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Snub dodecahedron

The snub dodecahedron or snub icosidodecahedron is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron.

Canonical coordinates for a snub dodecahedron are all the even permutations of (±2α, ±2, ±2β), (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)), (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)), (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)), with an even number of plusses, where α = ξ-1/ξ, and β = ξτ+τ2+τ/ξ, where τ = (1+√5)/2 is the golden mean and ξ is the real solution to ξ3-2ξ=τ, which is the horrible number 3√((τ+√(τ-5/27))/2)+3√((τ-√(τ-5/27))/2).

The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. In three-dimensional space, it has two distinct forms, which are mirror images (or "enantiomorphs") of each other. In higher-dimensional spaces, these are congruent.

The snub dodecahedron should not be confused with the truncated dodecahedron.

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