Medial hexagonal hexecontahedron

Medial hexagonal hexecontahedron

Type	Star polyhedron
Face
Elements	F = 60, E = 180 V = 104 (χ = −16)
Symmetry group	I, [5,3]⁺, 532
Index references	DU₄₆
dual polyhedron	Snub icosidodecadodecahedron

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 number. 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 $\arccos(\xi /(\xi +1))\approx 127.320\,132\,197\,62^{\circ }$ .