Great hexagonal hexecontahedron

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

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.

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.