Rank3
TypeUniform
Notation
Coxeter diagramx5/2o5x ()
Elements
Faces30 squares, 12 pentagons, 12 pentagrams
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge lengths (5–1)/2, 2, (1+5)/2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7}}{2}}\approx 1.32288}$
Volume${\displaystyle 19{\sqrt {5}}\approx 42.48529}$
Dihedral angles4–5/2: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
Central density3
Number of external pieces288
Level of complexity19
Related polytopes
ArmySemi-uniform Ti, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (pentagons) and ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (between ditrigons)
DualMedial deltoidal hexecontahedron
Convex coreChamfered dodecahedron
Abstract & topological properties
Flag count480
Euler characteristic-6
OrientableYes
Genus4
Properties
SymmetryH3, order 120
Flag orbits4
ConvexNo
NatureTame

The rhombidodecadodecahedron, or raded, is a uniform polyhedron. It consists of 30 squares, 12 pentagons, and 12 pentagrams. One pentagon, one pentagram, and two squares join at each vertex. It can be obtained by cantellation of the small stellated dodecahedron or great dodecahedron, or equivalently by expanding either polyhedron's faces outward and filling in the gaps with appropriate faces.

## Vertex coordinates

A rhombidodecadodecahedron of edge length 1 has vertex coordinates given by all permutations of

• ${\displaystyle \left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$

along with all even permutations of

• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm 1,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right).}$

## Related polyhedra

The rhombidodecadodecahedron is the colonel of a three-member regiment that also includes the icosidodecadodecahedron and the rhombicosahedron.

Oddly, it has the same circumradius as the cuboctatruncated cuboctahedron.