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May 8, 2026Open Access

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

Key Points

The aim is to formalize a unique ten-face triangular wing arrangement on a regular icosahedron and demonstrate geometric balance.
Formal geometric proofs were conducted to analyze the balance of the arrangement.
Closed-form radius calculation based on the golden ratio and edge length was established.
A reproducible geometric construction workflow was developed for CAD implementation.
The pole-opposite edge midpoints successfully form a perfectly balanced regular decagon.
Radius calculated as R = (phi/2)l, effectively linking geometry and the golden ratio.
The construction workflow can be used for practical applications in CAD and parametric geometry.

Abstract

This preprint formalizes a ten-face non-edge-sharing triangular wing arrangement on a regular icosahedron under a pole-anchored geometric framework. The work proves that the pole-opposite edge midpoints form a perfectly balanced regular decagon on the equatorial plane with closed-form radius R = (phi/2)l, where l is the icosahedron edge length and phi is the golden ratio. The paper establishes a reproducible geometric construction workflow suitable for CAD and parametric geometry implementations. Submitted to Discrete & Computational Geometry.

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Young June Jeon

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

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