Icositruncated dodecadodecahedron

In geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45.

Convex hull
Its convex hull is a nonuniform truncated icosidodecahedron.

Cartesian coordinates
Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of $$\begin{array}{crrlc} \Bigl(& \pm\bigl[2-\frac{1}{\varphi}\bigr],& \pm\,1,& \pm\bigl[2+\varphi\bigr] &\Bigr), \\ \Bigl(& \pm\,1,& \pm\,\frac{1}{\varphi^2},& \pm\bigl[3\varphi-1\bigr] &\Bigr), \\ \Bigl(& \pm\,2,& \pm\,\frac{2}{\varphi},& \pm\,2\varphi &\Bigr), \\ \Bigl(& \pm\,3,& \pm\,\frac{1}{\varphi^2},& \pm\,\varphi^2 &\Bigr), \\ \Bigl(& \pm\,\varphi^2,& \pm\,1,& \pm\bigl[3\varphi-2\bigr] &\Bigr), \end{array}$$

where $$\varphi = \tfrac{1+\sqrt 5}{2}$$ is the golden ratio.

Tridyakis icosahedron
The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.