Medial pentagonal hexecontahedron

The medial pentagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 irregular pentagons, each with two short, one medium, and two long edges. Its dual is the snub dodecadodecahedron.

If the pentagon faces have medium edge length 2, then the short edge length is $$1+\sqrt{\frac{1-\xi}{\phi^3-\xi}} ≈ 1.55076$$, and the long edge length is $$1+\sqrt{\frac{1-\xi}{-\phi^3-\xi}} ≈ 3.85415$$, where $$\xi ≈ -0.40903$$ is the smallest (most negative) real root of the polynomial $$8x^4-12x^3+5x+1$$. ​The pentagons have three interior angles of $$\arccos\left(\xi\right) ≈ 114.14440°$$, one of $$\arccos\left(\phi^2\xi+\phi\right) ≈ 56.82766°$$, and one of $$\arccos\left(\phi^{-2}\xi-\phi^{-1}\right) ≈ 140.73912°$$, where $$\phi$$ is the golden ratio.

The inradius R ≈ 1.07828 of the medial pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of $$8192x^4-12352x^3+3376x^2-104x+1$$.

A dihedral angle can be given as acos(α), where α ≈ -0.69216 is the negative of a real root of $$16x^4+x^3-9x^2-x+1$$.