Editing Octahedron

{{Short description|Polyhedron with eight triangular faces}}
{{distinguish|Octahedron (album){{!}}''Octahedron'' (album)}}
{{Use dmy dates|date=January 2020}}
{{split|date=August 2024|Square bipyramid|discuss=Talk:Octahedron#Create a square bipyramid or regular octahedron article}}

In [[geometry]], an '''octahedron''' ({{plural form}}: '''octahedra''' or '''octahedrons''') is a [[polyhedron]] with eight faces. One special case is the ''regular octahedron'', a [[Platonic solid]] composed of eight [[equilateral triangle]]s, four of which meet at each vertex. Regular octahedra occur in nature as [[crystal]] structures. Many types of irregular octahedra also exist, including both [[convex set|convex]] and non-convex shapes.

A regular octahedron is the three-dimensional case of the more general concept of a [[cross-polytope]].

== Regular octahedron ==
{{multiple image
 | image1 = Octahedron.jpg
 | image2 = Dual Cube-Octahedron.svg
 | footer = The regular octahedron and its [[dual polyhedron]], the [[cube]].
 | total_width = 300
}}
A '''regular octahedron''' is an octahedron that is a [[regular polyhedron]]. All the faces of a regular octahedron are [[equilateral triangle]]s of the same size, and exactly four triangles meet at each vertex. A regular octahedron is convex, meaning that for any two points within it, the [[line segment]] connecting them lies entirely within it.

It is one of the eight convex [[deltahedron|deltahedra]] because all of the faces are [[equilateral triangles]].{{r|trigg}} It is a [[composite polyhedron]] made by attaching two [[equilateral square pyramid]]s.{{r|timofeenko-2010|erickson}} Its [[dual polyhedron]] is the [[cube]], and they have the same [[Point groups in three dimensions| three-dimensional symmetry groups]], the octahedral symmetry <math> \mathrm{O}_\mathrm{h} </math>.{{r|erickson}}

=== As a Platonic solid ===
{{multiple image
 | image1 = Kepler Octahedron Air.jpg
 | caption1 = Sketch of a regular octahedron by [[Johannes Kepler]]
 | image2 = Mysterium Cosmographicum solar system model.jpg
 | caption2 = [[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]]
 | align = right
 | total_width = 300
}}
The regular octahedron is one of the [[Platonic solid]]s, a set of polyhedrons whose faces are [[Congruence (geometry)|congruent]] [[regular polygons]] and the same number of faces meet at each vertex.{{r|hs}} This ancient set of polyhedrons was named after [[Plato]] who, in his [[Timaeus (dialogue)|''Timaeus'']] dialogue, related these solids to nature. One of them, the regular octahedron, represented the [[classical element]] of [[Wind (classical element)|wind]].{{r|cromwell}}

Following its attribution with nature by Plato, [[Johannes Kepler]] in his ''[[Harmonices Mundi]]'' sketched each of the Platonic solids.{{r|cromwell}} In his ''[[Mysterium Cosmographicum]]'', Kepler also proposed the [[Solar System]] by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, [[regular icosahedron]], [[regular dodecahedron]], [[regular tetrahedron]], and [[cube]].{{r|livio}}

Like its dual, the regular octahedron has three properties: any two faces, two vertices, and two edges are transformed by rotation and reflection under the symmetry orbit, such that the appearance remains unchanged; these are [[isohedral]], [[isogonal figure|isogonal]], and [[isotoxal]] respectively. Hence, it is considered a [[regular polyhedron]]. Four triangles surround each vertex, so the regular octahedron is <math> 3.3.3.3 </math> by [[vertex configuration]] or <math> \{3,4\} </math> by [[Schläfli symbol]].{{r|wd}}

===As a square bipyramid===
[[File:Square bipyramid.png|thumb|upright=0.6|Square bipyramid]]
Many octahedra of interest are '''square bipyramids'''.{{r|oh}} A square bipyramid is a [[bipyramid]] constructed by attaching two square pyramids base-to-base. These pyramids cover their square bases, so the resulting polyhedron has eight triangular faces.{{r|trigg}}

A square bipyramid is said to be right if the square pyramids are symmetrically regular and both of their apices are on the line passing through the base's center; otherwise, it is oblique.{{r|polya}} The resulting bipyramid has [[Point groups in three dimensions|three-dimensional point group]] of [[dihedral group]] <math> D_{4\mathrm{h}} </math> of sixteen: the appearance is symmetrical by rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane.{{r|ak}} Therefore, this square bipyramid is [[face-transitive]] or isohedral.{{r|mclean}}

If the edges of a square bipyramid are all equal in length, then that square bipyramid is a regular octahedron. 

=== Metric properties and Cartesian coordinates ===
[[File:Octahedron.stl|thumb|3D model of regular octahedron]]
The surface area <math> A </math> of a regular octahedron can be ascertained by summing all of its eight equilateral triangles, whereas its volume <math> V </math> is twice the volume of a square pyramid; if the edge length is <math>a</math>,{{r|berman}}
<math display="block"> \begin{align}
 A &= 2\sqrt{3}a^2 &\approx 3.464a^2, \\
 V &= \frac{1}{3} \sqrt{2}a^3 &\approx 0.471a^3.
\end{align} </math>
The radius of a [[circumscribed sphere]] <math> r_u </math> (one that touches the octahedron at all vertices), the radius of an [[inscribed sphere]] <math> r_i </math> (one that tangent to each of the octahedron's faces), and the radius of a [[midsphere]] <math> r_m </math> (one that touches the middle of each edge), are:{{r|radii}}
<math display="block"> 
 r_u = \frac{\sqrt{2}}{2}a \approx 0.707a, \qquad
 r_i = \frac{\sqrt{6}}{6}a \approx 0.408a, \qquad
 r_m = \frac{1}{2}a = 0.5a.
</math>

The [[dihedral angle]] of a regular octahedron between two adjacent triangular faces is 109.47°. This can be obtained from the dihedral angle of an equilateral square pyramid: its dihedral angle between two adjacent triangular faces is the dihedral angle of an equilateral square pyramid between two adjacent triangular faces, and its dihedral angle between two adjacent triangular faces on the edge in which two equilateral square pyramids are attached is twice the dihedral angle of an equilateral square pyramid between its triangular face and its square base.{{r|johnson}}

An octahedron with edge length <math> \sqrt{2} </math> can be placed with its center at the origin and its vertices on the coordinate axes; the [[Cartesian coordinates]] of the vertices are:{{r|smith}}
<math display="block"> (\pm 1, 0, 0), \qquad (0, \pm 1, 0), \qquad (0, 0, \pm 1). </math>

=== Graph ===
[[File:Complex tripartite graph octahedron.svg|class=skin-invert-image|thumb|upright=0.8|The graph of a regular octahedron]]
The [[Skeleton (topology)|skeleton]] of a regular octahedron can be represented as a [[Graph (discrete mathematics)|graph]] according to [[Steinitz's theorem]], provided the graph is [[Planar graph|planar]]&mdash;its edges of a graph are connected to every vertex without crossing other edges&mdash;and [[k-vertex-connected graph|3-connected graph]]&mdash;its edges remain connected whenever two of more three vertices of a graph are removed.{{r|grunbaum-2003|ziegler}} Its graph called the '''octahedral graph''', a [[Platonic graph]].{{r|hs}}

The octahedral graph can be considered as [[Tripartite graph|complete tripartite graph]] <math> K_{2,2,2} </math>, a graph partitioned into three independent sets each consisting of two opposite vertices.{{r|negami}} More generally, it is a [[Turán graph]] <math> T_{6,3} </math>.

The octahedral graph is [[k-vertex-connected graph|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 polytope|simplicial]] [[well-covered graph|well-covered]] polyhedra, meaning that all of the [[maximal independent set]]s 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.{{r|fhnp}}

=== Related figures ===
[[File:Compound of two tetrahedra.png|right|thumb|upright=0.8|The octahedron represents the central intersection of two tetrahedra]]
The interior of the [[polyhedral compound|compound]] of two dual [[tetrahedra]] is an octahedron, and this compound&mdash;called the [[stella octangula]]&mdash;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. [[Rectification (geometry)|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 ratio|golden mean]] to define the vertices of a [[regular 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. Five octahedra define any given icosahedron in this fashion, and together they define a ''regular compound''. A regular icosahedron produced this way is called a ''snub octahedron''.{{r|kappraff}}

{{anchor|Trigonal antiprism}}The regular octahedron can be considered as the [[antiprism]], a [[Prism (geometry)|prism]] like polyhedron in which lateral faces are replaced by alternating equilateral triangles. It is also called ''trigonal antiprism''.{{sfnp|O'Keeffe|Hyde|2020|p=[https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141 141]}} Therefore, it has the property of [[Quasiregular polyhedron|quasiregular]], a polyhedron in which two different polygonal faces are alternating and meet at a vertex.{{r|maekawa}}

[[Tetrahedral-octahedral honeycomb|Octahedra and tetrahedra]] can be alternated to form a vertex, edge, and face-uniform [[tessellation of space]]. This and the regular tessellation of [[cube]]s are the only such [[uniform honeycomb]]s in 3-dimensional space.

The uniform [[tetrahemihexahedron]] is a [[tetrahedral symmetry]] [[faceting]] of the regular octahedron, sharing [[edge arrangement|edge]] and [[vertex arrangement]]. It has four of the triangular faces, and 3 central squares.

A regular octahedron is a [[n-ball|3-ball]] in the [[Taxicab geometry|Manhattan ({{math|''ℓ''}}{{sub|1}}) metric]].

=== Characteristic orthoscheme ===
Like all regular convex polytopes, the octahedron can be [[Dissection into orthoschemes|dissected]] into an integral number of disjoint [[orthoscheme]]s, all of the same shape characteristic of the polytope. A polytope's [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] is a [[fundamental domain|fundamental]] property because the polytope is generated by reflections in the [[Facet (geometry)|facets]] of its orthoscheme. The orthoscheme occurs in two [[chiral]] forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a [[Tetrahedron#Orthoschemes|quadrirectangular irregular tetrahedron]].

The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of [[Symmetry (geometry)|symmetry]]. 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, among its mirror planes, some that do not pass through any of its faces. The octahedron's [[Coxeter group|symmetry group]] is denoted [[Octahedral symmetry|B<sub>3</sub>]]. The octahedron and its [[dual polytope]], the [[cube]], have the same symmetry group but different characteristic tetrahedra. 

The '''characteristic tetrahedron of the regular octahedron''' can be found by a canonical dissection{{Sfn|Coxeter|1973|p=130|loc=§7.6 The symmetry group of the general regular polytope|ps=; "simplicial subdivision".}} of the regular octahedron {{Coxeter–Dynkin diagram|node_1|3|node|4|node}} which subdivides it into 48 of these characteristic orthoschemes {{Coxeter–Dynkin diagram|node|3|node|4|node}} surrounding the octahedron's center. Three left-handed orthoschemes and three right-handed orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a [[Tetrahedron#Trirectangular tetrahedron|trirectangular tetrahedron]]: a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron.{{Sfn|Coxeter|1973|pp=70-71|loc=Characteristic tetrahedra; Fig. 4.7A}}

{| class="wikitable floatright"
!colspan=6|Characteristics of the regular octahedron{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i); "Octahedron, 𝛽<sub>3</sub>"}}
|-
!align=right|
!align=center|edge
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral
|-
!align=right|𝒍
|align=center|<small><math>2</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|align=center|<small>109°28{{prime}}</small>
|align=center|<small><math>\pi - 2\psi</math></small>
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{4}{3}} \approx 1.155</math></small>
|align=center|<small>54°44{{prime}}8{{pprime}}</small>
|align=center|<small><math>\tfrac{\pi}{2} - \kappa</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>1</math></small>
|align=center|<small>45°</small>
|align=center|<small><math>\tfrac{\pi}{4}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>
|align=center|<small>35°15{{prime}}52{{pprime}}</small>
|align=center|<small><math>\kappa</math></small>
|align=center|<small>45°</small>
|align=center|<small><math>\tfrac{\pi}{4}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R/l</math></small>
|align=center|<small><math>\sqrt{2} \approx 1.414</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R/l</math></small>
|align=center|<small><math>1</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
|
|
|
|
|
|-
!align=right|<small><math>\kappa</math></small>
|align=center|
|align=center|<small>35°15{{prime}}52{{pprime}}</small>
|align=center|<small><math>\tfrac{\text{arc sec }3}{2}</math></small>
|align=center|
|align=center|
|}
If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{2}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small> (edges that are the ''characteristic radii'' of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a [[30-60-90 triangle|90-60-30 triangle]] which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a [[45-45-90 triangle|45-90-45 triangle]] with edges <small><math>1</math></small>, <small><math>\sqrt{2}</math></small>, <small><math>1</math></small>, a right triangle with edges <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, and a right triangle with edges <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>\sqrt{2}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>.

===Uniform colorings and symmetry===
There are 3 [[uniform coloring]]s of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.

The octahedron's [[symmetry group]] is O<sub>h</sub>, of order 48, the three dimensional [[hyperoctahedral group]].  This group's [[subgroup]]s include D<sub>3d</sub> (order 12), the symmetry group of a triangular [[antiprism]]; '''D<sub>4h</sub>''' (order 16), the symmetry group of a square [[bipyramid]]; and T<sub>d</sub> (order 24), the symmetry group of a [[Octahedron#Tetratetrahedron|rectified tetrahedron]]. These symmetries can be emphasized by different colorings of the faces.
{| class=wikitable
!Name
!Octahedron
![[Rectification (geometry)|Rectified]] [[tetrahedron]]<br>(Tetratetrahedron)
!Triangular [[antiprism]]
!Square [[bipyramid]]
!Rhombic fusil
|- align=center
!Image<br>(Face coloring)
|[[File:Uniform polyhedron-43-t2.png|100px]]<br>(1111)
|[[File:Uniform polyhedron-33-t1.svg|100px]]<br>(1212)
|[[File:Trigonal antiprism.png|100px]]<br>(1112)
|[[File:Square bipyramid.png|100px]]<br>(1111)
|[[File:Rhombic bipyramid.png|100px]]<br>(1111)
|- align=center
![[Coxeter diagram]]
|{{CDD|node_1|3|node|4|node}}
|{{CDD|node_1|3|node|4|node_h0}} = {{CDD|node_1|split1|nodes}}
|{{CDD|node_h|2x|node_h|6|node}}<br>{{CDD|node_h|2x|node_h|3|node_h}}
|{{CDD|node_f1|2x|node_f1|4|node}}
|{{CDD|node_f1|2x|node_f1|2x|node_f1}}
|- align=center
![[Schläfli symbol]]
|{3,4}
|r{3,3}
|s{2,6}<br>sr{2,3}
|ft{2,4}<br>{ } + {4}
|ftr{2,2}<br>{ } + { } + { }
|- align=center
![[Wythoff symbol]]
| 4 {{pipe}} 3 2
| 2 {{pipe}} 4 3
| 2 {{pipe}} 6 2 <br> {{pipe}} 2 3 2
| ||
|- align=center
![[List of spherical symmetry groups|Symmetry]]
|O<sub>h</sub>, [4,3], (*432)
|T<sub>d</sub>, [3,3], (*332)
|D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)<br>D<sub>3</sub>, [2,3]<sup>+</sup>, (322)
|D<sub>4h</sub>, [2,4], (*422)
|D<sub>2h</sub>, [2,2], (*222)
|- align=center
![[Group order|Order]]
|48
|24
|12<br>6
|16
|8
|}

== Other types of octahedra ==
[[File:Gyrobifastigium.png|thumb|upright=0.9|A regular faced convex polyhedron, the [[gyrobifastigium]].]]
An octahedron can be any polyhedron with eight faces. In a previous example, 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.<ref>{{Cite web |url=http://www.uwgb.edu/dutchs/symmetry/polynum0.htm |title=Enumeration of Polyhedra |access-date=2 May 2006 |archive-url=https://web.archive.org/web/20111010185122/http://www.uwgb.edu/dutchs/symmetry/polynum0.htm |archive-date=10 October 2011 |url-status=dead }}</ref> 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.<ref>{{cite web| url = http://www.numericana.com/data/polycount.htm| title = Counting polyhedra}}</ref><ref>{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm |title=Polyhedra with 8 Faces and 6-8 Vertices |access-date=14 August 2016 |url-status=dead |archive-url=https://web.archive.org/web/20141117072140/http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm |archive-date=17 November 2014  }}</ref> (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 of the polyhedrons do have eight faces aside from being square bipyramids in the following:
* [[Hexagonal prism]]: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
* Heptagonal [[Pyramid (geometry)|pyramid]]: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). All triangular faces can't 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 [[kite (geometry)|kites]].
* [[Gyrobifastigium]]: Two uniform [[triangular prisms]] glued over one of their square sides so that no triangle shares an edge with another triangle (Johnson solid 26).
* [[Truncated triangular trapezohedron]], also called Dürer's solid: Obtained by truncating two opposite corners of a cube or rhombohedron, this has six pentagon faces and two triangle faces.<ref>{{citation|last1=Futamura|first1=F.|author1-link=Fumiko Futamura|first2=M.|last2=Frantz|first3=A.|last3=Crannell|author3-link= Annalisa Crannell |title=The cross ratio as a shape parameter for Dürer's solid|journal=Journal of Mathematics and the Arts|volume=8|issue=3–4|year=2014|pages=111–119|doi=10.1080/17513472.2014.974483|arxiv=1405.6481|s2cid=120958490}}</ref>
* Octagonal [[hosohedron]]: degenerate in Euclidean space, but can be realized spherically.

[[File:Br2-anim.gif|thumb|upright=0.9|[[Bricard octahedron]] with an [[antiparallelogram]] as its equator. The axis of symmetry passes through the plane of the antiparallelogram.]]
The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it:
* Triangular [[antiprism]]s: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral.
* Tetragonal [[bipyramid]]s, 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 non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
* [[Bricard octahedron]], a non-convex self-crossing [[flexible polyhedron]]

==Octahedra in the physical world==

===Octahedra in nature===
[[Image:Fluorite octahedron.jpg|thumb|[[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 [[Coordination chemistry|coordinate]] six ligands in an octahedral or [[Jahn–Teller effect|distorted]] octahedral configuration.
* [[Widmanstätten pattern]]s in [[nickel]]-[[iron]] [[crystal]]s

===Octahedra in art and culture===
[[Image:Rubiks snake octahedron.jpg|thumb|Two identically formed [[Rubik's Snake]]s can approximate an octahedron.]]
* Especially in [[roleplaying game]]s, this solid is known as a "d8", one of the more common [[dice#Polyhedral dice|polyhedral dice]].
* If each edge of an octahedron is replaced by a one-[[ohm (unit)|ohm]] [[resistor]], the resistance between opposite vertices is {{sfrac|1|2}} ohm, and that between adjacent vertices {{sfrac|5|12}} ohm.<ref>{{cite journal |last=Klein |first=Douglas J. |year=2002 |title=Resistance-Distance Sum Rules |journal=Croatica Chemica Acta |volume=75 |issue=2 |pages=633–649 |url=http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |access-date=30 September 2006 |archive-url=https://web.archive.org/web/20070610165115/http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |archive-date=10 June 2007 |url-status=dead }}</ref>
* 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]].

===Tetrahedral octet truss===
A [[space frame]] of alternating tetrahedra and half-octahedra derived from the [[Tetrahedral-octahedral honeycomb]] was invented by [[Buckminster Fuller]] in the 1950s. It is commonly regarded as the strongest building structure for resisting [[cantilever]] stresses.

==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]].
{| class=wikitable
|[[File:Triangulated tetrahedron.png|120px]]
|[[File:Compound of two tetrahedra.png|120px]]
|-
![[tetrahedron]]
![[stellated octahedron]]
|}

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

{{Octahedral truncations}}

It is also one of the simplest examples of a [[hypersimplex]], a polytope formed by certain intersections of a [[hypercube]] with a [[hyperplane]].

The octahedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].
{{Triangular regular tiling}}

===Tetratetrahedron===
The regular octahedron can also be considered a ''[[rectification (geometry)|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:
{{Tetrahedron family}} <!-- This template shows too many figures. It needs replacing with the simple set described in the text -->

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'', {{sfrac|3|8}}, {{sfrac|1|2}}, {{sfrac|5|8}}, and ''s'', where ''r'' is any number in the range {{nowrap|0 < ''r'' ≤ {{sfrac|1|4}}}}, and ''s'' is any number in the range {{nowrap|{{sfrac|3|4}} ≤ ''s'' < 1}}.

The octahedron as a ''tetratetrahedron'' exists in a sequence of symmetries of quasiregular polyhedra and tilings with [[vertex configuration]]s (3.''n'')<sup>2</sup>, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With [[orbifold notation]] symmetry of *''n''32 all of these tilings are [[Wythoff construction]]s within a [[fundamental domain]] of symmetry, with generator points at the right angle corner of the domain.<ref>{{cite book |last=Coxeter |first=H.S.M. |author-link=Harold Scott MacDonald Coxeter |title-link=Regular Polytopes (book) |title=Regular Polytopes |edition=Third |date=1973 |publisher=Dover |isbn=0-486-61480-8 |at=Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction}}</ref><ref>{{citation |last=Huson |first=Daniel H. |title= Two Dimensional Symmetry Mutation |date=September 1998 |url=https://www.researchgate.net/publication/2422380}}</ref>
{{Quasiregular3 small table}}

===Trigonal antiprism===
As a trigonal [[antiprism]], the octahedron is related to the hexagonal dihedral symmetry family.
{{Hexagonal dihedral truncations}}

{{UniformAntiprisms}}

===Other related polyhedra===
Truncation of two opposite vertices results in a [[square bifrustum]].

The octahedron can be generated as the case of a 3D [[superellipsoid]] with all exponent values set to 1.

== See also ==
* [[Octahedral number]]
* [[Centered octahedral number]]
* [[:Image:Octahedron.gif|Spinning octahedron]]
* [[Stella octangula]]
* [[Triakis octahedron]]
* [[Disdyakis dodecahedron|Hexakis octahedron]]
* [[Truncated octahedron]]
* [[Octahedral molecular geometry]]
* [[Octahedral symmetry]]
* [[Octahedral graph]]
* [[Octahedral sphere]]

== Notes ==
{{notelist}}

== References ==
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<ref name=berman>{{cite journal
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<ref name=cromwell>{{cite book
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<ref name=grunbaum-2003>{{citation
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<ref name=hs>{{cite book
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<ref name=johnson>{{cite journal
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<ref name=kappraff>{{cite book
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<ref name=livio>{{cite book
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<ref name=maekawa>{{cite book
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<ref name=mclean>{{cite journal
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<ref name=negami>{{cite book
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<ref name=oh>{{cite book
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<ref name=polya>{{cite book
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<ref name=radii>{{harvtxt|Coxeter|1973}} Table I(i), pp. 292–293. See the columns labeled <math>{}_0\!\mathrm{R}/\ell</math>, <math>{}_1\!\mathrm{R}/\ell</math>, and <math>{}_2\!\mathrm{R}/\ell</math>, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses <math>2\ell</math> as the edge length (see p. 2).</ref>

<ref name=smith>{{cite book
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<ref name=timofeenko-2010>{{cite journal
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<ref name=wd>{{cite book
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<ref name=ziegler>{{cite book
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}}

== External links ==
* {{cite EB1911|wstitle=Octahedron |volume=19 |short=x}}
* {{mathworld |urlname=Octahedron |title=Octahedron}}
* {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o4o – oct}}
* [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1S2GJRuqXD7blH9ixbn1mPoTSPo6vkjWddm7xoNxFj&name=Octahedron#applet Editable printable net of an octahedron with interactive 3D view]
* [http://www.software3d.com/Octahedron.php Paper model of the octahedron]
* [http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra
** [http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] – Try: dP4

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