Rhombidodecadodecahedron

Rhombidodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 54, E = 120 V = 60 (χ = −6)
Faces by sides	30{4}+12{5}+12{^5/2}
Wythoff symbol	^5/2 5 | 2
Symmetry group	Ih, [5,3], *532
Index references	U38, C48, W76
Dual polyhedron	Medial deltoidal hexecontahedron
Vertex figure	4.^5/2.4.5
Bowers acronym	Raded

In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₃₈. It is given a Schläfli symbol t_0,2{5/2,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of

(±1/τ², 0, ±τ²))

(±1, ±1, ±(2τ−1))

(±2, ±1/τ, ±τ)

where τ = (1+√5)/2 is the golden ratio (sometimes written φ).

Related polyhedra

It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the icosidodecadodecahedron (having the pentagonal and pentagrammic faces in common) and the rhombicosahedron (having the square faces in common).

convex hull	Rhombidodecadodecahedron	Icosidodecadodecahedron
Rhombicosahedron	Compound of ten triangular prisms	Compound of twenty triangular prisms

Medial deltoidal hexecontahedron

Medial deltoidal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 54 (χ = −6)
Symmetry group	Ih, [5,3], *532
Index references	DU38
dual polyhedron	Rhombidodecadodecahedron

The medial deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.

See also

• List of uniform polyhedra

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208

External links

• Weisstein, Eric W., "Rhombidodecadodecahedron", MathWorld.
• Weisstein, Eric W., "Medial deltoidal hexecontahedron", MathWorld.
• Uniform polyhedra and duals