Icosahedral symmetry

The icosahedral group is the point group of the icosahedron, the dodecahedron (which is dual to the icosahedron ). It consists of the rotations and reflections that transform the icosahedron in itself and has order 120 It is to A5 x C2 isomorphic with the alternating group A5 of order 5 is ( group of even permutations of 5 objects) and C2, the cyclic group of order 2 (consisting of the identity and the space reflection at the center of the icosahedron ).

The isomorphic to A5 subgroup ( the icosahedral rotation group ) consists of the orientation-preserving motion symmetries of the icosahedron ( rotations ). One can realize, for example, as a group of even permutations of five a regular dodecahedron inscribed cube. A5 is the smallest simple non- commutative group and has order 60

The icosahedral fivefold contains twists and is thus incompatible with crystalline long-range order (see space group ). Other hand, quasicrystals have often icosahedral symmetry.

The character table of the icosahedral group contains the golden ratio and related numbers, which is a direct consequence of five-fold rotational symmetry.

Since football is derived from a truncated icosahedron, he also has the icosahedral group as a symmetry group, as well as the "Football molecule" C60 ( buckyball ).

The icosahedral group has numerous applications in mathematics, presented in the classic work of Felix Klein Lectures on the icosahedron and the solution of equations of the fifth degree. The fifth degree equation has no solution in Galois theory after the radicals, since A5 is not solvable ( it is a finite simple group ).