Great snub icosidodecahedron

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Great snub icosidodecahedron

Type	Uniform 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
Index references	U₅₇, C₈₈, W₁₁₆
3⁴.⁵/₂ (Vertex figure)	Great pentagonal hexecontahedron (dual polyhedron)

In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U₅₇.

This polyhedron can be considered a snub great icosahedron.

[edit] 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 solution to 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.