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

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.

Let ${\displaystyle \xi \approx -0.5055605785332548}$ be the largest (least negative) negative zero of the polynomial ${\displaystyle x^{3}+2x^{2}-\phi ^{-2}}$, where ${\displaystyle \phi }$ is the golden ratio. Let the point ${\displaystyle p}$ be given by

${\displaystyle p={\begin{pmatrix}\xi \\\phi ^{-2}-\phi ^{-2}\xi \\-\phi ^{-3}+\phi ^{-1}\xi +2\phi ^{-1}\xi ^{2}\end{pmatrix}}}$.

Let the matrix ${\displaystyle M}$ be given by

${\displaystyle M={\begin{pmatrix}1/2&-\phi /2&1/(2\phi )\\\phi /2&1/(2\phi )&-1/2\\1/(2\phi )&1/2&\phi /2\end{pmatrix}}}$.

${\displaystyle M}$ is the rotation around the axis ${\displaystyle (1,0,\phi )}$ by an angle of ${\displaystyle 2\pi /5}$, counterclockwise. Let the linear transformations ${\displaystyle T_{0},\ldots ,T_{11}}$ be the transformations which send a point ${\displaystyle (x,y,z)}$ to the even permutations of ${\displaystyle (\pm x,\pm y,\pm z)}$ with an even number of minus signs. The transformations ${\displaystyle T_{i}}$ constitute the group of rotational symmetries of a regular tetrahedron. The transformations ${\displaystyle T_{i}M^{j}}$ ${\displaystyle (i=0,\ldots ,11}$, ${\displaystyle j=0,\ldots ,4)}$ constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points ${\displaystyle T_{i}M^{j}p}$ are the vertices of a great snub icosahedron. The edge length equals ${\displaystyle -2\xi {\sqrt {1-\xi }}}$, the circumradius equals ${\displaystyle -\xi {\sqrt {2-\xi }}}$, and the midradius equals ${\displaystyle -\xi }$.

For a great snub icosidodecahedron whose edge length is 1, the circumradius is

${\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 0.6450202372957795}$

Its midradius is

${\displaystyle r={\frac {1}{2}}{\sqrt {\frac {1}{1-\xi }}}\approx 0.4074936889340787}$

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).

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

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.

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

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

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

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

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

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