# 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 | U$_{69}$, C$_{73}$, W$_{116}$ |
| Dual polyhedron | Great inverted pentagonal hexecontahedron |
| Vertex figure | 3$^4$.5/3 |
| Bowers acronym | Gisid |
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U$_{69}$. 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

Let $\xi \approx -0.5055605785332548$ be the largest (least negative) negative zero of the polynomial $x^{3}+2x^{2}-\phi ^{-2}$, where $\phi$ is the golden ratio. Let the point $p$ be given by
    $p={\begin{pmatrix}\xi \\\phi ^{-2}-\phi ^{-2}\xi \\-\phi ^{-3}+\phi ^{-1}\xi +2\phi ^{-1}\xi ^{2}\end{pmatrix}}$.
Let the matrix $M$ be given by
    $M={\begin{pmatrix}1/2&-\phi /2&1/(2\phi )\\\phi /2&1/(2\phi )&-1/2\\1/(2\phi )&1/2&\phi /2\end{pmatrix}}$.
$M$ is the rotation around the axis $(1,0,\phi )$ by an angle of $2\pi /5$, counterclockwise. Let the linear transformations $T_{0},\ldots ,T_{11}$ be the transformations which send a point $(x,y,z)$ to the even permutations of $(\pm x,\pm y,\pm z)$ with an even number of minus signs. The transformations $T_{i}$ constitute the group of rotational symmetries of a regular tetrahedron. The transformations $T_{i}M^{j}$ $(i=0,\ldots ,11$, $j=0,\ldots ,4)$ constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points $T_{i}M^{j}p$ are the vertices of a great snub icosahedron. The edge length equals $-2\xi {\sqrt {1-\xi }}$, the circumradius equals $-\xi {\sqrt {2-\xi }}$, and the midradius equals $-\xi$.
For a great snub icosidodecahedron whose edge length is 1, the circumradius is
    $R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 0.6450202372957795$
Its midradius is
    $r={\frac {1}{2}}{\sqrt {\frac {1}{1-\xi }}}\approx 0.4074936889340787$
The four positive real roots of the sextic in R$^2$, $4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0$ are the circumradii of the snub dodecahedron (U$_{29}$), great snub icosidodecahedron (U$_{57}$), great inverted snub icosidodecahedron (U$_{69}$), and great retrosnub icosidodecahedron (U$_{74}$).

## 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 | DU$_{69}$ |
| dual polyhedron | Great inverted snub icosidodecahedron |
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.