Octahedron

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Regular Octahedron
Octahedron
(Click here for rotating model)
TypePlatonic solid
ElementsF = 8, E = 12
V = 6 (χ = 2)
Faces by sides8{3}
Schläfli symbols{3,4}
r{3,3}
Wythoff symbol4 | 2 3
Coxeter diagramCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
SymmetryOh, BC3, [4,3], (*432)
Rotation groupO, [4,3]+, (432)
ReferencesU05, C17, W2
PropertiesRegular convex deltahedron
Dihedral angle109.47122° = arccos(-1/3)
Octahedron
3.3.3.3
(Vertex figure)
Hexahedron.png
Cube
(dual polyhedron)
Octahedron
Net

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.

An octahedron is the three-dimensional case of the more general concept of a cross polytope.

Regular octahedron[edit]

Dimensions[edit]

If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is

r_u = \frac{a}{2} \sqrt{2} \approx 0.7071067 \cdot a

and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is

r_i = \frac{a}{6} \sqrt{6}  \approx 0.4082482\cdot a

while the midradius, which touches the middle of each edge, is

r_m = \frac{a}{2} = 0.5\cdot a

Orthogonal projections[edit]

The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B2 and A2 Coxeter planes.

Orthogonal projections
Centered byEdgeFace
Normal
VertexFace
ImageCube t2 e.pngCube t2 fb.png3-cube t2 B2.svg3-cube t2.svg
Projective
symmetry
[2][2][4][6]

Cartesian coordinates[edit]

An octahedron with edge length sqrt(2) can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then

( ±1, 0, 0 );
( 0, ±1, 0 );
( 0, 0, ±1 ).

In an xyz Cartesian coordinate system, the octahedron with center coordinates (a, b, c) and radius r is the set of all points (x, y, z) such that

\left|x - a\right| + \left|y - b\right| + \left|z - c\right| = r.

Area and volume[edit]

The surface area A and the volume V of a regular octahedron of edge length a are:

A=2\sqrt{3}a^2 \approx 3.46410162a^2
V=\frac{1}{3} \sqrt{2}a^3 \approx 0.471404521a^3

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

If an octahedron has been stretched so that it obeys the equation:

\left|\frac{x}{x_m}\right|+\left|\frac{y}{y_m}\right|+\left|\frac{z}{z_m}\right| = 1

The formula for the surface area and volume expand to become:

A=4 \, x_m \, y_m \, z_m \times \sqrt{\frac{1}{x_m^2}+\frac{1}{y_m^2}+\frac{1}{z_m^2}}
V=\frac{4}{3}\,x_m\,y_m\,z_m

Additionally the inertia tensor of the stretched octahedron is:

 I = \begin{bmatrix}   \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\   0 & \frac{1}{10} m (x_m^2+z_m^2) & 0 \\    0 & 0 & \frac{1}{10} m (x_m^2+y_m^2) \end{bmatrix}

These reduce to the equations for the regular octahedron when:

x_m=y_m=z_m=a\,\frac{\sqrt{2}}{2}

Geometric relations[edit]

The octahedron represents the central intersection of two tetrahedra

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound.

Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs. Another is a tessellation of octahedra and cuboctahedra.

The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces.

Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid. Truncation of two opposite vertices results in a square bifrustum.

The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[1]

Uniform colorings and symmetry[edit]

There are 3 uniform colorings of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.

The octahedron's symmetry group is Oh, of order 48, the three dimensional hyperoctahedral group. This group's subgroups include D3d (order 12), the symmetry group of a triangular antiprism; D4h (order 16), the symmetry group of a square bipyramid; and Td (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different colorings of the faces.

NameOctahedronRectified tetrahedron
(Tetratetrahedron)
Triangular antiprismSquare bipyramidRhombic fusil
Image
(Face coloring)
Uniform polyhedron-43-t2.png
(1111)
Uniform polyhedron-33-t1.png
(1212)
Trigonal antiprism.png
(1112)
Square bipyramid.png
(1111)
Coxeter-DynkinCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
Schläfli symbol{3,4}t1{3,3}s{2,6}
sr{2,3}
fs{2,4}
{ } + {4}
fsr{2,2}
{ } + { } + { }
Wythoff symbol4 | 3 22 | 4 32 | 6 2
| 2 3 2
SymmetryOh, [4,3], (*432)Td, [3,3], (*332)D3d, [2+,6], (2*3)
D3, [2,3]+, (322)
D4h, [2,4], (*422)D2h, [2,2], (*222)
Symmetry order482412
6
168

Dual[edit]

The octahedron is the dual polyhedron to the cube.

Dual Cube-Octahedron.svg

Nets[edit]

It has eleven arrangements of nets.

Irregular octahedra[edit]

The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron.

Other convex octahedra[edit]

More generally, an octahedron can be any polyhedron with eight faces. The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges.[2] There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively.[3][4] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

Some better known irregular octahedra include the following:

Related polyhedra[edit]

A regular octahedron can be augmented into a tetrahedron by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the stellated octahedron.

Triangulated tetrahedron.pngCompound of two tetrahedra.png
tetrahedronstellated octahedron

The octahedron is one of a family of uniform polyhedra related to the cube.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432)[4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3}t{4,3}r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3}sr{4,3}h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{4,3}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svgUniform polyhedron-43-t01.svgUniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.pngUniform polyhedron-43-s012.pngUniform polyhedron-33-t0.pngUniform polyhedron-33-t2.pngUniform polyhedron-33-t01.pngUniform polyhedron-33-t12.pngUniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43V3.82V(3.4)2V4.62V34V3.43V4.6.8V34.4V33V3.62V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svgTriakisoctahedron.jpgRhombicdodecahedron.jpgTetrakishexahedron.jpgHexahedron.svgDeltoidalicositetrahedron.jpgDisdyakisdodecahedron.jpgPentagonalicositetrahedronccw.jpgTetrahedron.svgTriakistetrahedron.jpgPOV-Ray-Dodecahedron.svg

The octahedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

Trigonal dihedron.png
{3,2}
Uniform polyhedron-33-t2.png
{3,3}
Uniform polyhedron-43-t2.png
{3,4}
Uniform polyhedron-53-t2.png
{3,5}
Uniform polyhedron-63-t2.png
{3,6}
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 39-t0.png
{3,9}
...H2 tiling 23i-4.png
(3,∞}

Tetratetrahedron[edit]

The regular octahedron can also be considered a rectified tetrahedron – and can be called a tetratetrahedron. This can be shown by a 2-color face model. With this coloring, the octahedron has tetrahedral symmetry.

Compare this truncation sequence between a tetrahedron and its dual:

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332)[3,3]+, (332)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-t0.pngUniform polyhedron-33-t01.pngUniform polyhedron-33-t1.pngUniform polyhedron-33-t12.pngUniform polyhedron-33-t2.pngUniform polyhedron-33-t02.pngUniform polyhedron-33-t012.pngUniform polyhedron-33-s012.png
{3,3}t{3,3}r{3,3}t{3,3}{3,3}rr{3,3}tr{3,3}sr{3,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Tetrahedron.svgTriakistetrahedron.jpgHexahedron.svgTriakistetrahedron.jpgTetrahedron.svgRhombicdodecahedron.jpgTetrakishexahedron.jpgPOV-Ray-Dodecahedron.svg
V3.3.3V3.6.6V3.3.3.3V3.6.6V3.3.3V3.4.3.4V4.6.6V3.3.3.3.3

The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights r, 3/8, 1/2, 5/8, and s, where r is any number in the range (0,1/4], and s is any number in the range [3/4,1).

The tetratetrahedron can be seen in a sequence of quasiregular polyhedrons and tilings:

Dimensional family of quasiregular polyhedra and tilings: 3.n.3.n
Symmetry
*n32
[n,3]
SphericalEuclideanHyperbolic tiling
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
Quasiregular
figures
configuration
Uniform tiling 332-t1-1-.png
3.3.3.3
Uniform tiling 432-t1.png
3.4.3.4
Uniform tiling 532-t1.png
3.5.3.5
Uniform tiling 63-t1.png
3.6.3.6
Uniform tiling 73-t1.png
3.7.3.7
Uniform tiling 83-t1.png
3.8.3.8
H2 tiling 23i-2.png
3.∞.3.∞
Coxeter diagramCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Dual
(rhombic)
figures
configuration
Hexahedron.svg
V3.3.3.3
Rhombicdodecahedron.jpg
V3.4.3.4
Rhombictriacontahedron.svg
V3.5.3.5
Rhombic star tiling.png
V3.6.3.6
Order73 qreg rhombic til.png
V3.7.3.7
Uniform dual tiling 433-t01-yellow.png
V3.8.3.8
Ord3infin qreg rhombic til.png
V3.∞.3.∞
Coxeter diagramCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png

Trigonal antiprism[edit]

As a trigonal antiprism, the octahedron is related to the hexagonal dihedral symmetry family.

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)[6,2]+, (622)[1+,6,2], (322)[6,2+], (2*3)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel node h1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Hexagonal dihedron.pngDodecagonal dihedron.pngHexagonal dihedron.pngSpherical hexagonal prism.pngSpherical hexagonal hosohedron.pngSpherical truncated trigonal prism.pngSpherical dodecagonal prism2.pngSpherical hexagonal antiprism.pngTrigonal dihedron.pngSpherical trigonal antiprism.png
{6,2}t{6,2}r{6,2}2t{6,2}=t{2,6}2r{6,2}={2,6}rr{6,2}tr{6,2}sr{6,2}h{6,2}s{2,6}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node f1.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node f1.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel node fh.pngCDel 6.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel node fh.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node fh.pngCDel 2x.pngCDel node fh.png
Spherical hexagonal hosohedron.pngSpherical dodecagonal hosohedron.pngSpherical hexagonal hosohedron.pngSpherical hexagonal bipyramid.pngHexagonal dihedron.pngSpherical hexagonal bipyramid.pngSpherical dodecagonal bipyramid.pngSpherical hexagonal trapezohedron.pngSpherical trigonal hosohedron.pngSpherical trigonal trapezohedron.png
V62V122V62V4.4.6V26V4.4.6V4.4.12V3.3.3.6V32V3.3.3.3
Family of uniform antiprisms
23456789101112n
s{2,4}
sr{2,2}
s{2,6}
sr{2,3}
s{2,8}
sr{2,4}
s{2,10}
sr{2,5}
s{2,12}
sr{2,6}
s{2,14}
sr{2,7}
s{2,16}
sr{2,8}
s{2,18}
sr{2,9}
s{2,20}
sr{2,10}
s{2,22}
sr{2,11}
s{2,24}
sr{2,12}
s{2,2n}
sr{2,n}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 0x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 5.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 2x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 7.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 9.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 0x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 10.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 2x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 11.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 12.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel n.pngCDel node h.png
Digonal antiprism.pngTrigonal antiprism.pngSquare antiprism.pngPentagonal antiprism.pngHexagonal antiprism.pngAntiprism 7.pngOctagonal antiprism.pngEnneagonal antiprism.pngDecagonal antiprism.pngHendecagonal antiprism.pngDodecagonal antiprism.png
As spherical polyhedra
Spherical digonal antiprism.pngSpherical trigonal antiprism.pngSpherical square antiprism.pngSpherical pentagonal antiprism.pngSpherical hexagonal antiprism.pngSpherical heptagonal antiprism.pngSpherical octagonal antiprism.png

Square bipyramid[edit]

Family of bipyramids
23456789101112...
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 7.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 9.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 1x.pngCDel 0x.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 1x.pngCDel 1x.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 1x.pngCDel 2x.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel infin.pngCDel node.png
Triangular bipyramid.pngSquare bipyramid.pngPentagonale bipiramide.pngHexagonale bipiramide.pngOctagonal bipyramid.pngDecagonal bipyramid.pngBicone.svg
As spherical polyhedra
Spherical digonal bipyramid.pngSpherical trigonal bipyramid.pngSpherical square bipyramid.pngSpherical pentagonal bipyramid.pngSpherical hexagonal bipyramid.pngSpherical heptagonal bipyramid.pngSpherical octagonal bipyramid.pngSpherical enneagonal bipyramid.pngSpherical decagonal bipyramid.pngSpherical hendecagonal bipyramid.pngSpherical dodecagonal bipyramid.png

Tetrahemihexahedron[edit]

The regular octahedron shares its edges and vertex arrangement with one nonconvex uniform polyhedron: the tetrahemihexahedron, with which it shares four of the triangular faces.

Octahedron.png
Octahedron
Tetrahemihexahedron.png
Tetrahemihexahedron

Tetrahedral Truss[edit]

A framework of repeating tetrahedrons and octahedrons was invented by Buckminster Fuller in the 1950s, known as a space frame, commonly regarded as the strongest structure for resisting cantilever stresses.

Octahedra in the physical world[edit]

Octahedra in nature[edit]

Fluorite octahedron.

Octahedra in art and culture[edit]

Two identically formed rubik's snakes can approximate an octahedron.

See also[edit]

References[edit]

  1. ^ Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010). "On well-covered triangulations. III". Discrete Applied Mathematics 158 (8): 894–912. doi:10.1016/j.dam.2009.08.002. MR 2602814. 
  2. ^ [1]
  3. ^ Counting polyhedra
  4. ^ http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm
  5. ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved 2006-09-30. 

External links[edit]