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The Dodecahedron

This section needn’t be excessively long, because the dodecahedron is dual to the icosahedron upon exchanging edge midpoints and vertices. As such, the polynomial invariant for the vertices of a dodecahedron is

H(u,v)=u20−228(u15v5−u5v15)+494 u10v10+v20 ,H(u,v) =u^{20} - 228\left(u^{15}v^5 - u^5v^{15}\right)+494\,u^{10}v^{10}+v^{20} \,,
(1)

as before. Similary, the polynomial invariants for the dodecahedron’s 12 face centers and 30 edge midpoints are f(u,v)f(u, v) and T(u,v)T(u,v), respectively.

Introduction
Icosahedral Symmetry
Introduction
Solving Quintics