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arxiv: 2603.00017 · v2 · submitted 2026-02-02 · 🧮 math.GM

A Ten-Face Non-Edge-Sharing Wing Set on the Regular Icosahedron and a Decagonal Equatorial Balance

YoungJune Jeon This is my paper

Pith reviewed 2026-05-16 08:31 UTC · model grok-4.3

classification 🧮 math.GM
keywords icosahedrongolden ratiodecagonwing setpolyhedral geometrynon-edge-sharingequatorial balance
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The pith

Ten triangular wings on the icosahedron form a regular decagon whose radius is exactly half the golden ratio times the edge length.

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

The paper formalizes a construction for ten isosceles triangular faces placed on the vertices of a regular icosahedron. Each face has angles 36-36-108 degrees with the 36-degree angles fixed at the north or south pole, and the faces share vertices but no edges. The arrangement is defined around a north-south rotation axis using the vertex labels N, S, U1-U5 and L1-L5. This layout produces a perfectly regular decagon in the equatorial cross-section. The radius of the decagon is derived in closed form as R equals phi over 2 times the icosahedron edge length, where phi is the golden ratio.

Core claim

The ten-face non-edge-sharing wing set on the regular icosahedron yields a balanced regular decagon in the equatorial plane, with each face an isosceles 36-36-108 triangle anchored at a pole and the decagon radius given exactly by R = (phi/2) * ell.

What carries the argument

The vertex-labeled ten-face wing set with poles N and S, upper vertices U1-U5, lower vertices L1-L5, rotation axis NS, and the no-edge-sharing condition on the 36-36-108 triangles that enforces the regular decagon equator.

If this is right

  • The construction supplies a symmetry-consistent design principle for pole-anchored wing layouts on the icosahedron.
  • The closed-form radius permits exact geometric calculations without approximation.
  • A reproducible workflow is given for building such balanced wing sets.

Where Pith is reading between the lines

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

  • The exact radius formula may simplify computations in other icosahedral models that involve the golden ratio.
  • Analogous non-edge-sharing wing sets could be tested on other Platonic solids to check for regular equatorial polygons.
  • The no-edge-sharing property suggests modular surface designs where faces act as independent elements.

Load-bearing premise

The assumption that the specified vertex labeling N, S, U1-U5, L1-L5 and NS rotation axis allow ten 36-36-108 triangles to be realized without sharing edges while producing a perfectly regular decagon.

What would settle it

Construct the wing set on a physical or coordinate model of the regular icosahedron using the given vertex labels and check whether the equatorial vertices form a regular decagon with all radii exactly equal to (phi/2) times the edge length.

read the original abstract

We formalize a ten-face triangular wing set on a regular icosahedron under a vertex labeling N, S, U1-U5, L1-L5 with rotation axis NS. The wing faces satisfy: (i) each face is an isosceles 36-36-108 triangle with a 36-degree angle anchored at a pole (N or S); (ii) distinct faces may share vertices but share no edges; and (iii) a natural equatorial cross-section yields a perfectly balanced regular decagon. We derive a closed form for the decagon radius, R = (phi/2)*ell, where ell is the icosahedron edge length and phi is the golden ratio phi = (1 + sqrt(5))/2. Beyond the geometric results, we interpret the ten-face closure as a symmetry-consistent design principle for a pole-anchored wing layout and provide a reproducible construction workflow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript formalizes a ten-face triangular wing set on the regular icosahedron using vertex labels N, S, U1–U5, L1–L5 with NS as rotation axis. The faces are claimed to be isosceles 36-36-108 triangles that share vertices but no edges, inducing a regular equatorial decagon whose radius is derived in closed form as R = (φ/2)ℓ with φ the golden ratio and ℓ the edge length. A reproducible construction workflow and symmetry-based design interpretation are also provided.

Significance. A valid construction would supply a parameter-free closed-form radius together with an explicit, reproducible geometric workflow. The claimed result is therefore potentially useful for icosahedral symmetry studies if the set exists. However, the central geometric object is shown to be impossible by the structure of the face-adjacency graph, limiting the manuscript’s contribution to an illustration of an inconsistent selection criterion.

major comments (1)
  1. [Abstract and construction] The no-edge-sharing condition requires the selected 10 faces to form an independent set in the face-adjacency graph G (the dodecahedral graph on 20 vertices). Each selected face contributes three edges, all of which must connect to the complement; with |E(G)| = 30 this forces every edge to be a cut edge, so the complement must also be independent and G must be bipartite with equal parts of size 10. Yet G contains 5-cycles (the five faces incident to any icosahedral vertex), so G is not bipartite. This contradiction, which is load-bearing for the existence of the wing set and the subsequent radius derivation, appears in the construction described in the abstract and the full text.
minor comments (2)
  1. The vertex labeling N, S, U1–U5, L1–L5 is introduced without an accompanying diagram or coordinate table; a single figure showing the placement relative to the icosahedron would improve readability.
  2. The phrase “perfectly balanced regular decagon” is used without a precise definition of balance (e.g., centroid coincidence or moment equilibrium); a short clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for identifying the fundamental inconsistency in the proposed construction. The analysis of the face-adjacency graph correctly demonstrates that no such ten-face non-edge-sharing set can exist on the regular icosahedron. We accept this point and will revise the manuscript to remove the invalid claims and derivations.

read point-by-point responses

  1. Referee: [Abstract and construction] The no-edge-sharing condition requires the selected 10 faces to form an independent set in the face-adjacency graph G (the dodecahedral graph on 20 vertices). Each selected face contributes three edges, all of which must connect to the complement; with |E(G)| = 30 this forces every edge to be a cut edge, so the complement must also be independent and G must be bipartite with equal parts of size 10. Yet G contains 5-cycles (the five faces incident to any icosahedral vertex), so G is not bipartite. This contradiction, which is load-bearing for the existence of the wing set and the subsequent radius derivation, appears in the construction described in the abstract and the full text.

    Authors: We fully agree with this assessment. The referee's argument is correct: the requirement that the ten selected faces and their complement both be independent sets in the face-adjacency graph would imply that the graph is bipartite, which it is not due to the odd-length cycles corresponding to the five faces around each vertex. This means the described wing set cannot exist, and any derived properties such as the equatorial decagon radius are invalid. In the revised version of the manuscript, we will withdraw the construction, the radius formula, and related claims, and instead focus on valid subsets or alternative interpretations that do not violate the adjacency constraints. revision: yes

Circularity Check

0 steps flagged

No circularity: radius formula follows from external icosahedral geometry

full rationale

The claimed derivation of the decagon radius R = (phi/2)*ell relies on the standard embedding of the regular icosahedron in coordinates involving the golden ratio phi, which is an independent mathematical fact predating the paper. The vertex labeling N, S, U1-U5, L1-L5 and the no-edge-sharing condition are used to set up the geometry, but the algebraic steps that produce the closed form do not reduce the output to a redefinition or fit of the input parameters themselves. The existence of the ten-face set is a separate geometric claim whose validity can be checked externally; it does not render the radius derivation circular by construction. No self-citation chains, ansatz smuggling, or renaming of known results appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The construction rests on standard properties of the regular icosahedron and the golden ratio; the wing set itself is the principal new element introduced without external falsifiable evidence.

axioms (1)
  • standard math Regular icosahedron has 12 vertices, 20 faces, and edge length ell; its coordinates involve the golden ratio phi.
    Invoked implicitly when placing the labeled vertices N, S, U1-U5, L1-L5.
invented entities (1)
  • Ten-face triangular wing set no independent evidence
    purpose: Non-edge-sharing collection of 36-36-108 triangles anchored at the poles that produces a regular equatorial decagon.
    New design object defined by the three listed conditions; no independent evidence supplied beyond the construction.

pith-pipeline@v0.9.0 · 5457 in / 1352 out tokens · 60237 ms · 2026-05-16T08:31:30.445764+00:00 · methodology

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