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	|^3/2 ^5/3 2
Symmetry group	I, [5,3]^+, 532
Index references	U74, C90, W117
Dual polyhedron	Great pentagrammic hexecontahedron
Vertex figure	(3^4.^5/2)/2
Bowers acronym	Girsid

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

$\xi=\frac{\left(1+i \sqrt3\right)\left(\frac1{2 \tau}+\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13+ \left(1-i \sqrt3\right)\left(\frac1{2 \tau}-\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13}2$

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.

See also

• List of uniform polyhedra
• Great snub icosidodecahedron
• Great inverted snub icosidodecahedron

External links

• Weisstein, Eric W., "Great retrosnub icosidodecahedron", MathWorld.
• http://gratrix.net/polyhedra/uniform/summary