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 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 +\frac\bigr& \pm\,1,& \pm\,\frac &\Bigr)
\end

where $\varphi = \tfrac$ is the golden ratio. Using $\tfrac = 1 - \tfrac$ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to $10-\tfrac.$ 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.

See also

* List of uniform polyhedra

References

*

External links

*
*

Uniform polyhedra and duals

Uniform polyhedra

{{polyhedron-stub