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}
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

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as $U 74$ . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices.^[1] It is given a Schläfli symbol $sr{.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}3⁄2,5⁄3}.$

Cartesian coordinates

Let $\xi \approx -1.8934600671194555$ be the smallest (most 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 0.5800015046400155

Its midradius is

r={\frac {1}{2}}{\sqrt {\frac {1}{1-\xi }}}\approx 0.2939417380786233

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 (U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).