Bounding the density of spherical polygon packings

TL;DR

This paper develops a mathematical framework to determine optimal packings of regular spherical polygons, extending classical bounds using harmonic analysis and semidefinite programming.

Contribution

It introduces an algebraic criterion for polygon disjointness and extends the Lovász theta number to spherical packings on SO(3).

Findings

Derived bounds for spherical polygon packings.

Extended Lovász theta number to SO(3) Cayley graphs.

Reduced packing constraints to a sum-of-squares problem.

Abstract

We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group ${\rm SO}(3)$. To this end, we introduce an algebraic criterion characterizing when congruent regular spherical polygons have disjoint interiors, leading to a unified formulation of the packing constraints. Using harmonic analysis on ${\rm SO}(3)$, we reduce the theta number to a trigonometric sum-of-squares problem, which can be solved via semidefinite programming.