Rhombidodecadodecahedron

Rhombidodecadodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Raded
Coxeter diagram	x5/2o5x ()
Elements
Faces	30 squares, 12 pentagons, 12 pentagrams
Edges	60+60
Vertices	60
Vertex figure	Isosceles 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 angles	4–5/2: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	4–5: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°$
Central density	3
Number of pieces	288
Level of complexity	19
Related polytopes
Army	Semi-uniform Ti
Regiment	Raded
Dual	Medial deltoidal hexecontahedron
Conjugate	Rhombidodecadodecahedron
Convex core	Chamfered dodecahedron
Abstract properties
Euler characteristic	-6
Topological properties
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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 ()

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#42).

• Klitzing, Richard. "raded".

• Wikipedia Contributors. "Rhombidodecadodecahedron".
• McCooey, David. "Rhombidodecadodecahedron"