Great truncated icosidodecahedron

In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U68. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. It is given a Schläfli symbol $t_{0,1,2}\{{{sfrac|5|3}},3\},$ and Coxeter-Dynkin diagram,.

Cartesian coordinates
Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of $$\begin{array}{ccclc} \Bigl(& \pm\,\varphi,& \pm\,\varphi,& \pm \bigl[3-\frac{1}{\varphi}\bigr] &\Bigr),\\ \Bigl(& \pm\,2\varphi,& \pm\,\frac{1}{\varphi},& \pm\,\frac{1}{\varphi^3} &\Bigl), \\ \Bigl(& \pm\,\varphi,& \pm\,\frac{1}{\varphi^2},& \pm \bigl[1+\frac{3}{\varphi}\bigr] &\Bigr), \\ \Bigl(& \pm\,\sqrt{5},& \pm\,2,& \pm\,\frac{\sqrt{5}}{\varphi} &\Bigr), \\ \Bigl(& \pm\,\frac{1}{\varphi},& \pm\,3,& \pm\,\frac{2}{\varphi} &\Bigr), \end{array}$$

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

Great disdyakis triacontahedron
The great disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its faces are triangles.

Proportions
The triangles have one angle of $$\arccos\left(\tfrac{1}{6}+\tfrac{1}{15}\sqrt{5}\right) \approx 71.594\,636\,220\,88^{\circ}$$, one of $$\arccos\left(\tfrac{3}{4}+\tfrac{1}{10}\sqrt{5}\right) \approx  13.192\,999\,040\,74^{\circ}$$ and one of $$\arccos\left(\tfrac{3}{8}-\tfrac{5}{24}\sqrt{5}\right) \approx  95.212\,364\,738\,38^{\circ}.$$ The dihedral angle equals $$\arccos\left(\tfrac{-179+24\sqrt{5}}{241}\right) \approx  121.336\,250\,807\,39^{\circ}.$$ Part of each triangle lies within the solid, hence is invisible in solid models.