Here’s an identity I found. For arbitrary *r*, define,

and,

then,

Those two complicated expressions neatly wrap up into that last equation, doesn’t it? This is the *Brioschi quintic *form which the general quintic can be reduced into. Two of the polynomials are easily recognizable as icosahedral invariants, while,

is a polynomial invariant for the octahedron. This gave rise to the question *here*.

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