{{Short description|Polyhedron with eight triangular faces}}
{{distinguish|Octahedron (album){{!}}''Octahedron'' (album)}}
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In [[geometry]], an '''octahedron''' ({{plural form}}: '''octahedra''' or '''octahedrons''') is any [[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. Many types of irregular octahedra also exist, including both [[convex set|convex]] and non-convex shapes.

==Regular octahedron ==
{{main|Regular octahedron}}
[[File:Regular octahedron (jasmine color).svg|thumb|upright=0.9|A [[regular octahedron]]]]
The [[regular octahedron]] has eight [[equilateral triangle]] sides, six [[vertex (geometry)|vertices]] at which four sides meet, and twelve edges. Its [[dual polyhedron]] is a [[cube]].<ref>{{cite book
 | last = Erickson | first = Martin
 | year = 2011
 | title = Beautiful Mathematics
 | publisher = [[Mathematical Association of America]]
 | url = https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62
 | page = 62
 | isbn = 978-1-61444-509-8
}}</ref> It can be formed as the [[convex hull]] of the six axis-parallel [[unit vector]]s in three-dimensional [[Euclidean space]]. It is one of the five [[Platonic solid]]s,<ref>{{cite book
 | last1 = Herrmann | first1 = Diane L.
 | last2 = Sally | first2 = Paul J.
 | year = 2013
 | title = Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory
 | publisher = Taylor & Francis
 | isbn = 978-1-4665-5464-1
 | url = https://books.google.com/books?id=b2fjR81h6yEC&pg=PA252
 | page = 252
}}</ref> and the three-dimensional case of an infinite family of [[regular polytope]]s, the [[cross polytope]]s.<ref>{{cite book 
 | last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter
 | title = Regular Polytopes | title-link = Regular Polytopes (book)
 | publisher = Methuen and Co.
 | year = 1948
 | pages = 121–122
}}</ref> Although it does not tile space by itself, it can tile space together with the [[regular tetrahedron]] to form the [[tetrahedral-octahedral honeycomb]].<ref>{{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=PA233
 | pages = 233&ndash;234
}}</ref>

==Combinatorially equivalent to the regular octahedron==
[[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.<ref>{{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>
* 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.<ref>{{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>
* [[Schönhardt polyhedron]], a non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.<ref>{{cite journal
 | last = Schönhardt | first = E. | authorlink = Erich Schönhardt
 | journal = [[Mathematische Annalen]]
 | pages = 309–312
 | title = Über die Zerlegung von Dreieckspolyedern in Tetraeder
 | url = https://eudml.org/doc/159218
 | year = 1928
 | volume = 98 | doi = 10.1007/BF01451597
}}</ref>
* [[Bricard octahedron]], a non-convex self-crossing [[flexible polyhedron]].<ref>{{cite book
 | last = Connelly | first = Robert | author-link = Robert Connelly
 | editor-last = [[David A. Klarner|Klarner, David A.]] 
 | contribution = Flexing surfaces
 | doi = 10.1007/978-1-4684-6686-7_10
 | isbn = 978-1-4684-6688-1
 | pages = 79–89
 | publisher = Springer
 | title = The Mathematical Gardner
 | year = 1981}}.</ref><ref name="ft">{{citation
 | last1 = Fuchs | first1 = Dmitry
 | last2 = Tabachnikov | first2 = Serge | author2-link = Sergei Tabachnikov
 | doi = 10.1090/mbk/046
 | isbn = 978-0-8218-4316-1
 | location = Providence, RI
 | mr = 2350979
 | page = 347
 | publisher = American Mathematical Society
 | title = Mathematical Omnibus: Thirty lectures on classic mathematics
 | url = https://books.google.com/books?id=IiG9AwAAQBAJ&pg=PA347
 | year = 2007}}</ref>

==Other convex polyhedra==
{{Commons category|Polyhedra with 8 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.<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 |last=Michon |first=Gerard P. |title=Enumeration of Polyhedra - Numericana |url=http://www.numericana.com/data/polycount.htm |website=www.numericana.com}}</ref><ref name=":0">{{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  |last=Dutch|first=Steven}}</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.<ref name=":0" /> 

Notable eight-sided convex polyhedra include:

<gallery widths=180 heights=180>
File:Hexagonal Prism.svg | [[Hexagonal prism]]: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges. With all faces regular and all vertices symmetric to each other, this is a [[uniform polyhedron]].<ref>{{Cite journal | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | author-link2=Michael S. Longuet-Higgins | last3=Miller | first3=J. C. P. | author-link3=J. C. P. Miller| title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=[[Philosophical Transactions of the Royal Society A]]| issn=0080-4614 | volume=246 |issue=916 | pages=401–450 | doi=10.1098/rsta.1954.0003 | bibcode=<!-- 1954RSPTA.246..401C --> | s2cid=202575183 |url=http://rsta.royalsocietypublishing.org/content/roypta/246/916/401.full.pdf}}</ref> It tiles space by translation as a [[parallelohedron]].<ref>{{cite book|last=Alexandrov|first=A. D.|author-link=Aleksandr Danilovich Aleksandrov|page=349|publisher=Springer|title=Convex Polyhedra|title-link=Convex Polyhedra (book)|year=2005}}</ref> The hexagonal [[frustum]] is topologically equivalent.
File:Truncated tetrahedron (green).png | [[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. As a uniform polyhedron that is not a prism or [[antiprism]], this is an [[Archimedean solid]].{{r|kuchel|berman}}
File:Gyrobifastigium.png | [[Gyrobifastigium]]: Two uniform [[triangular prisms]] glued over one of their square sides so that no triangle shares an edge with another triangle. As a polyhedron whose faces are regular polygons, it is a [[Johnson solid]].{{r|berman}} It is a [[space-filling polyhedron]].<ref>{{cite book | last = Kepler | first = Johannes | author-link = Johannes Kepler | publisher = Paul Dry Books | year = 2010 | title = The Six-Cornered Snowflake | isbn = 9781589882850 | at = Footnote 18, [https://books.google.com/books?id=yE8yTUFWLXgC&pg=PA146 pp.&nbsp;146–147]}}</ref> Its [[dual polyhedron]] is also an octahedron.<ref>{{cite conference | last = Draghicescu | first = Mircea | editor1-last = Torrence | editor1-first = Eve | editor2-last = Torrence | editor2-first = Bruce | editor3-last = Séquin | editor3-first = Carlo | editor4-last = McKenna | editor4-first = Douglas | editor5-last = Fenyvesi | editor5-first = Kristóf | editor6-last = Sarhangi | editor6-first = Reza | contribution = Dual models: one shape to make them all | contribution-url = https://archive.bridgesmathart.org/2016/bridges2016-635.html | isbn = 978-1-938664-19-9 | location = Phoenix, Arizona | pages = 635–640 | publisher = Tessellations Publishing | title = Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture | year = 2016}}</ref>
File:Augmented triangular prism.png | [[Augmented triangular prism]]: The result of gluing a triangular prism to a [[square pyramid]], this has six equilateral triangle faces and two square faces. It is also a Johnson solid.{{r|berman}}
File:Triangular cupola.png | [[Triangular cupola]]: Another Johnson solid, this has one regular hexagon face, three square faces, and four equilateral triangle faces.{{r|berman}}
File:Tridiminished icosahedron.png| [[Tridiminished icosahedron]]: Another Johnson solid, obtained by removing three pentagonal pyramids from a regular icosahedron, resulting in three pentagonal and five triangular faces.<ref name=gailiunas>{{cite book | last = Gailiunas | first = Paul | contribution = A Polyhedral Byway | contribution-url = https://archive.bridgesmathart.org/2001/bridges2001-115.pdf | pages = 115&ndash;122 | title = Bridges: Mathematical Connections in Art, Music, and Science | year = 2001 | editor-last1 = Sarhangi | editor-first1 = Reza | editor-last2 = Jablan | editor-first2 = Slavik | publisher = Bridges Conference}}</ref>
File:Heptagonal pyramid.svg| Heptagonal [[Pyramid (geometry)|pyramid]]: One face is a [[heptagon]] (usually regular), and the remaining seven faces are triangles (usually [[isosceles triangle|isosceles]]).{{r|humble}} It is not possible for all triangular faces to be equilateral. It is a [[self-dual polyhedron]].
File:Trapezohedron 4 (green).png | [[Tetragonal trapezohedron]]: The eight faces are congruent [[kite (geometry)|kites]].{{r|dana}} Up to topological equivalence it is the only octahedron all of whose faces are [[quadrilateral]]s.{{r|bdg}}
File:Dual elongated triangular dipyramid.png | [[Triangular bifrustum]]: The dual of an [[elongated triangular bipyramid]] (a Johnson solid), this can be realized with six [[isosceles trapezoid]] faces and two equilateral triangle faces.
File:Triangular truncated trapezohedron.png | [[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>{{cite journal|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>
File:Elongated gyrobifastigium.png|[[Gabled rhombohedron]] with four pentagonal faces and four rectangular faces.<ref>{{cite journal|first1=Paul|last1=Gallagher|first2=Whan|last2=Ghang|first3=David|last3=Hu|first4=Zane|last4=Martin|first5=Maggie|last5=Miller|first6=Byron|last6=Perpetua|first7=Steven|last7=Waruhiu|volume=15|year=2014|issue=1|pages=210&ndash;236|journal=Rose-Hulman Undergraduate Mathematics Journal|title=Surface-area-minimizing <math>n</math>-hedral Tiles|url=https://scholar.rose-hulman.edu/rhumj/vol15/iss1/13/}}</ref> Like the gyrobifastigium, it is a space-filling polyhedron.<ref>{{cite journal | last = Goldberg | first = Michael | title = On the space-filling octahedra | journal = Geometriae Dedicata | volume = 10 | issue = 1 | pages = 323–335 | doi = 10.1007/BF01447431 | url = https://documents.mx/documents/on-the-space-filling-octahedra.html| year = 1981| archive-url = https://web.archive.org/web/20171222105633/https://documents.mx/documents/on-the-space-filling-octahedra.html | archive-date = 22 December 2017 }}</ref>
</gallery>

== References ==
<references>

<ref name="berman">{{cite journal
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971
 | issue = 5
 | bibcode = <!-- 1971FrInJ.291..329B -->
 }}</ref>

<ref name=bdg>{{cite journal
 | last1 = Broersma | first1 = H. J.
 | last2 = Duijvestijn | first2 = A. J. W.
 | last3 = Göbel | first3 = F.
 | doi = 10.1002/jgt.3190170508
 | issue = 5
 | journal = Journal of Graph Theory
 | mr = 1242180
 | pages = 613–620
 | title = Generating all 3-connected 4-regular planar graphs from the octahedron graph
 | volume = 17
 | year = 1993
 | url = https://research.utwente.nl/en/publications/9027937b-9128-4382-b4c6-14cb9b1d23a6
 }}</ref>

<ref name=dana>{{cite book|title= A Text-Book of Mineralogy: With an Extended Treatise on Crystallography and Physical Mineralogy|last1=Dana|first1=Edward Salisbury|last2=Ford|first2=W. E.|year=1922|edition=3rd|location=New York|publisher=Wiley|url=https://archive.org/details/textbookofminera00danauoft/page/88|page=89}}</ref>

<ref name="humble">{{cite book
 | last = Humble | first = Steve
 | year = 2016
 | title = The Experimenter's A-Z of Mathematics: Math Activities with Computer Support
 | page = 23
 | publisher = Taylor & Francis
 | isbn = 978-1-134-13953-8
 | url = https://books.google.com/books?id=S-80DwAAQBAJ&pg=PA23
}}</ref>

<ref name="kuchel">{{cite journal
 | last = Kuchel | first = Philip W.
 | year = 2012
 | title = 96.45 Can you 'bend' a truncated truncated tetrahedron?
 | journal = [[The Mathematical Gazette]]
 | volume = 96 | issue = 536 | pages = 317–323
 | doi = 10.1017/S0025557200004666
 | jstor = 23248575
}}</ref>

</references>

{{Polyhedra}}

{{Authority control}}

[[Category:Polyhedra]]