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Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It has the same shape as a football or a 60-carbon fullerene.

Canonical coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3τ), (±1,±3τ,0), (±3τ,0,±1) and the orthogonal bricks/3D-rectangles (±2,±(1+2τ),±τ), (±(1+2τ),±τ,±2), (±τ,±2,±(1+2τ)) along with the ortogonal bricks/3D-rectangles (±1,±(2+τ),±2τ), (±(2+τ),±2τ,±1), (±2τ,±1,±(2+τ)), where τ = (1+√5)/2 is the golden mean.

It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.

One easily verifies Euler's formula for polyhedra (see Leonhard Euler): 32 + 60 = 90 + 2.

A football is like this polyhedron except that it is more spherical, because the faces bulge due the pressure of the air inside.

It is also a model for the Buckminsterfullerene (C60) molecule. The diameter of the football and this buckyball are 22 cm and ca. 1 nm, respectively, hence the size ratio is 200,000,000 : 1.

See also: Icosahedron and Dodecahedron.

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