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 ${\displaystyle \phi }$, 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 ${\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 56.827\,663\,280\,94^{\circ }}$ and one of ${\displaystyle \arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 140.739\,123\,307\,76^{\circ }}$. Each face has one medium length edge, two short and two long ones. If the medium length is ${\displaystyle 2}$, then the short edges have length

${\displaystyle 1+{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 1.550\,761\,427\,20}$,

and the long edges have length

${\displaystyle 1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 3.854\,145\,870\,08}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 133.800\,984\,233\,53^{\circ }}$. The other real zero of the polynomial ${\displaystyle P}$ 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