Great hexagonal hexecontahedron

In geometry, the great hexagonal hexecontahedron (or great astroid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great snub dodecicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar pentagrammic faces.

Great hexagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 180 V = 104 (χ = −16)
Symmetry group	I, [5,3]+, 532
Index references	DU64
dual polyhedron	Great snub dodecicosidodecahedron

3D model of a great hexagonal hexecontahedron

Proportions

The faces are nonconvex hexagons. Denoting the golden ratio by ${\displaystyle \phi }$, the hexagons have one angle of ${\displaystyle \arccos(-\phi ^{-1})\approx 128.172\,707\,627\,01^{\circ }}$, one of ${\displaystyle 360^{\circ }-\arccos(-\phi ^{-1})\approx 231.827\,292\,372\,99^{\circ }}$, and four angles of ${\displaystyle 90^{\circ }}$. They have two long edges, two of medium length and two short ones. If the long edges have length ${\displaystyle 2}$, the medium ones have length ${\displaystyle 1+\phi ^{-3/2}\approx 1.485\,868\,271\,76}$ and the short ones ${\displaystyle 1-\phi ^{-3/2}\approx 0.514\,131\,728\,24}$. The dihedral angle equals ${\displaystyle 90^{\circ }}$.

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

External links

• Weisstein, Eric W. "Great hexagonal hexecontahedron". MathWorld.