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 (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.

See also

• dodecahedron
• icosahedron
• icosidodecahedron
• truncated dodecahedron

External links

• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Paper Models of Polyhedra Many links