On the icosahedron: from two to three dimensions

TL;DR

This paper explores the geometric and algebraic properties of the icosahedron, extending Klein's work on sphere quotients to three dimensions using group theory and representations of SU(2).

Contribution

It provides explicit coordinate expressions and relations for the quotient of the three-dimensional sphere by the icosahedral symmetry group, advancing understanding of 3D spherical quotients.

Findings

Explicit coordinate formulas for 3D sphere quotients

Relations satisfied by these coordinates

Extension of Klein's complex variable approach to three dimensions

Abstract

In his famous book, Felix Klein describes a complex variable for the quotients of the ordinary sphere by the finite groups of rotations and in particular for the most complex situation of the quotient by the symmetry group of the icosahedron. The purpose of this work and its sequels is to obtain similar results for the quotients of the three--dimensional sphere. Various properties of the group $SU(2)$ and of its representations are used to obtain explicit expressions for coordinates and the relations they satisfy.