Editing Octahedron (section)

== 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
|}