# Rhombicosahedron

(Redirected from Ri)
Rhombicosahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymRi
Elements
Faces30 squares, 20 hexagons
Edges60+60
Vertices60
Vertex figureButterfly, edge lengths 2 and 3 Measures (edge length 1)
Circumradius$\frac{\sqrt7}{2} ≈ 1.32288$ Dihedral angles4–6 #1: $\arccos\left(\frac{\sqrt3-\sqrt{15}}{6}\right) ≈ 110.90516^\circ$ 4–6 #2: $\arccos\left(\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 20.90516^\circ$ Central densityodd
Number of external pieces630
Level of complexity36
Related polytopes
ArmySemi-uniform ti, edge lengths $\frac{\sqrt5-1}{2}$ (pentagons), $\frac{3-\sqrt5}{2}$ (between ditrigons)
DualRhombicosacron
ConjugateRhombicosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count480
Euler characteristic–10
OrientableNo
Genus12
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The rhombicosahedron, or ri, is a uniform polyhedron. It consists of 30 squares and 20 hexagons. Two squares and two hexagons meet at each vertex.

It is a faceting of the rhombidodecadodecahedron, using its 30 squares along with the 20 hexagons of the icosidodecadodecahedron.

The truncated great icosidodecahedron (x5/2x3x) is a degenerate polyhedron with 20 hexagons, 30 squares, and 12 doubly-wound pentagons. If those pentagrams are blended out, the result is the rhombicosahedron.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the rhombidodecadodecahedron.