 Rank3
TypeUniform
SpaceSpherical
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$\frac{\sqrt7}{2} ≈ 1.32288$ Volume$19\sqrt5 ≈ 42.48529$ Dihedral angles4–5/2: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$ 4–5: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°$ Central density3
Number of pieces288
Level of complexity19
Related polytopes
ArmySemi-uniform Ti
DualMedial deltoidal hexecontahedron
Convex coreChamfered dodecahedron
Abstract properties
Euler characteristic-6
Topological properties
OrientableYes
Genus4
Properties
SymmetryH3, order 120
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

• $\left(±\frac{\sqrt5}{2},\,±\frac12,\,±\frac12\right),$ along with all even permutations of

• $\left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{3-\sqrt5}{4}\right),$ • $\left(±1,\,±\frac{\sqrt5-1}{4},\,±\frac{1+\sqrt5}{4}\right).$ ## 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.

o5o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great dodecahedron gad {5,5/2} x5o5/2o (     )
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o (     )
Dodecadodecahedron did r{5,5/2} o5x5/2o (     )
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x (     )
Small stellated dodecahedron sissid {5/2,5} o5o5/2x (     )
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x (     )
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x (     )
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s (     )