Great stellated truncated dodecahedron



In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol $t_{0,1}{5/3,3}.$

Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the small icosicosidodecahedron, the small ditrigonal dodecicosidodecahedron, and the small dodecicosahedron:

Cartesian coordinates
Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of $$\begin{array}{crclc} \Bigl(& 0,& \pm\,\varphi,& \pm \bigl[2-\frac{1}{\varphi}\bigr] &\Bigr) \\ \Bigl(& \pm\,\varphi,& \pm\,\frac{1}{\varphi},& \pm\,\frac{2}{\varphi} &\Bigr) \\ \Bigl(& \pm\,\frac{1}{\varphi^2},& \pm\,\frac{1}{\varphi},& \pm\,2 &\Bigr) \end{array}$$

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