Octahedron

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For the album by The Mars Volta, see Octahedron (album).

Regular Octahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F = 8, E = 12 V = 6 (χ = 2)
Faces by sides	8{3}
Schläfli symbols	{3,4}
Schläfli symbols	r{3,3}
Wythoff symbol	4 \| 2 3
Coxeter diagram
Symmetry	O_h, BC₃, [4,3], (*432)
Rotation group	O, [4,3]⁺, (432)
References	U₀₅, C₁₇, W₂
Properties	Regular convex deltahedron
Dihedral angle	109.47122° = arccos(-1/3)
3.3.3.3 (Vertex figure)	Cube (dual polyhedron)
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

Dimensions

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

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 by	Edge	Face Normal	Vertex	Face
Image
Projective symmetry	[2]	[2]	[4]	[6]

Cartesian coordinates

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 x–y–z 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

Area and volume

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

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

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 O_h, of order 48, the three dimensional hyperoctahedral group. This group's subgroups include D_3d (order 12), the symmetry group of a triangular antiprism; D_4h (order 16), the symmetry group of a square bipyramid; and T_d (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different colorings of the faces.

Name	Octahedron	Rectified tetrahedron (Tetratetrahedron)	Triangular antiprism	Square bipyramid	Rhombic fusil
Image (Face coloring)	(1111)	(1212)	(1112)	(1111)
Coxeter-Dynkin
Schläfli symbol	{3,4}	t₁{3,3}	s{2,6} sr{2,3}	fs{2,4} { } + {4}	fsr{2,2} { } + { } + { }
Wythoff symbol	4 \| 3 2	2 \| 4 3	2 \| 6 2 \| 2 3 2
Symmetry	O_h, [4,3], (*432)	T_d, [3,3], (*332)	D_3d, [2⁺,6], (2*3) D₃, [2,3]⁺, (322)	D_4h, [2,4], (*422)	D_2h, [2,2], (*222)
Symmetry order	48	24	12 6	16	8

Dual

The octahedron is the dual polyhedron to the cube.

Nets

It has eleven arrangements of nets.

Irregular octahedra

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.

Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles.
Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.
Schönhardt polyhedron, a nonconvex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.

Other convex octahedra

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:

Hexagonal prism: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
Heptagonal pyramid: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). It is not possible for all triangular faces to be equilateral.
Truncated tetrahedron: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated.
Tetragonal trapezohedron: The eight faces are congruent kites.

Related polyhedra

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.

tetrahedron	stellated 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{3^1,1}	t{3,4} t{3^1,1}	{3,4} {3^1,1}	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{3^1,1}

		=	=	=				= or	= or	=

Duals to uniform polyhedra
V4³	V3.8²	V(3.4)²	V4.6²	V3⁴	V3.4³	V4.6.8	V3⁴.4	V3³	V3.6²	V3⁵

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

{3,2}	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,9}	...	(3,∞}

Tetratetrahedron

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)


{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


V3.3.3	V3.6.6	V3.3.3.3	V3.6.6	V3.3.3	V3.4.3.4	V4.6.6	V3.3.3.3.3

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]	Spherical			Euclidean	Hyperbolic tiling
Symmetry *n32 [n,3]	*332 [3,3] T_d	*432 [4,3] O_h	*532 [5,3] I_h	*632 [6,3] p6m	*732 [7,3]	*832 [8,3]	*∞32 [∞,3]
Quasiregular figures configuration	3.3.3.3	3.4.3.4	3.5.3.5	3.6.3.6	3.7.3.7	3.8.3.8	3.∞.3.∞
Coxeter diagram
Dual (rhombic) figures configuration	V3.3.3.3	V3.4.3.4	V3.5.3.5	V3.6.3.6	V3.7.3.7	V3.8.3.8	V3.∞.3.∞
Coxeter diagram

Trigonal antiprism

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)


{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


V6²	V12²	V6²	V4.4.6	V2⁶	V4.4.6	V4.4.12	V3.3.3.6	V3²	V3.3.3.3

Family of uniform antiprisms
2	3	4	5	6	7	8	9	10	11	12	n
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}


As spherical polyhedra

Square bipyramid

Family of bipyramids
2	3	4	5	6	7	8	9	10	11	12	...	∞


As spherical polyhedra

Tetrahemihexahedron

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	Tetrahemihexahedron

Tetrahedral Truss

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

Octahedra in nature

Fluorite octahedron.

Natural crystals of diamond, alum or fluorite are commonly octahedral, as the space-filling tetrahedral-octahedral honeycomb.
The plates of kamacite alloy in octahedrite meteorites are arranged paralleling the eight faces of an octahedron.
Many metal ions coordinate six ligands in an octahedral or distorted octahedral configuration.
Widmanstätten patterns in nickel-iron crystals

Octahedra in art and culture

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

Especially in roleplaying games, this solid is known as a "d8", one of the more common non-cubical dice.
If each edge of an octahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.^[5]
Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see hexany.

References

↑ 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.
↑
↑ Counting polyhedra
↑ http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm
↑ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved 2006-09-30.

External links

Wikisource has the text of the 1911 Encyclopædia Britannica article Octahedron.

Weisstein, Eric W., "Octahedron", MathWorld.
Richard Klitzing, 3D convex uniform polyhedra, x3o4o - oct
Editable printable net of an octahedron with interactive 3D view
Paper model of the octahedron
K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra

v t e Polyhedra

Listed by number of faces

1–10 faces	Henahedron Dihedron Trihedron Tetrahedron Pentahedron Hexahedron Heptahedron Octahedron Enneahedron Decahedron

11–20 faces	Hendecahedron Dodecahedron Tridecahedron Tetradecahedron Pentadecahedron Hexadecahedron Heptadecahedron Octadecahedron Enneadecahedron Icosahedron

Others	Apeirohedron

Polyhedra

Platonic solids (regular)

Archimedean solids
(semiregular / uniform)

Catalan solids
(dual semiregular)

Dihedral regular

dihedron
hosohedron

Dihedral uniform

prisms antiprisms

Duals	bipyramids trapezohedra

Dihedral others

Degenerate polyhedra are in italics.

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	BC_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes

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