Truncated great icosahedron



In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol $t{3,5/2}$ or $t_{0,1}{3,5/2}$ as a truncated great icosahedron.

Cartesian coordinates
Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of

$$\begin{array}{crccc} \Bigl(& \pm\,1,& 0,& \pm\,\frac{3}{\varphi} &\Bigr) \\ \Bigl(& \pm\,2,& \pm\,\frac{1}{\varphi},& \pm\,\frac{1}{\varphi^3} &\Bigr) \\ \Bigl(& \pm \bigl[1+\frac{1}{\varphi^2}\bigr],& \pm\,1,& \pm\,\frac{2}{\varphi} &\Bigr) \end{array}$$

where $$\varphi = \tfrac{1+\sqrt 5}{2}$$ is the golden ratio. Using $$\tfrac{1}{\varphi^2} = 1 - \tfrac{1}{\varphi}$$ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to $$10-\tfrac{9}{\varphi}.$$ The edges have length 2.

Related polyhedra
This polyhedron is the truncation of the great icosahedron:

The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.

Great stellapentakis dodecahedron
The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.