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 φ, and let ξ ≈ − 0.409 037 788 014 42 {\displaystyle \xi \approx -0.409\,037\,788\,014\,42} be the smallest (most negative) real zero of the polynomial P = 8 x 4 − 12 x 3 + 5 x + 1. {\displaystyle P=8x^{4}-12x^{3}+5x+1.} Then each face has three equal angles of arccos ( ξ ) ≈ 114.144 404 470 43 ∘ , {\displaystyle \arccos(\xi )\approx 114.144\,404\,470\,43^{\circ },} one of arccos ( φ 2 ξ + φ ) ≈ 56.827 663 280 94 ∘ {\displaystyle \arccos(\varphi ^{2}\xi +\varphi )\approx 56.827\,663\,280\,94^{\circ }} and one of arccos ( φ − 2 ξ − φ − 1 ) ≈ 140.739 123 307 76 ∘ . {\displaystyle \arccos(\varphi ^{-2}\xi -\varphi ^{-1})\approx 140.739\,123\,307\,76^{\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
1 + 1 − ξ φ 3 − ξ ≈ 1.550 761 427 20 , {\displaystyle 1+{\sqrt {\frac {1-\xi }{\varphi ^{3}-\xi }}}\approx 1.550\,761\,427\,20,}
and the long edges have length
1 + 1 − ξ − φ − 3 − ξ ≈ 3.854 145 870 08. {\displaystyle 1+{\sqrt {\frac {1-\xi }{-\varphi ^{-3}-\xi }}}\approx 3.854\,145\,870\,08.}
The dihedral angle equals arccos ( ξ ξ + 1 ) ≈ 133.800 984 233 53 ∘ . {\displaystyle \arccos \left({\tfrac {\xi }{\xi +1}}\right)\approx 133.800\,984\,233\,53^{\circ }.} The other real zero of the polynomial P plays a similar role for the medial inverted pentagonal hexecontahedron.
References Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld. Uniform polyhedra and duals
