Convex Polyhedra in the $3$-Sphere and Tilings of the $2$-Sphere

TL;DR

This paper demonstrates a correspondence between convex polyhedral spheres in the 3-sphere and specific tilings of the 2-sphere, using Lie group structures and normal vectors.

Contribution

It introduces a novel method linking convex polyhedra in $S^3$ with tilings of $S^2$ via Lie group representations and normal vectors.

Findings

Existence of two canonical tilings of $S^2$ from convex polyhedral spheres in $S^3$

Use of Lie group $SU(2)$ and Maurer-Cartan forms in the construction

Connection between polyhedral faces and tiling structures

Abstract

We show that for every convex polyhedral sphere $P$ in $S^3$, there exist two canonical, non-edge-to-edge tilings of $S^{2}$ whose tiles are given by all the faces of $P$ and the dual convex polyhedral sphere $P^*$ to $P$. Under the identifications of $S^{3}$ with the Lie group $SU(2)$, and of $S^{2}$ with the unit sphere in the Lie algebra $su(2)$ of $SU(2)$, our result is obtained by considering the set $\widetilde P$ of outward unit normal vectors to $P$ and the maps from $\widetilde P$ to $S^{2}$ defined by using the left and right Maurer-Cartan forms on $SU(2)$.