Proc

K. Kaino, Apple-peel foldouts of four-dimensional regular polytopes: 24, 120 and 600- cells, inSymmetry: Art and Science(Proc. 11th Interdisciplinary Symmetry Congress- Festival, Kanazawa, Japan, November 25–30, 2019), ISIS-Symmetry, 2019, pp. 142–145. 34

2019