Medial pentagonal hexecontahedron explained
In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
Proportions
Denote the golden ratio by, and let
be the smallest (most negative) real zero of the polynomial
Then each face has three equal angles of
\arccos(\xi) ≈ 114.14440447043\circ,
one of
\arccos(\varphi2\xi+\varphi) ≈ 56.82766328094\circ
and one of
\arccos(\varphi-2\xi-\varphi-1) ≈ 140.73912330776\circ.
Each face has one medium length edge, two short and two long ones. If the medium length is 2, then the short edges have length
and the long edges have length
The
dihedral angle equals
\arccos\left(\tfrac{\xi}{\xi+1}\right) ≈ 133.80098423353\circ.
The other real zero of the polynomial plays a similar role for the medial inverted pentagonal hexecontahedron.
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