Nonconvex great rhombicosidodecahedron

Nonconvex great rhombicosidodecahedron
Type Uniform star polyhedron
Elements F = 62, E = 120
V = 60 (χ = 2)
Faces by sides 20{3}+30{4}+12{5/2}
Coxeter diagram
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
3.4.5/3.4
Bowers acronym Qrid

In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices.[1] It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol rr{53,3}. Its vertex figure is a crossed quadrilateral.

3D model of a nonconvex great rhombicosidodecahedron

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

Cartesian coordinates

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

where   is the golden ratio.

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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).

 
Nonconvex great rhombicosidodecahedron
 
Great dodecicosidodecahedron
 
Great rhombidodecahedron
 
Truncated great dodecahedron
 
Compound of six pentagonal prisms
 
Compound of twelve pentagonal prisms

Great deltoidal hexecontahedron

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

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

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References

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  1. ^ Maeder, Roman. "67: great rhombicosidodecahedron". MathConsult.
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