# Dictionary Definition

octahedron n : any polyhedron having eight plane
faces [also: octahedra
(pl)]

# User Contributed Dictionary

## English

### Noun

- a polyhedron with eight faces; the regular octahedron has regular triangles as faces and is one of the Platonic solids.

#### Translations

- Finnish: oktaedri
- French: octaèdre
- German: Oktaeder
- Hungarian: oktaéder
- Polish: ośmiościan foremny , oktaedr
- Portuguese: octaedro
- Swedish: oktaeder

#### Related terms

# Extensive Definition

An octahedron (plural: octahedra) is a polyhedron with eight faces.
A regular octahedron is a Platonic
solid composed of eight equilateral
triangles, four of which meet at each vertex.

The octahedron's symmetry
group is Oh, of order 48. This group's subgroups include D3d (order
12), the symmetry group of a triangular antiprism; D4h (order 16), the
symmetry group of a square bipyramid; and Td (order 24),
the symmetry group of a rectified
tetrahedron. These
symmetries can be emphasized by different decorations of the
faces.

It is a 3-dimensional cross
polytope.

## Cartesian coordinates

An octahedron can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then- ( ±1, 0, 0 );
- ( 0, ±1, 0 );
- ( 0, 0, ±1 ).

## Area and volume

The area A and the volume V of a regular octahedron of edge length a are:- A=2\sqrta^2 \approx 3.46410162a^2
- V=\frac \sqrta^3 \approx 0.471404521a^3

Thus the volume is four times that of a regular
tetrahedron with the
same edge length, while the surface area is twice (because we have
8 vs. 4 triangles).

## Geometric relations

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound.
Octahedra and tetrahedra can be alternated to form a vertex,
edge, and face-uniform tessellation
of space, called the octet truss
by Buckminster
Fuller. This is the only such tiling save the regular
tessellation of cubes, and
is one of the 28 convex
uniform honeycombs. Another is a tessellation of octahedra and
cuboctahedra.

The octahedron is unique among the Platonic
solids in having an even number of faces meeting at each vertex.
Consequently, it is the only member of that group to possess mirror
planes that do not pass through any of the faces.

Using the standard nomenclature for Johnson
solids, an octahedron would be called a square bipyramid.

## Related polyhedra

The octahedron can also be considered a rectified
tetrahedron - and can be called a tetratetrahedron. This can be
shown by a 2-color face model. With this coloring, the octahedron
has tetrahedral
symmetry.

Compare this truncation sequence between a
tetrahedron and its dual:

## Octahedra in the physical world

- Especially in roleplaying games, this solid is known as a d8, one of the more common Polyhedral dice.

## Octahedra in music

If you place notes on every vertex of an
octahedron, you can get a six note just intonation scale with
remarkable properties - it is highly symmetrical and has eight
consonant triads and twelve consonant diads. See hexany

## Other octahedra

The regular octahedron has 6 vertices and 12
edges, the minimum for an octahedron; nonregular octahedra may have
as many as 12 vertices and 18 edges.http://www.uwgb.edu/dutchs/symmetry/polynum0.htm

- Hexagonal prism: 6 squares, 2 hexagons
- Heptagonal pyramid: 7 triangles, 1 heptagon
- Tetragonal bipyramid: 8 triangles,
usually isosceles)
- The regular octahedron is a special case with equilateral triangles

- Truncated tetrahedron: 4 triangles, 4 hexagons
- Tetragonal trapezohedron - 8 kites

## See also

## References

## External links

- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Paper Models of Polyhedra Many links
- Octahedron - Jewellery Software

octahedron in Azerbaijani: Oktaedr

octahedron in Catalan: Octàedre

octahedron in Czech: Osmistěn

octahedron in Danish: Oktaeder

octahedron in German: Oktaeder

octahedron in Estonian: Korrapärane
oktaeeder

octahedron in Spanish: Octaedro

octahedron in Esperanto: Okedro

octahedron in Basque: Oktaedro

octahedron in French: Octaèdre

octahedron in Hebrew: תמניון

octahedron in Korean: 정팔면체

octahedron in Italian: Ottaedro

octahedron in Latvian: Oktaedrs

octahedron in Dutch: Octaëder

octahedron in Japanese: 正八面体

octahedron in Norwegian: Oktaeder

octahedron in Polish: Ośmiościan foremny

octahedron in Portuguese: Octaedro

octahedron in Russian: Октаэдр

octahedron in Simple English: Octahedron

octahedron in Serbian: Октаедар

octahedron in Finnish: Oktaedri

octahedron in Swedish: Oktaeder

octahedron in Tamil: எண்முக முக்கோணகம்

octahedron in Thai: ทรงแปดหน้า

octahedron in Chinese: 正八面體