Snub icosidodecadodecahedron

In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.

Cartesian coordinates
Let $$\rho\approx 1.3247179572447454$$ be the real zero of the polynomial $$x^3-x-1$$. The number $$\rho$$ is known as the plastic ratio. Denote by $$\phi$$ the golden ratio. Let the point $$p$$ be given by
 * $$p=

\begin{pmatrix} \rho \\ \phi^2\rho^2-\phi^2\rho-1\\ -\phi\rho^2+\phi^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$$ are the vertices of a snub icosidodecadodecahedron. The edge length equals $$2\sqrt{\phi^2\rho^2-2\phi-1}$$, the circumradius equals $$\sqrt{(\phi+2)\rho^2+\rho-3\phi-1}$$, and the midradius equals $$\sqrt{\rho^2+\rho-\phi}$$.

For a snub icosidodecadodecahedron whose edge length is 1, the circumradius is
 * $$R = \frac12\sqrt{\rho^2+\rho+2} \approx 1.126897912799939$$

Its midradius is
 * $$r = \frac12\sqrt{\rho^2+\rho+1} \approx 1.0099004435452335$$

Medial hexagonal hexecontahedron
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.