Rhombicosahedron

Rhombicosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Ri
Elements
Faces	30 squares, 20 hexagons
Edges	60+60
Vertices	60
Vertex figure	Butterfly, edge lengths √2 and √3
Measures (edge length 1)
Circumradius	${\frac {\sqrt {7}}{2}}\approx 1.32288$
Dihedral angles	4–6 #1: $\arccos \left({\frac {{\sqrt {3}}-{\sqrt {15}}}{6}}\right)\approx 110.90516^{\circ }$
	4–6 #2: $\arccos \left({\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }$
Central density	odd
Number of external pieces	630
Level of complexity	36
Related polytopes
Army	Semi-uniform ti, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (pentagons), ${\frac {3-{\sqrt {5}}}{2}}$ (between ditrigons)
Regiment	Raded
Dual	Rhombicosacron
Conjugate	Rhombicosahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–10
Orientable	No
Genus	12
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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.

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#44).

• Klitzing, Richard. "ri".
• Wikipedia contributors. "Rhombicosahedron".
• McCooey, David. "Rhombicosahedron"