Octahedron

The ( also a v. Austrian: the ) octahedra [ ɔktae dɐ ː ] (from Greek oktáedron, octahedron ') is one of the five Platonic solids, or more precisely a regular polyhedron ( polyhedron ) with

• Eight ( congruent ) equilateral triangles as faces
• Twelve ( equally divided) edges and
• Six corners in which meet four areas

The octahedron is both an equilateral four-sided bipyramid ( with a square base ) and an equilateral antiprism ( with an equilateral triangle as the base ).

Symmetry

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

• Three four-fold axes of rotation ( through opposite corners)
• Four threefold rotational axes ( through the centers of opposite faces )
• Six -fold rotation axes ( through the center of opposite edges )
• Nine planes of symmetry ( three levels by four corners, six levels by two corners and two edges midpoints )

And is

• Point symmetry ( centrally symmetric)

Overall, the symmetry group of the octahedron - the octahedron or cube group - 48 elements.

Relations with other polyhedra

The octahedron is the hexahedron (cube ) dual polyhedron (and vice versa).

If, on the sides of the octahedron tetrahedron, there arises a star tetrahedron.

Use of octahedron and cube numerous body can be constructed which also have the cube group as a symmetry group. Thus, for example, receives

• The truncated octahedron with 8 hexagons and 6 squares
• The cuboctahedron with 8 triangles and 6 squares, ie, with 14 faces and 12 vertices
• The truncated cube with 8 triangles and 6 octagons

As averages of an octahedron with a cube (see Archimedes' body) and

• The rhombic dodecahedron ( 8 6 = 14 vertices and 12 rhombi as faces)

As a convex hull of a union of an octahedron with a cube.

Formulas

Generalization

The analogs of the octahedron are in arbitrary dimension n called the (n- dimensional ) cross-polytopes and are also regular polytopes. The n- dimensional cross-polytope has corners and is of (n -1 )-dimensional simplexes limited ( as facets ). The four-dimensional cross-polytope has 8 corners, 24 edges of equal length, 32 equilateral triangles as faces and 16 tetrahedra as facets. ( The one-dimensional cross-polytope is a stretch, the two-dimensional cross-polytope is the square. )

A model for the n-dimensional cross-polytope is the unit sphere with respect to the L1 norm

In the vector space Rn. Namely, the (closed ) cross-polytope therefore

• The amount

• The convex hull of the 2n vertices, wherein the unit vectors.
• The average of the 2 n half-spaces defined by the hyperplanes of the mold

The volume of the n-dimensional Kreuzpolytops is where r > 0 is the radius of the sphere around the origin with respect to the L1 norm. The relationship can be proved by recursion and the Fubini's theorem.

Applications

In chemistry can in the prediction of molecular geometries according to the VSEPR model resulting octahedral molecules. In the crystal structures, such as the face-centered cubic sodium chloride structure ( coordination number of 6), dipping the octahedra in the unit cell; just as in coordination chemistry, if encamp 6 ligands around a central atom.

Some naturally occurring minerals, such as alum, crystallize in octahedral form.

In role-playing games octahedral dice are used and referred to therein as " W8 ", ie a cube with eight faces.