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. It is given a Schläfli symbol $sr{ 3/2,5/3}.$

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
 * $$R = \frac12\sqrt{\frac{2-\xi}{1-\xi}} \approx 0.5800015046400155 $$

Its midradius is
 * $$r=\frac{1}{2}\sqrt{\frac{1}{1-\xi}} \approx 0.2939417380786233$$

The four positive real roots of the sextic in $R2$, $$4096R^{12} - 27648R^{10} + 47104R^8 - 35776R^6 + 13872R^4 - 2696R^2 + 209 = 0$$ are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).