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}.

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

Cartesian coordinates

Let ${\displaystyle \xi \approx -1.8934600671194555}$ be the smallest (most 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

${\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 0.5800015046400155}$

Its midradius is

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

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