Great retrosnub icosidodecahedron

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.