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

TL;DR

This paper formalizes a unique ten-face wing set on a regular icosahedron with specific geometric and symmetry properties, providing a closed-form expression for the decagon radius and a reproducible design workflow.

Contribution

It introduces a novel geometric configuration on the icosahedron, deriving a closed-form for the decagon radius and establishing a symmetry-based design principle.

Findings

Derived a closed-form expression for the decagon radius R = (phi/2)*ell.

Defined a ten-face wing set with specific isosceles triangular faces.

Provided a reproducible construction workflow for the geometric configuration.

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.