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 Rhombidodecadodecahedron vertfig.png
Bowers acronym Raded

In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices.[1] It is given a Schläfli symbol t0,2{​52,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.

3D model of a rhombidodecadodecahedron

Cartesian coordinatesEdit

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

(±1/τ2, 0, ±τ2)
(±1, ±1, ±5)
(±2, ±1/τ, ±τ)

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

Related polyhedraEdit

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
Compound of ten triangular prisms
Compound of twenty triangular prisms

Medial deltoidal hexecontahedronEdit

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

The medial deltoidal hexecontahedron (or midly lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.

See alsoEdit


  1. ^ Maeder, Roman. "38: rhombidodecadodecahedron". MathConsult.

External linksEdit