Great retrosnub icosidodecahedron

Template:Short description Template:Uniform polyhedra db File:Great retrosnub icosidodecahedron.stl

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as Template:Math. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol Template:Math

Cartesian coordinates

Let $\xi \approx -1.8934600671194555$ be the smallest (most negative) zero of the polynomial $x^3+2x^2-{\varphi }^{-2}$, where $\varphi$ is the golden ratio. Let the point $p$ be given by

$p=\left(\begin{array}{c}\xi \\ {\varphi }^{-2}-{\varphi }^{-2}\xi \\ -{\varphi }^{-3}+{\varphi }^{-1}\xi +2{\varphi }^{-1}{\xi }^2\end{array}\right)$.

Let the matrix $M$ be given by

$M=\left(\begin{array}{ccc}1/2& -\varphi /2& 1/(2\varphi )\\ \varphi /2& 1/(2\varphi )& -1/2\\ 1/(2\varphi )& 1/2& \varphi /2\end{array}\right)$.

$M$ is the rotation around the axis $(1,0,\varphi )$ by an angle of $2\pi /5$, counterclockwise. Let the linear transformations $T_0,\dots ,T_{11}$ 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 $T_i{M}^j$ $(i=0,\dots ,11$, $j=0,\dots ,4)$ 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 Template:Math, $$4096R^{12}-27648R^{10}+47104R^8-35776R^6+13872R^4-2696R^2+209=0$$ 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

External links

• Template:Mathworld

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