Octahedron

An Octahedron is the three-dimensional cross polytope. It is the dual of a cube - in other words, it is a cube where all vertices have been replaced with faces and all faces have been replaced with vertices. It is also the triangular antiprism, a fact that can clearly be seen from its graph. In addition it is a square bipyramid, meaning it is composed of two square pyramids stuck together at their bases. The dihedral angle of an octahedron is the arcos(-13). The octahedron has four triangle meeting at a vertex.

Other Names
Jonathan Bowers calls the octahedron an Oct, a shortened form of its name. The Higher Dimensions wiki sometimes calls this shape an Aerohedron, from aero- referring to a cross polytope and -hedron referring to three dimensions.

It can also be called a rectified tetrahedron or tetratetrahedron, as it can be made by truncating a tetrahedron to the midpoints of the edges. In this form it can be represented as $$r\{3,3\}$$.

Structure
The octahedron is composed of 8 triangles, with four of them meeting at each vertex. It is formed of two suare pyramids joined together. It is also a triangular antiprism, composed of two triangles, in dual orientations, joined by a band of 6 triangles.

Hypervolumes

 * vertex count = $$6$$
 * edge length = $$12l$$
 * surface area = $$2\sqrt { 3 } { l }^{ 2 }$$
 * surcell volume = $$\frac { \sqrt { 2 } }{ 3 } { l }^{ 3 }$$

Subfacets

 * 6 points (0D)
 * 12 lines segments (1D)
 * 8 triangles (2D)

For more information on the shape:
Go clicc the lincc