Invariant polynomials

We can construct Γ-invariant homogeneous polynomials on ℂ²:

• V, of degree 12, vanishing on the 1d subspaces corresponding to icosahedron vertices.

• E, of degree 30, vanishing on the 1d subspaces corresponding to icosahedron edge midpoints.

• F, of degree 20, vanishing on the 1d subspaces corresponding to icosahedron face centers.

Three polynomials on a 2d space must obey a relation, and they do:

V⁵ + E² + F³ = 0

after we normalize them suitably.

A point in ℂP¹ is a 1d subspace of ℂ², so each icosahedron vertex defines a 1d subspace in ℂ², and there's a linear function on ℂ², unique up to a constant factor, that vanishes on this 1d subspace. The icosahedron has 12 vertices, so we get 12 linear functions this way. Multiplying them gives V, a homogeneous polynomial of degree 12 that vanishes on all the 1d subspaces corresponding to icosahedron vertices!

The same trick gives E, which has degree 30 because the icosahedron has 30 edges, and F, which has degree 20 because the icosahedron has 20 faces.

For details, again see Jerry Shurman's Geometry of the Quintic and Oliver Nash's On Klein's icosahedral solution of the quintic.