Great snub icosidodecahedron

Great snub icosidodecahedron
Type	Uniform star polyhedron
Elements	F = 92, E = 150 V = 60 (χ = 2)
Faces by sides	(20+60){3}+12{^5/2}
Wythoff symbol	|2 ^5/2 3
Symmetry group	I, [5,3]^+, 532
Index references	U57, C88, W116
Dual polyhedron	Great pentagonal hexecontahedron
Vertex figure	3^4.^5/2
Bowers acronym	Gosid

In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U₅₇. It can be represented by a Schläfli symbol sr{5/2,3}, and Coxeter-Dynkin diagram .

This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),

(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),

(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),

(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and

(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ

and

β = −ξ/τ+1/τ²−1/(ξτ),

where τ = (1+√5)/2 is the golden mean and ξ is the negative real root of ξ³−2ξ=−1/τ, or approximately −1.5488772. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

Related polyhedra

Great pentagonal hexecontahedron

Great pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 92 (χ = 2)
Symmetry group	I, [5,3]^+, 532
Index references	DU57
dual polyhedron	Great snub icosidodecahedron

The great pentagonal hexecontahedron is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.

See also

• List of uniform polyhedra
• Great inverted snub icosidodecahedron
• Great retrosnub icosidodecahedron

References

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

External links

• Weisstein, Eric W., "Great pentagonal hexecontahedron", MathWorld.
• Weisstein, Eric W., "Great snub icosidodecahedron", MathWorld.
• Uniform polyhedra and duals