Truncated great icosahedron explained

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.^[1] It is given a Schläfli symbol or 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

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

where

\varphi=\tfrac{1+\sqrt5}{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.

External links

- - Uniform polyhedra and duals

Notes and References

Web site: 55: great truncated icosahedron. Maeder. Roman. MathConsult.