Great pentagrammic hexecontahedron

Great pentagrammic hexecontahedron

Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 92 (χ = 2)
Symmetry group	I, [5,3]⁺, 532
Index references	DU₇₄
dual polyhedron	Great retrosnub icosidodecahedron

In geometry, the great pentagrammic hexecontahedron (or great dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its 60 faces are irregular pentagrams.

Proportions

Denote the golden ratio by $\phi$ . Let $\xi \approx 0.946\,730\,033\,56$ be the largest positive zero of the polynomial $P=8x^{3}-8x^{2}+\phi ^{-2}$ . Then each pentagrammic face has four equal angles of $\arccos(\xi )\approx 18.785\,633\,958\,24^{\circ }$ and one angle of $\arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 104.857\,464\,167\,03^{\circ }$ . Each face has three long and two short edges. The ratio $l$ between the lengths of the long and the short edges is given by

l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.774\,215\,864\,94

.

The dihedral angle equals $\arccos(\xi /(\xi +1))\approx 60.901\,133\,713\,21^{\circ }$ . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial $P$ play a similar role in the description of the great pentagonal hexecontahedron and the great inverted pentagonal hexecontahedron.