Rhombic hexecontahedron

Rhombic hexecontahedron
(3D model)
Rank	3
Space	Spherical
Elements
Faces	60 golden rhombi
Edges	60 + 60
Vertices	12 + 20 + 30
Measures (edge length 1)
Volume	$10+2\sqrt{5}$
Surface area	$24\sqrt{5}$
Central density	1
Related polytopes
Convex hull	Dodecahedron
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No

The rhombic hexecontahedron is a polyhedron with 60 golden rhombus faces.

Construction

Golden rhombohedra

An acute golden rhombohedron.

The rhombic hexecontahedron can be constructed by joining 20 acute golden rhombohedra around a single point. This gives an easy way to determine the volume of rhombic hexecontahedron given the volume of the acute golden rhombohedron. The volume of a single acute golden rhombohedron of edge length 1 is $\frac{5+\sqrt{5}}{10}$ thus the volume of the rhombic hexecontahedron of edge length 1 is 20 times that making it $10+2\sqrt{5}$. The surface area can also be calculated by similar means.

Stellation

The rhombic hexecontahedron is a stellation of the rhombic triacontahedron.

Its stellation is given by the following stellation diagram:

Yellow represents the original face, red represents the new faces, green represents space internal to the new polyhedron and dark blue represents space external to the new polyhedron. Since the faces form coplanar pairs two faces appear in the stellation diagram.

Related polytopes

The rhombic hexecontahedron is topologically equivalent to the deltoidal hexecontahedron which has kites for faces instead of rhombi.

Grünbaum's new rhombic hexecontahedron is another stellation of the rhombic triacontahedron with 60 golden rhombus faces.

External links

• Wikipedia Contributors. "Rhombic hexecontahedron".
• Weisstein, Eric W. "Rhombic Hexecontahedron" at MathWorld.
• McCooey, David. "Rhombic Hexecontahedron"

Bibliography

• Grünbaum, Branko (1996a), A New Rhombic Hexecontahedron-Once More (PDF)
• Grünbaum, Branko (1996b), A New Rhombic Hexecontahedron (PDF)