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.

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 $r\{3,3\}$ .

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.

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 line segments (1D)
8 triangles (2D)
1 octahedron (3D)

Radii[]

Vertex radius: $\frac{\sqrt{2}{2}}l$
Edge radius: $1/2l$
Face radius: $\frac{\sqrt{6}{6}}l$

Angles[]

Dihedral angle: $\arccos(\frac{-1}{3})$

Equations[]

The surface of an octahedron with side

$\sqrt{2}$ can be given by

$|x|+|y|+|z| = 1$

Vertex coordinates[]

The coordinates of an octahedron of side

$\sqrt{2}$ are all permuations of (±1,0,0).

Related shapes[]

Dual: Cube
Vertex figure: Square, side length 1
Diminishings/Segments: Square pyramid (vertex first)
Regiment members: 2 (other member: Tetrahemihexahedron)

See also[]

Polytope Wiki. "Octahedron".
Bowers, Jonathan. "Polyhedraon Category 1: Regulars".

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 $\{3^{n-1}\}$	Null polytope $)($ $\emptyset$	Point $()$ $\mathbb{B}^0$	Line segment $\{\}$ $\mathbb{B}^1$	Triangle $\{3\}$	Tetrahedron $\{3^2\}$	Pentachoron $\{3^3\}$	Hexateron $\{3^4\}$	Heptapeton $\{3^5\}$	Octaexon $\{3^6\}$	Enneazetton $\{3^7\}$	Decayotton $\{3^8\}$	Hendecaxennon $\{3^9\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	...	Omegasimplex
Cross $\{3^{n-2},4\}$				Square $\{4\}$	Octahedron $\{3, 4\}$	Hexadecachoron $\{3^2, 4\}$	Pentacross $\{3^3, 4\}$	Hexacross $\{3^4, 4\}$	Heptacross $\{3^5, 4\}$	Octacross $\{3^6, 4\}$	Enneacross $\{3^7, 4\}$	Dekacross $\{3^8, 4\}$	Hendekacross $\{3^9, 4\}$	Dodekacross $\{3^{10}, 4\}$	Tridekacross $\{3^{11}, 4\}$	Tetradekacross $\{3^{12}, 4\}$	Pentadekacross $\{3^{13}, 4\}$	Hexadekacross $\{3^{14}, 4\}$	...	Omegacross
Hydrotopes $\{3^{n-2}, 5\}$				Pentagon $\{5\}$	Icosahedron $\{3, 5\}$	Hexacosichoron $\{3^2, 5\}$	Order-5 pentachoric tetracomb $\{3^3, 5\}$	Order-5 hexateric pentacomb $\{3^4, 5\}$	...
Hypercube $\{4, 3^{n-2}\}$				Square $\{4\}$	Cube $\{4,3\}$	Tesseract $\{4, 3^2\}$	Penteract $\{4, 3^3\}$	Hexeract $\{4, 3^4\}$	Hepteract $\{4, 3^5\}$	Octeract $\{4, 3^6\}$	Enneract $\{4, 3^7\}$	Dekeract $\{4, 3^8\}$	Hendekeract $\{4, 3^9\}$	Dodekeract $\{4, 3^{10}\}$	Tridekeract $\{4, 3^{11}\}$	Tetradekeract $\{4, 3^{12}\}$	Pentadekeract $\{4, 3^{13}\}$	Hexadekeract $\{4, 3^{14}\}$	...	Omegeract
Cosmotopes $\{5, 3^{n-2}\}$				Pentagon $\{5\}$	Dodecahedron $\{5, 3\}$	Hecatonicosachoron $\{5, 3^2\}$	Order-3 hecatonicosachoric tetracomb $\{5, 3^3\}$	Order-3-3 hecatonicosachoric pentacomb $\{5, 3^4\}$	...
Hyperball $\mathbb B^n$				Disk $\mathbb B^2$	Ball $\mathbb B^3$	Gongol $\mathbb B^4$	Pentorb $\mathbb B^5$	Hexorb $\mathbb B^6$	Heptorb $\mathbb B^7$	Octorb $\mathbb B^8$	Enneorb $\mathbb B^9$	Dekorb $\mathbb B^{10}$	Hendekorb $\mathbb B^{11}$	Dodekorb $\mathbb B^{12}$	Tridekorb $\mathbb B^{13}$	Tetradekorb $\mathbb B^{14}$	Pentadekorb $\mathbb B^{15}$	Hexadekorb $\mathbb B^{16}$	...	Omegaball $\mathbb B^{\aleph_0}$


$\{2,4\}$	$\{3, 4\}$	$\{4,4\}$	$\{5,4\}$	$\{6,4\}$	$\{7,4\}$	$\{8,4\}$	...	$\{\infty,4\}$	$\{\frac{\pi i}{\lambda},4\}$
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


$\{3,2\}$	$\{3, 3\}$	$\{3, 4\}$	$\{3, 5\}$	$\{3,6\}$	$\{3,7\}$	$\{3,8\}$	...	$\{3,\infty\}$	$\{3,\frac{\pi i}{\lambda}\}$
Trigonal dihedron	Tetrahedron	Octahedron	Icosahedron	Triangular tiling	Order-7 triangular tiling	Order-8 triangular tiling	...	Infinite-order triangular tiling	Imaginary-order triangular tiling


Regular $t_0 \{4,3\}$	Rectified $t_1 \{4,3\}$	Birectified $t_2 \{4,3\}$	Truncated $t_{0,1} \{4,3\}$	Bitruncated $t_{1,2} \{4,3\}$	Cantellated $t_{0,2} \{4,3\}$	Cantitruncated $t_{0,1,2} \{4,3\}$
Cube	Cuboctahedron	Octahedron	Truncated cube	Truncated octahedron	Rhombicuboctahedron	Great rhombicuboctahedron


Regular $t_0 \{3,3\}$	Rectified $t_1 \{3,3\}$	Birectified $t_2 \{3,3\}$	Truncated $t_{0,1} \{3,3\}$	Bitruncated $t_{1,2} \{3,3\}$	Cantellated $t_{0,2} \{3,3\}$	Cantitruncated $t_{0,1,2} \{3,3\}$
Tetrahedron	Octahedron	Tetrahedron	Truncated tetrahedron	Truncated tetrahedron	Cuboctahedron	Truncated octahedron


$t_1 \{2,3\}$	$t_1 \{3,3\}$	$t_1 \{4,3\}$	$t_1 \{5,3\}$	$t_1 \{6,3\}$	$t_1 \{7,3\}$	$t_1 \{8,3\}$	...	$t_1 \{\infty,3\}$	$t_1 \{\frac{i\pi}{\lambda},3\}$
Trigonal dihedron	Octahedron	Cuboctahedron	Icosidodecahedron	Trihexagonal tiling	Triheptagonal tiling	Trioctagonal tiling	...	Triapeirogonal tiling	Tripseudogonal tiling