Great pentagrammic hexecontahedron explained

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 ≈ 0 . 94673003356 be the largest positive zero of the polynomial P = 8 x ^{3-8x} ^{2+\phi} ^{-2} . Then each pentagrammic face has four equal angles of \arccos(\xi) ≈ 18 . 78563395824 ^{\circ} and one angle of \arccos( - \phi ^{-1} + \phi ^{-2} \xi) ≈ 104 . 85746416703 ^{\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 byl = (2-4 \xi ^{2})/(1-2 \xi) ≈ 1 . 77421586494 . The dihedral angle equals \arccos(\xi/(\xi +1 )) ≈ 60 . 90113371321 ^{\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