Truncated icosahedron

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Truncated icosahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F=32, E=90, V=60 (χ=2)
Faces by sides	12{5}+20{6}
Schläfli symbol	t{3,5}
Wythoff symbol	2 5 \| 3
Symmetry group	I_h
Index references	U₂₅, C₂₇, W₉
Dual	Pentakis dodecahedron
Properties	Semiregular convex
Vertex figure 5.6.6

A colored model

Net (polyhedron)

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.

This polyhedron can be constructed from an icosahedron with the 12 vertices truncated (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.

1 Canonical coordinates
2 Geometric relations
3 Applications
4 Truncated icosahedra in the arts
5 See also
6 References
7 External links

[edit] Canonical coordinates

Canonical coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3φ), (±1,±3φ,0), (±3φ,0,±1) and the orthogonal cuboids (±2,±(1+2φ),±φ), (±(1+2φ),±φ,±2), (±φ,±2,±(1+2φ)) along with the orthogonal cuboids (±1,±(2+φ),±2φ), (±(2+φ),±2φ,±1), (±2φ,±1,±(2+φ)), where φ = (1+√5)/2 is the golden mean. Using φ² = φ + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9φ + 10. The edges have length 2.

[edit] Geometric relations

The truncated icosahedron easily verifies the Euler characteristic:

32 + 60 − 90 = 2.

With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons).

[edit] Applications

Compared to a football (soccer ball).

A football (soccer ball) comprises the same pattern of regular pentagons and regular hexagons, but is more spherical due to the pressure of the air inside and the elasticity of the ball.

This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs (Richard Rhodes. Dark Sun: The Making of the Hydrogen Bomb, ISBN 0-684-82414-0. Touchstone Books, 1996., p. 195).

The truncated icosahedron can also be described as a model of the Buckminsterfullerene (C₆₀) molecule. The diameter of the football and the Buckminsterfullerene molecule are 22 cm and ca. 1 nm, respectively, hence the size ratio is 200,000,000 : 1.

[edit] Truncated icosahedra in the arts

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

[edit] See also

[edit] References

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)