Rhombicosahedron

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Rhombicosahedron
Rank3
TypeUniform
Notation
Bowers style acronymRi
Elements
Faces30 squares, 20 hexagons
Edges60+60
Vertices60
Vertex figureButterfly, edge lengths 2 and 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7}}{2}}\approx 1.32288}$
Dihedral angles4–6 #1: ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {15}}}{6}}\right)\approx 110.90516^{\circ }}$
4–6 #2: ${\displaystyle \arccos \left({\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }}$
Central densityodd
Number of external pieces630
Level of complexity36
Related polytopes
ArmySemi-uniform ti, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (pentagons), ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (between ditrigons)
RegimentRaded
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.