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 = $12 l$
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:[]

[1]Go clicc the lincc

See also[]


	Zeroth	First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
Simplex	Point	Line	Triangle	Tetrahedron	Pentachoron	Hexateron	Heptapeton	Octaexon	Enneazetton	Decayotton	Hendecaxennon
Hypercube	Point	Line	Square	Cube	Tesseract	Penteract	Hexeract	Hepteract	Octeract	Enneract	Dekeract
Cross	Point	Line	Square	Octahedron	Hexadecachoron	Pentarss	Hexarss	Heptarss	Octarss	Ennearss	Decarss
Hypersphere	Point	Line	Circle	Sphere	Glome	Hyperglome	Hexaphere	Heptaphere	Octaphere	Enneaphere	Decaphere