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Truncated icosahedron

Truncated icosahedron
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Type	Archimedean
Faces	12 pentagons 20 hexagons
Edges	90
Vertices	60
Vertex configuration	5,6,6
Symmetry group	icosahedral (Ih)
Dual polyhedron	pentakis dodecahedron
Properties	convex, semi-regular (vertex-uniform)

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 the Euler characteristic:

32 + 60 - 90 = 2.

A football (soccer ball) 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

• dodecahedron
• icosahedron
• icosidodecahedron
• truncated dodecahedron