Great retrosnub icosidodecahedron

Great retrosnub icosidodecahedron
Type Uniform star polyhedron
Elements F = 92, E = 150
V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5/2}
Wythoff symbol(s) |3/2 5/3 2
Symmetry group I, [5,3]+, 532
Index references U74, C90, W117
Bowers acronym Girsid

(34.5/2)/2
(Vertex figure)

Great pentagrammic hexecontahedron
(dual polyhedron)

In geometry, the great 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

$\xi ={\frac {\left(1+i{\sqrt 3}\right)\left({\frac 1{2\tau }}+{\sqrt {{\frac {\tau ^{{-2}}}4}-{\frac 8{27}}}}\right)^{{\frac 13}}+\left(1-i{\sqrt 3}\right)\left({\frac 1{2\tau }}-{\sqrt {{\frac {\tau ^{{-2}}}4}-{\frac 8{27}}}}\right)^{{\frac 13}}}2}$

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.