Truncated icosahedron

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In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygon.

It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.

Contents

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, 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.

Cartesian coordinates

Cartesian 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.

Area and volume

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

Geometric relations

The truncated icosahedron easily verifies the Euler characteristic:

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

Applications

The balls used in association football and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. [1] The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball. This ball type was introduced in 1970; starting with the 2006 World Cup, the design has been superseded by newer patterns.

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