Truncated great icosahedron

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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 Template:Math or Template:Math 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

$\begin{matrix} ( & \pm 1, & 0, & \pm \frac{3}{φ} & ) \\ ( & \pm 2, & \pm \frac{1}{φ}, & \pm \frac{1}{φ^{3}} & ) \\ ( & \pm [1 + \frac{1}{φ^{2}}], & \pm 1, & \pm \frac{2}{φ} & ) \end{matrix}$

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

Name	Great stellated dodecahedron	Truncated great stellated dodecahedron	Great icosidodecahedron	Truncated great icosahedron	Great icosahedron
Coxeter-Dynkin diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
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Great stellapentakis dodecahedron

Template:Uniform polyhedra db File:Great stellapentakis dodecahedron.stl The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.