Regular Octahedron  

(Click here for rotating model) 

Type  Platonic solid 
Elements  F = 8, E = 12 V = 6 (χ = 2) 
Faces by sides  8{3} 
Schläfli symbol  {3,4} 
Wythoff symbol  4  2 3 
CoxeterDynkin  
Symmetry  O_{h} or (*432) 
References  U_{05}, C_{17}, W_{2} 
Properties  Regular convex deltahedron 
Dihedral angle  109.47122° = arccos(1/3) 
3.3.3.3 (Vertex figure) 
Cube (dual polyhedron) 
Net 
In geometry, 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.
An octahedron is the threedimensional case of the more general concept of a cross polytope.
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If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is
and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is
while the midradius, which touches the middle of each edge, is
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
The area A and the volume V of a regular octahedron of edge length a are:
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).
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 faceuniform 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.
There are 3 uniform colorings of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.
The octahedron's symmetry group is O_{h}, of order 48, the three dimensional hyperoctahedral group. This group's subgroups include D_{3d} (order 12), the symmetry group of a triangular antiprism; D_{4h} (order 16), the symmetry group of a square bipyramid; and T_{d} (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different colorings of the faces.
Name  Octahedron  Rectified tetrahedron  Triangular antiprism  Square bipyramid 

CoxeterDynkin  
Schläfli symbol  {3,4}  t_{1}{3,3}  s{3,2}  
Wythoff symbol  4  3 2  2  4 3  
Symmetry  O_{h} (*432) 
T_{d} (*332) 
D_{3d} (2*3) 
D_{4h} (*322) 
Symmetry order  48  24  12  16 
Image (uniform coloring) 
(1111) 
(1212) 
(1112) 
The octahedron is the dual polyhedron to the cube. If the original octahedron has edge length 1, its dual cube has edge length .
It has eleven arrangements of nets.
For example:
The regular octahedron can also be considered a rectified tetrahedron  and can be called a tetratetrahedron. This can be shown by a 2color face model. With this coloring, the octahedron has tetrahedral symmetry.
Compare this truncation sequence between a tetrahedron and its dual:
Tetrahedron 
Truncated tetrahedron 
octahedron 
Truncated tetrahedron 
Tetrahedron 
The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract. If this diagonal is oriented vertically with a height of 1, then the five slices above occur at heights r, 3/8, 1/2, 5/8, and s, where r is any number in the range (0,1/4], and s is any number in the range [3/4,1).
The regular octahedron shares its edges and vertex arrangement with one nonconvex uniform polyhedra: the tetrahemihexahedron, with which it shares four of the triangular faces.
Octahedron 
Tetrahemihexahedron 
Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see hexany.
The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond oneforone with the features of a regular octahedron.
More generally, an octahedron can be any polyhedron with eight faces. 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.[1] Other nonregular octahedra include the following.



Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
OCTAHEDRON (Gr. 6KT6, eight, Spa, base), a solid bounded by eight triangular faces; it has 6 vertices and 12 edges. The regular octahedron has for its faces equilateral triangles; it is the reciprocal of the cube. Octahedra having triangular faces other than equilateral occur as crystal forms. See Polyhedron and Crystallography.
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An octahedron is a polyhedron (a 3D shape) with eight sides, which are all equilateral triangles, four of which meet at each corner. It has 12 edges and 6 corners. It is one of the 5 Platonic solids.
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