AskDefine | Define octahedron

Dictionary Definition

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

User Contributed Dictionary



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


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

  • If each edge of an octahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.

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.


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
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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
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