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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 .
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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/ξand
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.
Great pentagonal hexecontahedron
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.