Truncated Icosahedron - just the facts

Truncated Icosahedron - just the facts

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

Construction
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, & leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.

Icosahedron

Area & volume
The area A & the volume V of the truncated icosahedron of edge length a are:

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 & 52 for the hexagons, together 73 (see areas of regular polygons).

Applications

Compared to a football (soccer ball).A football (soccer) ball is an example of a truncated icosahedron found in everyday life. The ball comprises the same pattern of regular pentagons & regular hexagons, but is more spherical due to the pressure of the air inside & 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 & 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 & the Buckminsterfullerene molecule are 22 cm & ca. 1 nm, respectively, hence the size ratio is 200,000,000 : 1.

It is also the shape of buckyball, the Carbon thing.

Truncated icosahedra in the arts
A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion

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