Rhombidodecadodecahedron

Rhombidodecadodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
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	${\frac {\sqrt {7}}{2}}\approx 1.32288$
Volume	$19{\sqrt {5}}\approx 42.48529$
Dihedral angles	4–5/2: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$
Central density	3
Number of external pieces	288
Level of complexity	19
Related polytopes
Army	Semi-uniform Ti, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (pentagons) and ${\frac {3-{\sqrt {5}}}{2}}$ (between ditrigons)
Regiment	Raded
Dual	Medial deltoidal hexecontahedron
Conjugate	Rhombidodecadodecahedron
Convex core	Chamfered dodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	-6
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Flag orbits	4
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

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

along with all even permutations of

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

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.

External links

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

• Klitzing, Richard. "raded".
• Wikipedia contributors. "Rhombidodecadodecahedron".
• McCooey, David. "Rhombidodecadodecahedron"