Medial pentagonal hexecontahedron

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.

Medial pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]+, 532
Index references	DU40
dual polyhedron	Snub dodecadodecahedron

Proportions

Denote the golden ratio by φ, and let ${\displaystyle \xi \approx -0.409\,037\,788\,014\,42}$ be the smallest (most negative) real zero of the polynomial ${\displaystyle P=8x^{4}-12x^{3}+5x+1.}$ Then each face has three equal angles of ${\displaystyle \arccos(\xi )\approx 114.144\,404\,470\,43^{\circ },}$ one of ${\displaystyle \arccos(\varphi ^{2}\xi +\varphi )\approx 56.827\,663\,280\,94^{\circ }}$ and one of ${\displaystyle \arccos(\varphi ^{-2}\xi -\varphi ^{-1})\approx 140.739\,123\,307\,76^{\circ }.}$ 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 $${\displaystyle 1+{\sqrt {\frac {1-\xi }{\varphi ^{3}-\xi }}}\approx 1.550\,761\,427\,20,}$$ and the long edges have length $${\displaystyle 1+{\sqrt {\frac {1-\xi }{-\varphi ^{-3}-\xi }}}\approx 3.854\,145\,870\,08.}$$ The dihedral angle equals ${\displaystyle \arccos \left({\tfrac {\xi }{\xi +1}}\right)\approx 133.800\,984\,233\,53^{\circ }.}$ 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