Medial pentagonal hexecontahedron

In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

Proportions
Denote the golden ratio by $&phi;$, and let $$\xi\approx -0.409\,037\,788\,014\,42$$ be the smallest (most negative) real zero of the polynomial $$P=8x^4-12x^3+5x+1.$$ Then each face has three equal angles of $$\arccos(\xi)\approx 114.144\,404\,470\,43^{\circ},$$ one of $$\arccos(\varphi^2\xi+\varphi)\approx 56.827\,663\,280\,94^{\circ}$$ and one of $$\arccos(\varphi^{-2}\xi-\varphi^{-1})\approx 140.739\,123\,307\,76^{\circ}.$$ Each face has one medium length edge, two short and two long ones. If the medium length is 2, then the short edges have length $$1 + \sqrt{\frac{1-\xi}{\varphi^3-\xi}} \approx 1.550\,761\,427\,20,$$ and the long edges have length $$1 + \sqrt{ \frac{1-\xi}{-\varphi^{-3}-\xi}}\approx 3.854\,145\,870\,08.$$ The dihedral angle equals $$\arccos\left(\tfrac{\xi}{\xi+1}\right) \approx 133.800\,984\,233\,53^{\circ}.$$ The other real zero of the polynomial $P$ plays a similar role for the medial inverted pentagonal hexecontahedron.