1,176
pages

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.

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 .

Symbols

The octahedron's Dynkin based symbols include:

• o4o3x (regular)
• o3x3o (rectified tetrahedron)
• s2s6o (triangular antiprism)
• s2s3s
• qo ox4oo&#zx (square dipyramid)
• qo ox ox&#zx (recangular dipyramid)
• qoo oqo ooq&#zx (rhombic dipyramid)
• oxo4ooo&#xt (join of two square pyramids)
• oxo oxo&#xt (two rectangle pyramids)
• xo3ox&#x (triangle antipodium)
• xox oqo&#xt (trapezoid bipyramid)
• oxox&#xr

Structure

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

Sections

When seen vertex first, it looks like a point expanding to a square of the same side as its edge length, then shrinking back. When seen face first, it appears as a triangle that expands through various hexagons, then back to a triangle in dual orientation.

Angles

• Dihedral angle:

Equations

The surface of an octahedron with side

can be given by

Vertex coordinates

The coordinates of an octahedron of side

are all permuations of (±1,0,0).

Related shapes

Regular polyhedra
Convex regular polyhedra: tet · cube · oct · doe · ike

Self-intersecting regular polyhedra: gad · sissid · gike · gissid

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

Null polytope

Point

Line segment

Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Tridecahendon

... Omegasimplex
Cross

Square

Octahedron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

... Omegaball

...
Square hosohedron Octahedron Square tiling Order-4 pentagonal tiling Order-4 hexagonal tiling Order-4 heptagonal tiling Order-4 octagonal tiling ... Order-4 apeirogonal tiling Order-4 pseudogonal tiling
Regular
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
Cube Cuboctahedron Octahedron Truncated cube Truncated octahedron Rhombicuboctahedron Great rhombicuboctahedron
Regular
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
Tetrahedron Octahedron Tetrahedron Truncated tetrahedron Truncated tetrahedron Cuboctahedron Truncated octahedron
...
Trigonal dihedron Octahedron Cuboctahedron Icosidodecahedron Trihexagonal tiling Triheptagonal tiling Trioctagonal tiling ... Triapeirogonal tiling Tripseudogonal tiling