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# Truncated icosahedron

Truncated icosahedron

Click on picture for large version.
TypeArchimedean
Faces12 pentagons
20 hexagons
Edges90
Vertices60
Vertex configuration5,6,6
Symmetry groupicosahedral (Ih)
Dual polyhedronpentakis dodecahedron
Propertiesconvex, semi-regular (vertex-uniform)

The truncated icosahedron is an Archimedean solid. It has the same shape as a football (US: soccer ball) 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 orthogonal 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.

This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Richard Rhodes. Dark Sun: The Making of the Hydrogen Bomb, ISBN 0684824140. Touchstone Books, 1996., p. 195).

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.

## Truncated icosahedra in the arts

A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion.