Medial hexagonal hexecontahedron

The medial hexagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 asymmetric nonconvex hexagons.

It is the dual of the Snub icosidodecadodecahedron.

Each hexagon has two long edges, two of medium length and two short ones.

If the medium edges have unit length the short edge length is $$\frac{1}{2}-\frac{\sqrt{(1-\xi)/(\phi^{3}-\xi)}}{2}\approx 0.226\,793\,780$$ and the long edge length is $$\frac{1}{2}+\frac{\sqrt{(1-\xi)/(-\phi^{-3}-\xi)}}{2}\approx 2.060\,724\,408$$.

Each hexagon has four equal angles of $$\arccos(\xi)\approx 112.175\,128\,045\,27^{\circ}$$, one of $$\arccos(\phi^2\xi+\phi)\approx 50.958\,265\,917\,31^{\circ}$$ and one of $$360^{\circ}-\arccos(\phi^{-2}\xi-\phi^{-1})\approx 220.341\,221\,901\,59^{\circ}$$.

Each dihedral angle is equal to $$\arccos(\xi/(\xi+1))\approx 127.320\,132\,197\,62^{\circ}$$,

where $$\phi$$ is the golden ratio,

$$\xi\approx -0.377\,438\,833\,12$$ is the the only real root of the polynomial $$8x^3-4x^2+1$$.

$$\xi$$ can also be written as $$-\frac{1}{2\rho}$$, where ρ is the plastic number.