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 line segments (1D)
- 8 triangles (2D)

### See also

Zeroth | First | Second | Third | Fourth | Fifth | Sixth | Seventh | Eighth | Ninth | Tenth | Eleventh | Twelfth | Thirteenth | Fourteenth | Fifteenth | Sixteenth | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Simplex | Point | Line | Triangle | Tetrahedron | Pentachoron | Hexateron | Heptapeton | Octaexon | Enneazetton | Decayotton | Hendecaxennon | Dodecadakon | Tredecahendakon | Quattuordecadokon | Quindecatradakon | Sexdecateradakon | Septendecapetadakon |

Hypercube | Point | Line | Square | Cube | Tesseract | Penteract | Hexeract | Hepteract | Octeract | Enneract | Dekeract | Undekeract | Dodekeract | Tredekeract | Quattuordekeract | Quindekeract | Sexdekeract |

Cross | Point | Line | Square | Octahedron
| Hexadecachoron | Pentarss | Hexarss | Heptarss | Octarss | Ennearss | Decarss | Hendecarss | Dodecarss | Tredecarss | Quattuordecarss | Quindecatrarss | Sexdecaterarss |

Hypersphere | Point | Line | Circle | Sphere | Glome | Hyperglome | Hexaphere | Heptaphere | Octaphere | Enneaphere | Decaphere | Hendecaphere | Dodecaphere | Tredecaphere | Quattuordecaphere | Quindecatraphere | Sexdecateraphere |