Dodecahedron

The dodecahedron [ ˌ dodekaʔe dər ː ] (from Greek Zwölfflächner ) is a body with twelve faces. This generally is a Platonic solid meant, namely the (regular ) pentagonal dodecahedron, a body with

The regular dodecahedron

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

• Fivefold six axes of rotation ( through opposite centroids )
• Ten threefold axes of rotation ( through opposite corners)
• Fifteen -fold rotation axes ( through the center of opposite edges )
• Fifteen planes of symmetry ( by opposing - and parallel - edges)

And is

• Inversion symmetry ( point reflection with respect to the Dodekaedermittelpunkts )

Overall, the symmetry group of the dodecahedron - the dodecahedral or icosahedral group - 120 elements. The 60 orientation-preserving symmetries correspond to the alternating group. Sometimes this subgroup is called " icosahedral ". The full symmetry group is isomorphic to the direct product. That the product is right, you can see the fact that the point reflection at the center commutes with the rotations.

The symmetry of the dodecahedron is occurring here by the five-fold symmetry axes with a periodic spatial structure not compatible (see tiling ). It can therefore no crystal lattice with icosahedral symmetry type (but see quasicrystals ).

On the structure

The icosahedron is the dodecahedron to the dual polyhedron (and vice versa).

With the help of dodecahedron and icosahedron numerous body can be constructed which also have the dodecahedron as a symmetry group. Thus, for example, receives

• The truncated dodecahedron with 12 vertices and 20 triangles Ten ( by blunting of the corners of a dodecahedron )
• The icosahedron with 12 pentagons and 20 triangles
• The truncated icosahedron with 12 pentagons and 20 hexagons as the average of a dodecahedron with icosahedron (similar to a soccer ball, see also Archimedean body, fullerenes )
• And the Rhombentriakontaeder with 12 20 = 32 corners and 30 rhombi as faces ( It is made by blending straight pyramids on the dodecahedron, of which two sides complement each other, that is, lie in one plane and an edge have in common. )

From the edges of the dodecahedron can be three pairs of opposite ( ie a total of six ) select edges so that these pairs spanning three congruent to each other pairwise orthogonal rectangles. The remaining eight corners then form the corners of a ( the dodecahedron registered ) cube. There are five such positions, each edge of the dodecahedron belongs to exactly such a position, and each corner is inscribed vertex of two dice. The symmetry group of the dodecahedron causes all 5! = 120 permutations of these five positions or cubes.

Since the edges of the inscribed cube are diagonals of the pentagons, corresponds to the ratio of the lengths of the edges of the dodecahedron and that of an inscribed cube the golden section.

Formulas

Applications

• Some geodesic domes are polyhedra that are derived from the dodecahedron by the pentagons are further subdivided into ( isosceles ) triangles.
• There are dodekaederförmige dice.
• Dodecahedron be used as original recycling container (for example, in Paris).
• In building acoustics dodekaederförmige speakers are used to obtain the best possible omni.
• Instead of a glass ball crystal Zwölfflächner be used for room illumination.
• The purpose of the Roman dodecahedron is unclear.
• A dodecahedron can be used as an annual calendar: each month gets its own pentagon.
• Both Megaminx as well as Alexander 's Star are variations of Rubik's Cubes in the shape of a dodecahedron as a three-dimensional puzzle.

Others

• In the science-fiction movie Contact ( 1997), the transport sphere is embedded in a lattice, which has the shape of a dodecahedron.

The cubic pentagonal dodecahedron

The cubic pentagonal dodecahedron can be externally easily confused with the regular pentagonal dodecahedron. It also has 12 faces, 20 vertices and 30 edges. But the faces ( pentagons ) are not equal to each other. Each of the 12 areas has four shorter and a longer edge. Overall, the polyhedron has 24 short and 6 long edges. While it has cubic symmetry. In nature, pyrite is sometimes ( FeS2 ) is present in the form of cubic pentagonal dodecahedron tetrahedra. Therefore, the cubic pentagonal dodecahedron also pyrite - dodecahedron or Pyritoeder is called. In crystals fivefold axes are impossible, as the regular pentagonal dodecahedron owns them, because there is no continuous periodic surface filling with five-fold symmetry. Only if not strictly periodic " crystals ", ie quasicrystals, a regular dodecahedron is conceivable.

Other dodecahedron

Other regular dodecahedron are eg:

• The rhombic dodecahedron has 12 congruent rhombi as faces, 14 vertices and 24 edges. It is the typical crystal form of the grenade.
• The trigondodecahedron has 12 congruent equilateral triangles as faces, 8 vertices and 18 edges.
• Great dodecahedron