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Great retrosnub icosidodecahedron

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Great retrosnub icosidodecahedron

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It is given a Schläfli symbol s{3/2,5/3}.

Cartesian coordinates

Cartesian coordinates for the vertices of a great retrosnub 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 smaller positive real root of ξ3−2ξ=−1/τ, namely

ξ = ( 1 + i 3 ) ( 1 2 τ + τ 2 4 8 27 ) 1 3 + ( 1 i 3 ) ( 1 2 τ τ 2 4 8 27 ) 1 3 2

or approximately 0.3264046. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

References

Great retrosnub icosidodecahedron Wikipedia