View source for Regular octahedron

{{Short description|Solid with eight equal triangular faces}}
{{Use dmy dates|date=January 2020}}

{{infobox polyhedron
 | name = Regular octahedron
 | image = Octahedron.jpg
 | type = [[Bipyramid]],<br>[[Deltahedron]],<br>[[Octahedron]],<br>[[Platonic solid]],<br>[[Regular polyhedron]]
 | vertices = 6
 | edges = 12
 | faces = 8
 | schläfli = <math> \{3,4\} </math>
 | symmetry = [[octahedral symmetry]]
 | dual = [[cube]]
 | net = [[Image:Octahedron flat.svg|frameless|upright=1.0]]
 | angle = 109.47°
 | properties = [[composite polyhedron|composite]],<br>[[convex polyhedron|convex]],<br>[[isohedral]],<br>[[isogonal figure|isogonal]],<br>[[isotoxal]]
}}
In [[geometry]], a '''regular octahedron''' is a highly symmetrical type of [[octahedron]] (eight-sided [[polyhedron]]) with eight [[equilateral triangle]]s as its [[face (geometry)|faces]], four of which meet at each [[vertex (geometry)|vertex]]. It is a type of [[square bipyramid]] or [[triangular antiprism]] with equal-length [[edge (geometry)|edges]]. Regular octahedra occur in nature as [[crystal]] structures. Other types of octahedra also exist, with various amounts of symmetry.

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

== Description ==
The regular octahedron is one of the [[Platonic solid]]s, a set of [[convex polyhedron|convex polyhedra]] 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 [[classical element]]s, with the octahedron representing [[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}}

{{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]]
 | image3 = Dual Cube-Octahedron.svg
 | caption3 = The dual of a regular octahedron is cube
 | align = center
 | total_width = 420
}}
A regular octahedron is the [[cross-polytope]] in 3-dimensional space. It can be oriented and scaled so that its axes align with [[Cartesian coordinate]] axes and its vertices have coordinates{{r|smith}}
<math display="block"> (\pm 1, 0, 0), \qquad (0, \pm 1, 0), \qquad (0, 0, \pm 1). </math>
Such an octahedron has edge length {{tmath|\sqrt2}}.

The regular octahedron's [[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}} 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}}

== Metrical properties ==
[[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 <math>2\arctan{\sqrt{2}}</math>, which is about 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}}

== As other special cases ==

The regular octahedron is one of the eight convex [[deltahedron|deltahedra]], polyhedra whose faces are all [[equilateral triangles]].{{r|trigg}}

{{multiple image
 | image1 = Square bipyramid.png
 | caption1 = Square bipyramid
 | image2 = Square prism.svg
 | caption2 = The dual of a square bipyramid, the [[square prism]]
 | total_width = 320
}}
{{anchor|Square bipyramid}}A regular octahedron is a type of square [[bipyramid]],{{r|oh}} a [[composite polyhedron]] constructed by attaching two [[equilateral square pyramid]]s base-to-base.{{r|timofeenko-2010|erickson}} The [[dual polyhedron]] of a bipyramid in general is the [[prism (geometry)|prism]], and vice versa; the regular octahedron's dual, the cube, is a type of [[square prism]].{{r|sibley}}

{{anchor|Trigonal antiprism}}The regular octahedron is a type of trigonal [[antiprism]],{{sfnp|O'Keeffe|Hyde|2020|p=[https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141 141]}} formed by taking a trigonal [[Prism (geometry)|prism]] with equilateral triangle bases and rectangular lateral faces, and replacing the rectangles by alternating isosceles triangles. In the case of the regular octahedron, all of the resulting faces are congruent equilateral triangles.

The regular octahedron can also be considered a [[rectification (geometry)|rectified]] tetrahedron, sometimes called a '''tetratetrahedron''' (by analogy to the [[cuboctahedron]] and [[icosidodecahedron]]); if alternate faces are considered to have different types (e.g. different colors or orientations), the octahedron can be considered a type of [[quasiregular polyhedron]], a polyhedron in which two different types of polygonal faces alternate around each vertex.{{r|maekawa}} It 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.{{r|coxeter|huson}}
{{Quasiregular3 small table}}

== Graph ==
[[File:Complex tripartite graph octahedron.svg|class=skin-invert-image|thumb|upright=0.8|The graph of a regular octahedron]]
The [[Graph of a polytope|skeleton]] of a regular octahedron is the (undirected) [[Undirected graph|graph]] formed by its vertices and edges. [[Steinitz's theorem]] guarantees that the this graph can be drawn [[Planar graph|with no edge crossing another]], and all remaining vertices [[k-vertex-connected graph|remain connected]] when any two 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}} Additionally, 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}}

== Other related figures ==
{{multiple image
 | image1 = Compound of two tetrahedra.png
 | caption1 = The regular octahedron represents the [[Stellated octahedron|central intersection of two tetrahedra]]
 | image2 = Truncatedoctahedron.svg
 | caption2 = The [[truncated octahedron]] by removing the vertices of a regular octahedron
 | image3 = Diamant1.gif
 | caption3 = The [[triakis octahedron]] by attaching triangular pyramids on each face
 | total_width = 500
}}
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]].{{sfnp|Cromwell|1997|p=171, 261}} 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.

Several constructions of polyhedra commence from the regular octahedron. The [[truncated octahedron]] is an [[Archimedean solid]], constructed by removing all of the regular octahedron's vertices, resulting in six squares and eight hexagons, leaving out six square pyramids.{{r|diudea}} The [[triakis octahedron]] is a [[Catalan solid]], the [[Kleetope]] of a regular octahedron, by attaching triangular pyramids onto its faces, [[Topology|topologically]] similar to the stellated octahedron.{{r|bpv}} 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 three central squares.{{r|pmtsgsd}}

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}}

{{multiple image
 | total_width = 360
 | image1 = HC P1-P3.png
 | caption1 = [[Tetrahedral-octahedral honeycomb]] by regular octahedra and tetrahedra
 | image2 = Rectified cubic honeycomb.jpg
 | caption2 = [[Rectified cubic honeycomb]] by regular octahedra and cuboctahedra
}}
Regular octahedra and regular tetrahedra can be alternated to form a vertex, edge, and face-uniform [[tessellation of space]], which is named [[tetrahedral-octahedral honeycomb]].{{sfnp|Posamentier et al.|2022|p=[https://books.google.com/books?id=DGxYEAAAQBAJ&pg=PA233 232&ndash;234]}} [[R. Buckminster Fuller]] in the 1950s applied these alternating polyhedra as a [[space frame]], which developed the strongest building structure for resisting [[cantilever]] stresses.{{r|mp}} Another honeycomb is tesselating the regular octahedra alternately with [[cuboctahedron|cuboctahedra]], named the [[rectified cubic honeycomb]].{{sfnp|Posamentier et al.|2022|p=[https://books.google.com/books?id=DGxYEAAAQBAJ&pg=PA235 234&ndash;235]}}

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

== Other appearances ==

{{glossary}}
{{term|Nature and science}}
{{defn|[[Image:Fluorite octahedron.jpg|thumb|upright=0.6|[[Fluorite]] with octahedral structure]]
<p>The natural crystals with octahedral structures are commonly discovered in [[diamond]],{{r|wells}} [[alum]],{{r|glr}} and [[fluorite]]. 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.
</p>

<p>[[Octahedral molecular geometry]] is a chemical molecule resembling a regular octahedron in [[stereochemistry]]. This structure has a [[main-group element]] without an active [[lone pair]], which can be described by a model that predicts the geometry of molecules known as [[VSEPR theory]].{{r|phh}}
</p>

<p>If each edge of an octahedron is replaced by a one-[[ohm (unit)|ohm]] [[resistor]], the resistance between opposite vertices is <math display="inline"> \frac{1}{2} </math> ohm, and that between adjacent vertices <math display="inline"> \frac{5}{12} </math> ohm.{{r|klein}}
</p>
}}

{{term|Popular culture}}
{{defn|[[Image:Rubiks snake octahedron.jpg|thumb|Two identically formed [[Rubik's Snake]]s can approximate an octahedron.]]
<p>In [[roleplaying game]]s, this solid is known as a "d8", one of the more common [[dice#Polyhedral dice|polyhedral dice]].{{r|br}}
</p>
}}

{{term|Music theory}}
{{defn|The [[hexany]] is the octahedron's [[orthogonal projection]]. 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.{{r|terumi}}
</p>
}}

{{glossary end}}

== 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>.

=== Notes ===
{{notelist}}

== See also ==
* [[25 great circles of the spherical octahedron]], arrangement of [[great circles]] within a spherical octahedron;
* [[Centered octahedral number]], [[figurate number]] that counts the points of a three-dimensional integer lattice that lie inside a regular octahedron centered at the origin;
* [[Disdyakis dodecahedron|Hexakis octahedron]], another polyhedron's construction involving the regular octahedron's commencement;
* [[Octahedral number]], [[figurate number]] that represents the number of spheres in a regular octahedron formed from [[close-packed spheres]];
* [[Octahedral sphere]], spherical shape of a cross-polytope;
* [[Skewb Diamond]], an octahedral version of [[Rubik's cube]]
* [[Superellipsoid]], a solid whose horizontal sections are of the same squareness.

== References ==
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}}</ref>

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<ref name=grunbaum-2003>{{citation
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<ref name=hs>{{cite book
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}}</ref>

<ref name=huson>{{cite web
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}}</ref>

<ref name=johnson>{{cite journal
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 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603| doi-access = free
}}</ref>

<ref name=kappraff>{{cite book
 | last = Kappraff | first = Jay
 | year = 1991
 | edition = 2nd
 | title = Connections: The Geometric Bridge Between Art and Science
 | publisher = [[World Scientific]]
 | url = https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475
 | page = 475
 | isbn = 978-981-281-139-4
}}</ref>

<ref name=klein>{{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>

<ref name=livio>{{cite book
 | last = Livio | first = Mario | author-link = Mario Livio
 | title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
 | url = https://books.google.com/books?id=bUARfgWRH14C
 | orig-year = 2002
 | edition = First trade paperback
 | year = 2003
 | publisher = [[Random House|Broadway Books]]
 | location = New York City
 | isbn = 0-7679-0816-3
 | pages = 70–71
}}</ref>

<ref name=maekawa>{{cite book
 | last = Maekawa | first = Jun
 | year = 2022
 | title = Art & Science of Geometric Origami: Create Spectacular Paper Polyhedra, Waves, Spirals, Fractals, and More!
 | url = https://books.google.com/books?id=Kq-kEAAAQBAJ&pg=PA42
 | page = 42
 | publisher = [[Tuttle Publishing|Tuttle]]
 | isbn = 978-1-4629-2398-4
}}</ref>

<ref name=mp>{{cite book
 | last1 = Miura | first1 = Koryo
 | last2 = Pellegrino | first2 = Sergio
 | year = 2020
 | doi = 10.1017/9781139048569
 | isbn = 9781139048569
 | title = Forms and Concepts for Lightweight Structures
 | url = https://books.google.com/books?id=v9LQDwAAQBAJ&pg=PA28
 | page = 28
 | publisher = [[Cambridge University Press]]
}}</ref>

<ref name=negami>{{cite book
 | last = Negami | first = S.
 | year = 2016
 | contribution = Faithful Embeddings of Planar Graphs on Orientable Closed Surfaces
 | contribution-url = https://books.google.com/books?id=HarWCwAAQBAJ&pg=PA250
 | page = 250
 | editor-last1 = Širáň | editor-first1 = Jozef
 | editor-last2 = Jajcay | editor-first2 = Robert
 | title = Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, West Malvern, UK, July 2014
 | series = Springer Proceedings in Mathematics & Statistics
 | volume = 159
 | publisher = Springer
 | doi = 10.1007/978-3-319-30451-9
 | isbn = 978-3-319-30451-9
}}</ref>

<ref name=oh>{{cite book
 | last1 = O'Keeffe | first1 = Michael
 | last2 = Hyde | first2 = Bruce G.
 | title = Crystal Structures: Patterns and Symmetry
 | year = 2020
 | url = https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141
 | page = 141
 | publisher = [[Dover Publications]]
 | isbn = 978-0-486-83654-6
}}</ref>

<ref name=pmtsgsd>{{cite book
 | last1 = Posamentier | first1 = Alfred S.
 | last2 = Maresch | first2 = Guenter
 | last3 = Thaller | first3 = Bernd
 | last4 = Spreitzer | first4 = Christian
 | last5 = Geretschlager | first5 = Robert
 | last6 = Stuhlpfarrer | first6 = David
 | last7 = Dorner | first7 = Christian
 | year = 2022
 | title = Geometry In Our Three-dimensional World
 | publisher = [[World Scientific]]
 | isbn = 9789811237126
 | url = https://books.google.com/books?id=DGxYEAAAQBAJ&pg=PA268
 | pages = 267&ndash;268
 | ref = {{harvid|Posamentier et al.|2022}}
}}</ref>

<ref name=phh>{{cite book
 | last1 = Petrucci | first1 = Ralph H.
 | last2 = Harwood | first2 = William S.
 | last3 = Herring | first3 = F. Geoffrey
 | title = General Chemistry: Principles and Modern Applications
 | volume = 1
 | year = 2002
 | url = https://books.google.com/books?id=EZEoAAAAYAAJ&pg=PA413
 | pages = 413–414
 | publisher = [[Prentice Hall]]
 | isbn = 978-0-13-014329-7
}} See Table 11.1.</ref>

<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=sibley>{{cite book
 | last = Sibley | first = Thomas Q.
 | year = 2015
 | title = Thinking Geometrically: A Survey of Geometries
 | publisher = Mathematical Association of American
 | page = 53
 | url = https://books.google.com/books?id=EUh2CgAAQBAJ&pg=PA53
 | isbn =978-1-939512-08-6
}}</ref>

<ref name=smith>{{cite book
 | last = Smith | first = James
 | year = 2000
 | title = Methods of Geometry
 | url = https://books.google.com/books?id=B0khWEZmOlwC&pg=PA392
 | page = 392
 | publisher = [[John Wiley & Sons]]
 | isbn = 978-1-118-03103-2
}}</ref>

<ref name=terumi>{{cite book
 | title = Microtonality and the Tuning Systems of Erv Wilson
 | first = Terumi | last = Narushima
 | year = 2018
 | publisher = Taylor & Francis
 | url = https://books.google.com/books?id=VmkPEAAAQBAJ&pg=PA151
 | page = 151&ndash;155
 | isbn = 978-1-317-51343-8
}}</ref>

<ref name=timofeenko-2010>{{cite journal
 | last = Timofeenko | first = A. V.
 | year = 2010
 | title = Junction of Non-composite Polyhedra
 | journal = St. Petersburg Mathematical Journal
 | volume = 21 | issue = 3 | pages = 483–512
 | doi = 10.1090/S1061-0022-10-01105-2
 | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
}}</ref>

<ref name=trigg>{{cite journal
 | last = Trigg | first = Charles W. | author-link = Charles W. Trigg
 | issue = 1
 | journal = Mathematics Magazine
 | jstor = 2689647
 | pages = 55–57
 | title = An Infinite Class of Deltahedra
 | volume = 51
 | year = 1978
 | doi = 10.1080/0025570X.1978.11976675
}}</ref>

<ref name=wd>{{cite book
 | last1 = Walter | first1 = Steurer
 | last2 = Deloudi | first2 = Sofia
 | year = 2009
 | title = Crystallography of Quasicrystals: Concepts, Methods and Structures
 | series = Springer Series in Materials Science
 | volume = 126
 | url = https://books.google.com/books?id=nVx-tu596twC&pg=PA50
 | page = 50
 | isbn = 978-3-642-01898-5
 | doi = 10.1007/978-3-642-01899-2
}}</ref>

<ref name=wells>{{cite book
 | last = Wells | first = Alexander Frank
 | year = 1984
 | title = Structural Inorganic Chemistry
 | url = https://books.google.com/books?id=uR77AAAAQBAJ&pg=PA127
 | page = 127
 | publisher = [[Oxford University Press]]
 | edition = 5th
 | isbn = 978-0-19-965763-6
}}</ref>

<ref name=ziegler>{{cite book
 | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler
 | contribution = Chapter 4: Steinitz' Theorem for 3-Polytopes
 | isbn = 0-387-94365-X
 | pages = 103–126
 | publisher = Springer-Verlag
 | series = [[Graduate Texts in Mathematics]]
 | title = Lectures on Polytopes
 | volume = 152
 | year = 1995
}}</ref>
</references>

== 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

{{Polyhedra}}
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[[Category:Deltahedra]]
[[Category:Individual graphs]]
[[Category:Platonic solids]]
[[Category:Prismatoid polyhedra]]
[[Category:Bipyramids]]