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Spherical harmonics and the icosahedron

Authors
  • Hitchin, Nigel
Type
Preprint
Publication Date
Jun 01, 2007
Submission Date
Jun 01, 2007
Identifiers
arXiv ID: 0706.0088
Source
arXiv
License
Unknown
External links

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.

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