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}
Coxeter diagram
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
Bowers acronym	Girsid

3D model of a great retrosnub icosidodecahedron

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron 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

$${\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha ,&\pm \,2,&\pm \,2\beta &{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi -{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]},&\pm {\bigl [}-\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta +\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]},&\pm {\bigl [}\alpha +\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}-\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]}&{\Bigr )},\\\end{array}}}$$

with an even number of plus signs, where

$${\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }},\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }},\end{aligned}}}$$

where ${\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}$ is the golden ratio and ξ is the smaller positive real root of

$${\displaystyle {\begin{aligned}\xi ^{3}-2\xi =-{\frac {1}{\varphi }}\quad \implies \quad \xi &={\frac {\left(1+i{\sqrt {3}}\right){\sqrt[{3}]{{\frac {1}{2\varphi }}+{\sqrt {{\frac {1}{4\varphi ^{2}}}-{\frac {8}{27}}}}}}+\left(1-i{\sqrt {3}}\right){\sqrt[{3}]{{\frac {1}{2\varphi }}-{\sqrt {{\frac {1}{4\varphi ^{2}}}-{\frac {8}{27}}}}}}}{2}}\\[4pt]&\approx 0.3264046\end{aligned}}}$$

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

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

where x is the appropriate root of ${\displaystyle x^{3}+2x^{2}={\Big (}{\tfrac {1\pm {\sqrt {5}}}{2}}{\Big )}^{2}.}$ The four positive real roots of the sextic in R²,

$${\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0}$$

are the circumradii of the snub dodecahedron (U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).

See also

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

References

External links

• Weisstein, Eric W. "Great retrosnub icosidodecahedron". MathWorld.