Medial pentagonal hexecontahedron

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

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 $\phi$, and let $\xi \approx -0.409\,037\,788\,014\,42$ be the smallest (most negative) real zero of the polynomial $P=8x^{4}-12x^{3}+5x+1$. Then each face has three equal angles of $\arccos(\xi )\approx 114.144\,404\,470\,43^{\circ }$, one of $\arccos(\phi ^{2}\xi +\phi )\approx 56.827\,663\,280\,94^{\circ }$ and one of $\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 140.739\,123\,307\,76^{\circ }$. Each face has one medium length edge, two short and two long ones. If the medium length is $2$, then the short edges have length

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

and the long edges have length

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

The dihedral angle equals $\arccos(\xi /(\xi +1))\approx 133.800\,984\,233\,53^{\circ }$. 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