Regular octahedron  

(Click here for rotating model)  
Type  Platonic solid 
Elements  F = 8, E = 12 V = 6 (χ = 2) 
Faces by sides  8{3} 
Conway notation  O aT 
Schläfli symbols  {3,4} 
r{3,3} or {}+{}+{}=3{}  
Face configuration  V4.4.4 
Wythoff symbol  4  2 3 
Coxeter diagram  
Symmetry  O_{h}, BC_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
References  U_{05}, C_{17}, W_{2} 
Properties  regular, convexdeltahedron, Hanner polytope 
Dihedral angle  109.47122° = arccos(−1⁄3) 
3.3.3.3 (Vertex figure) 
Cube (dual polyhedron) 
Net 
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.
An octahedron is the threedimensional case of the more general concept of a cross polytope.
A regular octahedron is a 3ball in the Manhattan (ℓ_{1}) metric.
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
The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B_{2} and A_{2} Coxeter planes.
Centered by  Edge  Face Normal 
Vertex  Face 

Image  
Projective symmetry 
[2]  [2]  [4]  [6] 
The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Orthographic projection  Stereographic projection 

An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then
In an x–y–z Cartesian coordinate system, the octahedron with center coordinates (a, b, c) and radius r is the set of all points (x, y, z) such that
The surface 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 rather than 4 triangles).
If an octahedron has been stretched so that it obeys the equation
the formulas for the surface area and volume expand to become
Additionally the inertia tensor of the stretched octahedron is
These reduce to the equations for the regular octahedron when
Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid.
The octahedron is the dual polyhedron of the cube.
If an octahedron of edge length is inscribed in a cube, then the length of an edge of the cube .
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. An icosahedron produced this way is called a snub octahedron.
Octahedra and tetrahedra can be alternated to form a vertex, edge, and faceuniform tessellation of space. This and the regular tessellation of cubes are the only such uniform honeycombs in 3dimensional space.
Like all regular convex polytopes, the octahedron can be dissected into an integral number of disjoint orthoschemes, all of the same shape characteristic of the polytope. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the facets of its orthoscheme. The orthoscheme occurs in two chiral forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a quadrirectangular irregular tetrahedron.
The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of symmetry. 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, among its mirror planes, some that do not pass through any of its faces. The octahedron's symmetry group is denoted B_{3}. The octahedron and its dual polytope, the cube, have the same symmetry group but different characteristic tetrahedra.
The characteristic tetrahedron of the regular octahedron can be found by a canonical dissection^{[1]} of the regular octahedron which subdivides it into 48 of these characteristic orthoschemes surrounding the octahedron's center. Three lefthanded orthoschemes and three righthanded orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a trirectangular tetrahedron: a triangular pyramid with the octahedron face as its equilateral base, and its cubecornered apex at the center of the octahedron.^{[2]}
Characteristics of the regular octahedron^{[3]}  

edge  arc  dihedral  
𝒍  90°  109°28′  
𝟀  54°44′8″  90°  
𝝓  45°  60°  
𝟁  35°15′52″  45°  
If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths , , (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁 of the octahedron), plus , , (edges that are the characteristic radii of the octahedron). The 3edge path along orthogonal edges of the orthoscheme is , , , first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 906030 triangle which is onesixth of an octahedron face. The three faces interior to the octahedron are: a 459045 triangle with edges , , , a right triangle with edges , , , and a right triangle with edges , , .
The octahedron is 4connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4connected simplicial wellcovered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.^{[4]}
The regular octahedron has eleven arrangements of nets.
The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces, and 3 central squares.
Octahedron 
Tetrahemihexahedron 
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 (Tetratetrahedron) 
Triangular antiprism  Square bipyramid  Rhombic fusil 

Image (Face coloring) 
(1111) 
(1212) 
(1112) 
(1111) 
(1111) 
Coxeter diagram  =  

Schläfli symbol  {3,4}  r{3,3}  s{2,6} sr{2,3} 
ft{2,4} { } + {4} 
ftr{2,2} { } + { } + { } 
Wythoff symbol  4  3 2  2  4 3  2  6 2  2 3 2 

Symmetry  O_{h}, [4,3], (*432)  T_{d}, [3,3], (*332)  D_{3d}, [2^{+},6], (2*3) D_{3}, [2,3]^{+}, (322) 
D_{4h}, [2,4], (*422)  D_{2h}, [2,2], (*222) 
Order  48  24  12 6 
16  8 
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; irregular octahedra may have as many as 12 vertices and 18 edges.^{[5]} There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively.^{[6]}^{[7]} (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Some better known irregular octahedra include the following:
A space frame of alternating tetrahedra and halfoctahedra derived from the Tetrahedraloctahedral honeycomb was invented by Buckminster Fuller in the 1950s. It is commonly regarded as the strongest building structure for resisting cantilever stresses.
A regular octahedron can be augmented into a tetrahedron by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the stellated octahedron.
tetrahedron  stellated octahedron 

The octahedron is one of a family of uniform polyhedra related to the cube.
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2)  
{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  




 
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
It is also one of the simplest examples of a hypersimplex, a polytope formed by certain intersections of a hypercube with a hyperplane.
The octahedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
*n32 symmetry mutation of regular tilings: {3,n}  

Spherical  Euclid.  Compact hyper.  Paraco.  Noncompact hyperbolic  
3.3  3^{3}  3^{4}  3^{5}  3^{6}  3^{7}  3^{8}  3^{∞}  3^{12i}  3^{9i}  3^{6i}  3^{3i} 
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:
Family of uniform tetrahedral polyhedra  

Symmetry: [3,3], (*332)  [3,3]^{+}, (332)  
{3,3}  t{3,3}  r{3,3}  t{3,3}  {3,3}  rr{3,3}  tr{3,3}  sr{3,3} 
Duals to uniform polyhedra  
V3.3.3  V3.6.6  V3.3.3.3  V3.6.6  V3.3.3  V3.4.3.4  V4.6.6  V3.3.3.3.3 
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 first five slices above occur at heights r, 3/8, 1/2, 5/8, and s, where r is any number in the range 0 < r ≤ 1/4, and s is any number in the range 3/4 ≤ s < 1.
The octahedron as a tetratetrahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)^{2}, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are Wythoff constructions within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.^{[9]}^{[10]}
*n32 orbifold symmetries of quasiregular tilings: (3.n)^{2}  

Construction 
Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Quasiregular figures 

Vertex  (3.3)^{2}  (3.4)^{2}  (3.5)^{2}  (3.6)^{2}  (3.7)^{2}  (3.8)^{2}  (3.∞)^{2} 
As a trigonal antiprism, the octahedron is related to the hexagonal dihedral symmetry family.
Uniform hexagonal dihedral spherical polyhedra  

Symmetry: [6,2], (*622)  [6,2]^{+}, (622)  [6,2^{+}], (2*3)  
{6,2}  t{6,2}  r{6,2}  t{2,6}  {2,6}  rr{6,2}  tr{6,2}  sr{6,2}  s{2,6}  
Duals to uniforms  
V6^{2}  V12^{2}  V6^{2}  V4.4.6  V2^{6}  V4.4.6  V4.4.12  V3.3.3.6  V3.3.3.3 
Antiprism name  Digonal antiprism  (Trigonal) Triangular antiprism 
(Tetragonal) Square antiprism 
Pentagonal antiprism  Hexagonal antiprism  Heptagonal antiprism  Octagonal antiprism  Enneagonal antiprism  Decagonal antiprism  Hendecagonal antiprism  Dodecagonal antiprism  ...  Apeirogonal antiprism 

Polyhedron image  ...  
Spherical tiling image  Plane tiling image  
Vertex config.  2.3.3.3  3.3.3.3  4.3.3.3  5.3.3.3  6.3.3.3  7.3.3.3  8.3.3.3  9.3.3.3  10.3.3.3  11.3.3.3  12.3.3.3  ...  ∞.3.3.3 
Bipyramid name  Digonal bipyramid  Triangular bipyramid (See: J_{12}) 
Square bipyramid (See: O) 
Pentagonal bipyramid (See: J_{13}) 
Hexagonal bipyramid  Heptagonal bipyramid  Octagonal bipyramid  Enneagonal bipyramid  Decagonal bipyramid  ...  Apeirogonal bipyramid 

Polyhedron image  ...  
Spherical tiling image  Plane tiling image  
Face config.  V2.4.4  V3.4.4  V4.4.4  V5.4.4  V6.4.4  V7.4.4  V8.4.4  V9.4.4  V10.4.4  ...  V∞.4.4 
Coxeter diagram  ... 
Truncation of two opposite vertices results in a square bifrustum.
The octahedron can be generated as the case of a 3D superellipsoid with all exponent values set to 1.