Icosahedron

The ( also a v. Austrian: the ) icosahedron [ ikosae dər ː ] ( from Ancient Greek εἰκοσάεδρος eikosáedros "Twenty flat " ) is a polyhedron ( a polyhedron ), more precisely one of the five Platonic solids: with

  • Twenty ( congruent ) equilateral triangles as faces
  • Thirty ( equally divided) edges and
  • Twelve corners in which meet five areas.

Symmetry

Due to its high symmetry - all corners, edges and surfaces are equal to each other like - is the icosahedron is a regular polyhedron. It has:

  • Six fivefold axes of rotation ( through opposite corners)
  • Ten threefold axes of rotation ( through the centers of opposite faces )
  • Fifteen -fold rotation axes ( through the center of opposite edges )
  • Fifteen planes of symmetry ( by opposing - and parallel - edges)

And is

  • Centrally symmetric ( point reflection at the center of the polyhedron ).

Overall, the symmetry group of the icosahedron - the icosahedron or dodecahedron - 120 elements ( icosahedral ). The subgroup of rotations of the icosahedron has order 60 and is the smallest non-abelian simple group (A5, Alternating group of order 5). The symmetry of the icosahedron is not (due to occur with him five-fold symmetry ) with a periodic spatial structure compatible. It can therefore no crystal lattice with icosahedral symmetry type (but see quasicrystals ).

Cartesian coordinates

The following Cartesian coordinates define the vertices of an icosahedron with edge length a = 2, centered at the origin:

With equal to the golden ratio.

Relations with other polyhedra

The dodecahedron is the icosahedron to the dual polyhedron (and vice versa). With the help of the icosahedron and dodecahedron numerous body can be constructed which also have the icosahedral group as a symmetry group. Thus, for example, receives

  • A truncated icosahedron with 20 hexagons and 12 pentagons, like a football ( see below), see also fullerene. It arises from the icosahedron by the corners are cut off so that always occurs a regular pentagon as a cutting surface.
  • An icosahedron with 20 triangles and 12 pentagons
  • A truncated dodecahedron with 20 triangles and 12 ten corners as a means of an icosahedron with a dodecahedron (see Archimedean bodies )
  • A Rhombentriakontaeder with 20 12 = 32 corners and 30 rhombi as faces as convex hull of a union of an icosahedron with a dodecahedron and
  • A Ikosaederstern by all edges of an icosahedron be extended beyond its corners addition, to intersect three of them at one point.

On the structure of the icosahedron

As the graph below shows, there are three pairs may, under the edges of the icosahedron opposite ( ie a total of six ) select edges so that these pairs spanning three congruent to each other pairwise orthogonal rectangles. ( The lengths of the sides of these rectangles correspond to the golden ratio because they are sides or diagonals of regular pentagons. ) The icosahedron can therefore be inscribed in a cube that these six edges lie in the six faces of the cube and parallel to the edges of the cube are.

The 24 remaining edges limit 8 triangles ( among the 20 faces of the icosahedron ), which in the faces of a - circumscribed the icosahedron - octahedron lie, the corners of the icosahedron are on its edges.

Altogether there are five such positions, each edge of the icosahedron to exactly such a group of orthogonal pairs of part edges, while each surface is twice the area of ​​a circumscribed octahedron. The symmetry group of the icosahedron causes all 5! / 2 = 60 even permutations of these five positions.

The edges of the icosahedron contain twelve planar pentagons, each edge to two and each corner is one of five of these pentagons. You can use this property to construct a wire model.

Formulas

Importance of the icosahedron in the cluster physics

Great importance is the icosahedral form in clusters ( collections of atoms in the order 3-50000 atoms ) from a magnitude of more than 7 atoms. This is because the rule of Friedel, indicating that that structure has the lowest energy, for which the number of the nearest neighbor bonds, is a maximum. With many free clusters, this occurs from 7 atoms, where there also are exceptions, however, and other structures are preferred (such as cubes ).

Furthermore, there is in the cluster physics called magic numbers, which are closely related to the so-called Mackayschen icosahedron. Here shell closures provide (ie perfect atom icosahedron ) for particularly stable clusters. This occurs in clusters on the magic number 1, 13, 55, 147, 309, 561, 923 and 1415. This rather old findings of Alan Mackay play a significant role in the current cluster physics.

The cluster numbers can be prepared by the following formula:

C = total number of atoms in the cluster n = the number of atoms per edge

Applications

  • The capsids of many viruses have an icosahedral symmetry. This is explained by the fact that viruses package their nucleic acid optimal. The icosahedron is favorable in this respect because the icosahedron of all regular polyhedra with a given diameter has the largest volume.
  • Rudolf von Laban had used the icosahedron for its space harmonics intensive and thus influenced the modern dance. This is done today in the Laban movement studies.
  • Stafford Beer had worked out in his cybernetic management theory, the icosahedral structure as a model for optimal networking of employees in teams.
  • In many role-playing games icosahedron than twenty -sided dice ( d20 ) can be used.
  • A placed in the globe icosahedron forms the core of the lattice structure when weather prediction model GME of the German Weather Service ( see Numerical weather prediction ).
  • The Dogic is a variant of the Rubik's Cube in the form of an icosahedron as a three dimensional, mechanical puzzle.
  • Inside a Magic 8 Ball is an icosahedron, on which are the possible answers. It floats in a dark blue liquid inside the ball.
408916
de