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	(3⁴.⁵/₂)/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.

Great retrosnub icosidodecahedron

Cartesian coordinates

See also

External links