Great inverted snub icosidodecahedron

Great inverted snub 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	| 5/3 2 3
Symmetry group	I, [5,3]+, 532
Index references	U69, C73, W116
Dual polyhedron	Great inverted pentagonal hexecontahedron
Vertex figure	34.5/3
Bowers acronym	Gisid

3D model of a great inverted snub icosidodecahedron

In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U₆₉. It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.

Cartesian coordinates

Cartesian coordinates for the vertices of a great inverted snub icosidodecahedron are all the even permutations of

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

with an even number of plus signs, where

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

where $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden ratio and ξ is the greater positive real solution to:

$$\xi ^{3}-2\xi =-{\tfrac {1}{\varphi }}\quad \implies \quad \xi \approx 1.2224727.$$

Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

The circumradius for unit edge length is

$$R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.816081\dots$$

where x is the appropriate root of

$$x^{3}+2x^{2}=\left({\frac {1\pm {\sqrt {5}}}{2}}\right)^{2}.$$

The four positive real roots of the sextic in R²,

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

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

Related polyhedra

Great inverted pentagonal hexecontahedron

Great inverted pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 92 (χ = 2)
Symmetry group	I, [5,3]+, 532
Index references	DU69
dual polyhedron	Great inverted snub icosidodecahedron

3D model of a great inverted pentagonal hexecontahedron

The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron. It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices.

It is the dual of the uniform great inverted snub icosidodecahedron.

Proportions

Denote the golden ratio by $\phi$. Let $\xi \approx 0.252\,780\,289\,27$ be the smallest positive zero of the polynomial $P=8x^{3}-8x^{2}+\phi ^{-2}$. Then each pentagonal face has four equal angles of $\arccos(\xi )\approx 75.357\,903\,417\,42^{\circ }$ and one angle of $360^{\circ }-\arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 238.568\,386\,330\,33^{\circ }$. Each face has three long and two short edges. The ratio $l$ between the lengths of the long and the short edges is given by

$l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 3.528\,053\,034\,81$.

The dihedral angle equals $\arccos(\xi /(\xi +1))\approx 78.359\,199\,060\,62^{\circ }$. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial $P$ play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

See also

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

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 126

External links

• Weisstein, Eric W. "Great inverted pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Great inverted snub icosidodecahedron". MathWorld.