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
}

{\displaystyle 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

{\displaystyle 6}
* edge length =

12
l

{\displaystyle 12 l}
* surface area =

2

3

l

2

{\displaystyle 2\sqrt { 3 } { l }^{ 2 }}
* surcell volume =

2

3

l

3

{\displaystyle \frac { \sqrt { 2 } }{ 3 } { l }^{ 3 }}

### Subfacets

* 6 points (0D)
* 12 line segments (1D)
* 8 triangles (2D)
* 1 octahedron (3D)

### Radii

* Vertex radius:

2

2

l

{\displaystyle \frac{\sqrt{2}{2}}l}
* Edge radius:

1

/

2
l

{\displaystyle 1/2l}
* Face radius:

6

6

l

{\displaystyle \frac{\sqrt{6}{6}}l}

### Angles

* Dihedral angle:

arccos
⁡
(

−
1

3

)

{\displaystyle \arccos(\frac{-1}{3})}

### Equations

The surface of an octahedron with side

2

{\displaystyle \sqrt{2}}

can be given by

x

y

z

=
1

### Vertex coordinates

The coordinates of an octahedron of side

2

{\displaystyle \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".

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

{
2
,
4
}

{\displaystyle \{2,4\}}

{
3
,
4
}

{\displaystyle \{3, 4\}}

{
4
,
4
}

{\displaystyle \{4,4\}}

{
5
,
4
}

{\displaystyle \{5,4\}}

{
6
,
4
}

{\displaystyle \{6,4\}}

{
7
,
4
}

{\displaystyle \{7,4\}}

{
8
,
4
}

{\displaystyle \{8,4\}}

{
∞
,
4
}

{\displaystyle \{\infty,4\}}

{

π
i

λ

,
4
}

{\displaystyle \{\frac{\pi i}{\lambda},4\}}

{
3
,
2
}

{\displaystyle \{3,2\}}

{
3
,
3
}

{\displaystyle \{3, 3\}}

{
3
,
4
}

{\displaystyle \{3, 4\}}

{
3
,
5
}

{\displaystyle \{3, 5\}}

{
3
,
6
}

{\displaystyle \{3,6\}}

{
3
,
7
}

{\displaystyle \{3,7\}}

{
3
,
8
}

{\displaystyle \{3,8\}}

{
3
,
∞
}

{\displaystyle \{3,\infty\}}

{
3
,

π
i

λ

}

{\displaystyle \{3,\frac{\pi i}{\lambda}\}}

t
0

{
4
,
3
}

{\displaystyle t\_0 \{4,3\}}

t
1

{
4
,
3
}

{\displaystyle t\_1 \{4,3\}}

t
2

{
4
,
3
}

{\displaystyle t\_2 \{4,3\}}

t

0
,
1

{
4
,
3
}

{\displaystyle t\_{0,1} \{4,3\}}

t

1
,
2

{
4
,
3
}

{\displaystyle t\_{1,2} \{4,3\}}

t

0
,
2

{
4
,
3
}

{\displaystyle t\_{0,2} \{4,3\}}

t

0
,
1
,
2

{
4
,
3
}

{\displaystyle t\_{0,1,2} \{4,3\}}

t
0

{
3
,
3
}

{\displaystyle t\_0 \{3,3\}}

t
1

{
3
,
3
}

{\displaystyle t\_1 \{3,3\}}

t
2

{
3
,
3
}

{\displaystyle t\_2 \{3,3\}}

t

0
,
1

{
3
,
3
}

{\displaystyle t\_{0,1} \{3,3\}}

t

1
,
2

{
3
,
3
}

{\displaystyle t\_{1,2} \{3,3\}}

t

0
,
2

{
3
,
3
}

{\displaystyle t\_{0,2} \{3,3\}}

t

0
,
1
,
2

{
3
,
3
}

{\displaystyle t\_{0,1,2} \{3,3\}}

t
1

{
2
,
3
}

{\displaystyle t\_1 \{2,3\}}

t
1

{
3
,
3
}

{\displaystyle t\_1 \{3,3\}}

t
1

{
4
,
3
}

{\displaystyle t\_1 \{4,3\}}

t
1

{
5
,
3
}

{\displaystyle t\_1 \{5,3\}}

t
1

{
6
,
3
}

{\displaystyle t\_1 \{6,3\}}

t
1

{
7
,
3
}

{\displaystyle t\_1 \{7,3\}}

t
1

{
8
,
3
}

{\displaystyle t\_1 \{8,3\}}

t
1

{
∞
,
3
}

{\displaystyle t\_1 \{\infty,3\}}

t
1

{

i
π

λ

,
3
}

{\displaystyle t\_1 \{\frac{i\pi}{\lambda},3\}}