Medial hexagonal hexecontahedron

In geometry, the medial hexagonal hexecontahedron (or midly dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

Proportions
The faces of the medial hexagonal hexecontahedron are irregular nonconvex hexagons. Denote the golden ratio by $$\phi$$, and let $$\xi\approx -0.377\,438\,833\,12$$ be the real zero of the polynomial $$8x^3-4x^2+1$$. The number $$\xi$$ can be written as $$\xi=-1/(2\rho)$$, where $$\rho$$ is the plastic ratio. Then each face 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 face has two long edges, two of medium length and two short ones. If the medium edges have length $$2$$, the long ones have length $$1+\sqrt{(1-\xi)/(-\phi^{-3}-\xi)}\approx 4.121\,448\,816\,41$$ and the short ones $$1-\sqrt{(1-\xi)/(\phi^{3}-\xi)}\approx 0.453\,587\,559\,98$$. The dihedral angle equals $$\arccos(\xi/(\xi+1))\approx 127.320\,132\,197\,62^{\circ}$$.