Nonconvex great rhombicosidodecahedron

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Nonconvex great rhombicosidodecahedron
Uniform great rhombicosidodecahedron.png
Type Uniform star polyhedron
Elements F = 62, E = 120
V = 60 (χ = 2)
Faces by sides 20{3}+30{4}+12{5/2}
Wythoff symbol 5/3 3 | 2
5/2 3/2 | 2
Symmetry group Ih, [5,3], *532
Index references U67, C84, W105
Dual polyhedron Great deltoidal hexecontahedron
Vertex figure Uniform great rhombicosidodecahedron vertfig.png
3.4.5/3.4
Bowers acronym Qrid

In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol t0,2{5/3,3}. Its vertex figure is a crossed quadrilateral.

This model shares the name with the convex great rhombicosidodecahedron, also known as the truncated icosidodecahedron.

Cartesian coordinates[edit]

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

(±1/τ2, 0, ±(2−1/τ))
(±1, ±1/τ3, ±1)
(±1/τ, ±1/τ2, ±2/τ)

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

Related polyhedra[edit]

It shares its vertex arrangement with the truncated great dodecahedron, and with the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the great dodecicosidodecahedron (having the triangular and pentagrammic faces in common), and the great rhombidodecahedron (having the square faces in common).

Uniform great rhombicosidodecahedron.png
Nonconvex great rhombicosidodecahedron
Great dodecicosidodecahedron.png
Great dodecicosidodecahedron
Great rhombidodecahedron.png
Great rhombidodecahedron
Great truncated dodecahedron.png
Truncated great dodecahedron
UC34-6 pentagonal prisms.png
Compound of six pentagonal prisms
UC35-12 pentagonal prisms.png
Compound of twelve pentagonal prisms

Great deltoidal hexecontahedron[edit]

Great deltoidal hexecontahedron
DU67 great strombic hexecontahedron.png
Type Star polyhedron
Face DU67 facets.png
Elements F = 60, E = 120
V = 62 (χ = 2)
Symmetry group Ih, [5,3], *532
Index references DU67
dual polyhedron Nonconvex great rhombicosidodecahedron

The great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the Great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices.

It is also called a great strombic hexecontahedron.

See also[edit]

References[edit]

External links[edit]