 Octahedron

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 symbol {3,4} Wythoff symbol 4  2 3 CoxeterDynkin Symmetry O_{h}, [4,3], (*432) References U_{05}, C_{17}, W_{2} Properties Regular convex deltahedron Dihedral angle 109.47122° = arccos(1/3)
3.3.3.3
(Vertex figure)
Cube
(dual polyhedron)
NetIn 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.
An octahedron is the threedimensional case of the more general concept of a cross polytope.
Contents
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
and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is
while the midradius, which touches the middle of each edge, is
Cartesian coordinates
Orthographic projections An octahedron 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 ).
Area and volume
The surface area A and the volume V of a regular octahedron of edge length a are:
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).
Geometric relations
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 faceuniform 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 4connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4connected simplicial wellcovered 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 CoxeterDynkin Schläfli symbol {3,4} t_{1}{3,3} s{3,2} Wythoff symbol 4  3 2 2  4 3 2  6 2 Symmetry O_{h}
[4,3]
(*432)T_{d}
[3,3]
(*332)D_{3d}
[2^{+},6]
(2*3)D_{4h}
[2,3]
(*322)Symmetry order 48 24 12 16 Image
(uniform coloring)
(1111)
(1212)
(1112)Dual
The octahedron is the dual polyhedron to the cube.
Nets
It has eleven arrangements of nets.
This example depicts it as both a dipyramid and an antiprism:
Related polyhedra
Tetratetrahedron
The regular octahedron can also be considered a rectified tetrahedron  and can be called a tetratetrahedron. This can be shown by a 2color face model. With this coloring, the octahedron has tetrahedral symmetry.
Compare this truncation sequence between a tetrahedron and its dual:
Tetrahedron
Truncated tetrahedron
Octahedron
Truncated tetrahedron
TetrahedronThe 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 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).
This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,9}Name tetrahedron rectified tetrahedron
(octahedron)truncated tetrahedron cantellated tetrahedron
(cuboctahedron)omnitruncated tetrahedron
(truncated octahedron)Snub tetrahedron
(icosahedron)Schläfli {3,3} t_{1}{3,3} t_{0,1}{3,3} t_{0,2}{3,3} t_{0,1,2}{3,3} s{3,3} CoxeterDynkin Graph
(A_{3})Graph
(A_{2})Solid Tiling 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
TetrahemihexahedronOctahedra in the physical world
 Especially in roleplaying games, this solid is known as a "d8", one of the more common noncubical 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.^{[2]}
 Natural crystals of diamond, alum or fluorite are commonly octahedral, as the space=filling tetrahedraloctahedral 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.
Octahedra in music
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.
Other octahedra
The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond oneforone 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.
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; nonregular octahedra may have as many as 12 vertices and 18 edges.[1] Other nonregular 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.
See also
 Spinning octahedron
 Stella octangula
 Triakis octahedron
 Hexakis octahedron
 Truncated octahedron
 Octahedral molecular geometry
 Octahedral symmetry
 Octahedral graph
References
 ^ Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010). "On wellcovered triangulations. III". Discrete Applied Mathematics 158 (8): 894–912. doi:10.1016/j.dam.2009.08.002. MR2602814.
 ^ Klein, Douglas J. (2002). "ResistanceDistance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. http://jagor.srce.hr/ccacaa/CCAPDF/cca2002/v75n2/CCA_75_2002_633_649_KLEIN.pdf. Retrieved 20060930.
External links
 Weisstein, Eric W., "Octahedron" from 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 SemiRegular Polyhedra
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
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 Tetrahedron
 Pentahedron
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 Dodecahedron
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 Icosahedron
Polyhedron navigator Platonic solids (regular) Archimedean solids
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(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform Dihedral others Degenerate polyhedra are in italics.Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: Deltahedra
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