Nonconvex great rhombicosidodecahedron

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 t0,2{53,3}. Its vertex figure is a crossed quadrilateral.

Nonconvex great rhombicosidodecahedron
TypeUniform star polyhedron
ElementsF = 62, E = 120
V = 60 (χ = 2)
Faces by sides20{3}+30{4}+12{5/2}
Wythoff symbol2
5/2 3/2 | 2
Symmetry groupIh, [5,3], *532
Index referencesU67, C84, W105
Dual polyhedronGreat deltoidal hexecontahedron
Vertex figure
3.4.5/3.4
Bowers acronymQrid
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

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).

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

Great deltoidal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 62 (χ = 2)
Symmetry groupIh, [5,3], *532
Index referencesDU67
dual polyhedronNonconvex 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

References

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