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| {{Uniform polyhedra db|Uniform polyhedron stat table|Girsid}}
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| In [[geometry]], the '''great retrosnub icosidodecahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>74</sub>. It is given a [[Schläfli symbol]] s{3/2,5/3}.
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| == Cartesian coordinates ==
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| [[Cartesian coordinates]] for the vertices of a great retrosnub icosidodecahedron are all the [[even permutation]]s of
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| : (±2α, ±2, ±2β), | |
| : (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
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| : (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
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| : (±(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
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| : α = ξ−1/ξ
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| and
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| : β = −ξ/τ+1/τ<sup>2</sup>−1/(ξτ),
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| where τ = (1+√5)/2 is the [[golden ratio|golden mean]] and
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| ξ is the smaller positive real [[root of a function|root]] of ξ<sup>3</sup>−2ξ=−1/τ, namely
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| : <math>\xi=\frac{\left(1+i \sqrt3\right)\left(\frac1{2 \tau}+\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13+
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| \left(1-i \sqrt3\right)\left(\frac1{2 \tau}-\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13}2</math>
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| or approximately 0.3264046.
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| Taking the [[odd permutation]]s of the above coordinates with an odd number of plus signs gives another form, the [[Chirality (mathematics)|enantiomorph]] of the other one.
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| == See also ==
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| * [[List of uniform polyhedra]]
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| * [[Great snub icosidodecahedron]]
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| * [[Great inverted snub icosidodecahedron]]
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| == External links ==
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| * {{mathworld | urlname = GreatRetrosnubIcosidodecahedron| title = Great retrosnub icosidodecahedron}}
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| * http://gratrix.net/polyhedra/uniform/summary
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| {{Polyhedron-stub}}
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| [[Category:Uniform polyhedra]]
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