Spherical harmonics and the icosahedron

TL;DR

This paper introduces a sextic invariant on spherical harmonics that characterizes when the nodal set contains vertices of two icosahedra, linking algebraic invariants with geometric configurations.

Contribution

It defines a new invariant on spherical harmonics and connects it to geometric structures like icosahedra and algebraic surfaces, providing a novel characterization.

Findings

Invariant J is positive iff the nodal set contains two icosahedra vertices.

The proof involves advanced algebraic geometry and vector bundle classification.

Links spherical harmonics with geometric configurations of polyhedra.

Abstract

We define a sextic invariant J on the seven-dimensional space of degree three spherical harmonics and show that J is positive if and only if the nodal set of the spherical harmonic contains the vertices of exactly two regular icosahedra. The proof uses the geometry of the Clebsch diagonal cubic surface, Atiyah's classification of vector bundles on an elliptic curve and a Fano threefold introduced by Mukai.