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	\| 2 3/2 5/3
Symmetry group	I, [5,3]⁺, 532
Index references	U₇₄, C₉₀, W₁₁₇
Dual polyhedron	Great pentagrammic hexecontahedron
Vertex figure	(3⁴.5/2)/2
	Girsid

3D model of a great retrosnub icosidodecahedron

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron or Azathoth is a nonconvex uniform polyhedron, indexed as U₇₄. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices.^[1] It is given a Schläfli symbol sr{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 {1}{3}}+\left(1-i{\sqrt {3}}\right)\left({\frac {1}{2\tau }}-{\sqrt {{\frac {\tau ^{-2}}{4}}-{\frac {8}{27}}}}\right)^{\frac {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. Taking the odd permutations with an even number of plus signs or vice versa results in the same two figures rotated by 90 degrees.

The circumradius for unit edge length is

R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.580002\dots

where $x$ is the appropriate root of $x^{3}+2x^{2}={\Big (}{\tfrac {1\pm {\sqrt {5}}}{2}}{\Big )}^{2}$ . The four positive real roots of the sextic in $R^{2},$