The Silent Golden Station: The Harmonic Fingerprint of the Cube–Octahedron Sweep Finds Icosaxes, and Only Icosaxes

A solid can be listened to: its corners, seen from the centre, are notes on a sphere, and each degree of spherical harmonic is a tone. Cubic solids ring in the fourth tone, icosahedral ones never do. Sliding from the octahedron to the cube, the fourth tone falls silent once, at exactly the golden station, Icosaxes, because its corners are two icosahedra. The sound finds the golden point with no geometry put in. Abstract. Give every corner of a solid seen from its centre a unit weight on the sphere and expand in spherical harmonics: the strength Ql of each degree l is the solid's harmonic fingerprint, the bond-orientational order of Steinhardt, Nelson and Ronchetti [1], which physicists use to tell the cubic, hexagonal and icosahedral order of atoms apart. We apply it to #90's sweep Pμ = C ∩ μC* of the cube and its polar octahedron, from the octahedron (μ = 1) to the cube (μ = 3). Main theorem. (A) Q4 vanishes at exactly one point of the sweep, μ = φ, Icosaxes; there the corner condition is a⁴ − 3a² + 1 = (a² + a − 1)(a² − a − 1) = 0 with a = μ − 1. (B) At Icosaxes Ql vanishes exactly for l = 2, 4, 8, 14, the even degrees with no icosahedral harmonic: its 24 corners are the two icosahedra of the pyritohedral pair (#90), so the one cubic solid on the sweep sounds icosahedral. (C) Q8 vanishes at exactly two points, μ = φ and μ = (√13 − 1)/2, the reciprocals of the golden and the bronze ratio. (D) On the second dual sweep, the cuboctahedron cut by the rhombic dodecahedron, Q4 vanishes once, at a point with no golden number, 1 + √((6√30 − 31)/17). The proofs are short polynomial identities; every value is checked numerically against the classical fingerprints of the cubic, hexagonal and body-centred packings and the icosahedron.