# Rhombic hexecontahedron

Rhombic hexecontahedron Rank3
SpaceSpherical
Elements
Faces60 golden rhombi
Edges60 + 60
Vertices12 + 20 + 30
Measures (edge length 1)
Volume$10+2\sqrt{5}$ Surface area$24\sqrt{5}$ Central density1
Related polytopes
Convex hullDodecahedron
Abstract & topological properties
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo

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

## Construction

### Golden rhombohedra

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.