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 $$\frac{1}{2}-\frac{\sqrt{(1-\xi)/(\phi^{3}-\xi)}}{2} ≈ 0.22679$$ and the long edge length is $$\frac{1}{2}+\frac{\sqrt{(1-\xi)/(-\phi^{-3}-\xi)}}{2}\approx 2.06072$$, where $$\xi ≈ -0.37744$$ is the the only real root of the polynomial $$8x^3-4x^2+1$$, and $$\phi$$ is the golden ratio.

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

Each hexagon has four equal angles of $$\arccos\left(\xi\right) ≈ 112.17513°$$, one of $$\arccos\left(\phi^2\xi+\phi\right) ≈ 50.95827°$$, and one of $$360°-\arccos\left(\phi^{-2}\xi-\phi^{-1}\right) ≈ 220.34122°$$.

A dihedral angle is equal to $$\arccos\left(\frac{\xi}{\xi+1}\right) ≈ 127.32013°$$.

The inradius R ≈ 0.90505 of a medial hexagonal hexecontahedron with unit edge length is equal to $$\frac{\sqrt{66\left(78+\sqrt[3]{12\left(27765+539\sqrt{69}\right)}+\sqrt[3]{12\left(27765-539\sqrt{69}\right)}\right)}}{132}$$.