The Icosahedron reference article from the English Wikipedia on 24-Jul-2004
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Icosahedron

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Icosahedron
Icosahedron

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Type Platonic
Face polygon triangle
Faces 20
Edges 30
Vertices 12
Faces per vertex 5
Vertices per face 3
Symmetry group icosahedral (Ih)
Dual polyhedron dodecahedron
Properties regular, convex
An icosahedron [ˌaıkəsə'hiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces. The faces of a regular icosahedron are equilateral triangless. [Etymology: 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedral adjective

In geometry, the regular icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges. Its dual polyhedron is the dodecahedron.

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

Golden rectangles in an icosahedron
Canonical coordinates for the vertices of an icosahedron centered at the origin are {(0,±1,±φ), (±1,±φ,0), (±φ,0,±1)}, where φ = (1+√5)/2 is the golden mean — note these form three mutually orthogonal golden rectangles. The 20 edges of an octahedron can be partitioned in the golden mean so that the resulting vertices define a regular icosahedron; the five octahedra defining any given icosahedron form a regular polyhedral compound.

There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry than the tetrahedron. The icosahedron has a large number of stellations, including one of the Kepler-Poinsot solids and some of the regular compounds, which could be discussed here.

Many viruses, including HIV, have the shape of an icosahedron. Viruses need to be small, and the icosahedron is the regular polyhedron with largest volume per diameter. A regular polyhedron is used because it can be built from a single basic unit protein that's reused over and over again; that saves space in the viral genome.

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