View source for Truncated octahedron

{{Short description|Archimedean solid}}
{{infobox polyhedron
 | name = Truncated octahedron
 | image = Truncatedoctahedron.jpg
 | type = [[Archimedean solid]],<br>[[Parallelohedron]],<br>[[Permutohedron]],<br>[[Plesiohedron]],<br>[[Zonohedron]]
 | faces = 14
 | edges = 36
 | vertices = 24
 | symmetry = [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>
 | dual = [[tetrakis hexahedron]]
 | net = Polyhedron truncated 8 net.svg
 | vertex_figure = Polyhedron truncated 8 vertfig.svg
}}
In [[geometry]], the '''truncated octahedron''' is the [[Archimedean solid]] that arises from a regular [[octahedron]] by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular [[hexagon|hexagons]] and 6 [[Square (geometry)|squares]]), 36 edges, and 24 vertices. Since each of its faces has [[point symmetry]] the truncated octahedron is a '''6'''-[[zonohedron]]. It is also the [[Goldberg polyhedron]] G<sub>IV</sub>(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a [[permutohedron]].

The truncated octahedron was called the "mecon" by [[Buckminster Fuller]].<ref>{{cite web |url=http://mathworld.wolfram.com/TruncatedOctahedron.html |website=Wolfram Mathworld |title=Truncated Octahedron}}</ref>

Its [[dual polyhedron]] is the [[tetrakis hexahedron]]. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths {{sfrac|9|8}}{{sqrt|2}} and {{sfrac|3|2}}{{sqrt|2}}.

== Classifications ==
=== As an Archimedean solid ===
A truncated octahedron is constructed from a [[regular octahedron]] by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving out six [[square pyramid]]s. Setting the edge length of the regular octahedron equal to <math> 3a </math>, it follows that the length of each edge of a square pyramid (to be removed) is <math> a </math> (the square pyramid is has four [[Equilateral square pyramid|equilateral]] triangles as faces, the first [[Johnson solid]]). From the equilateral square pyramid's property, its volume is <math display="inline"> \tfrac{\sqrt{2}}{6}a^3 </math>. Because six equilateral square pyramids are removed by truncation, the volume of a truncated octahedron <math> V </math> is obtained by subtracting the volume of those six from that of a regular octahedron:{{r|berman}}
<math display="block"> V = \frac{\sqrt{2}}{3} (3a)^3 - 6 \cdot \frac{\sqrt{2}}{6} a^3 = 8\sqrt{2}a^3 \approx 11.3137 a^3. </math>
The surface area of a truncated octahedron can be obtained by summing all polygonals' area, six squares and eight hexagons. Considering the edge length <math> a </math>, this is:{{r|berman}}
<math display="block"> (6 + 12\sqrt{3})a^2 \approx 26.7846a^2. </math>

[[File:Truncated octahedron.stl|thumb|left|3D model of a truncated octahedron]]
The truncated octahedron is one of the thirteen [[Archimedean solid]]s. In other words, it has a highly symmetric and semi-regular polyhedron with two or more different regular polygonal faces that meet in a vertex.{{r|diudea}} The [[dual polyhedron]] of a truncated octahedron is the [[tetrakis hexahedron]]. They both have the same three-dimensional symmetry group as the regular octahedron does, the [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>.{{r|kocakoca}} A square and two hexagons surround each of its vertex, denoting its [[vertex figure]] as <math> 4 \cdot 6^2 </math>.{{r|williams}}

The dihedral angle of a truncated octahedron between square-to-hexagon is <math display="inline"> \arccos(-1/\sqrt{3}) \approx 125.26^\circ </math>, and that between adjacent hexagonal faces is <math display="inline"> \arccos (-1/3) \approx 109.47^\circ </math>.{{r|johnson}}

The [[Cartesian coordinates]] of the vertices of a truncated octahedron with edge length 1 are all permutations of{{cn|date=December 2024}}
<math display="block">
\bigl(\pm\sqrt{2}, \pm\tfrac{\sqrt{2}}{2}, 0\bigr).
</math>

=== As a space-filling polyhedron ===
{{multiple image
 | image1 = Symmetric group 4; permutohedron 3D; transpositions (1-based).png
 | caption1 = Truncated octahedron as a permutahedron of order 4
 | image2 = Truncated octahedra.png
 | caption2 = Truncated octahedra tiling space
 | total_width = 400
}}
The truncated octahedron can be described as a [[permutohedron]] of order 4 or '''4-permutohedron''', meaning it can be represented with even more symmetric coordinates in four dimensions: all permutations of <math> (1, 2, 3, 4) </math> form the vertices of a truncated octahedron in the three-dimensional subspace <math> x + y + z + w = 10 </math>.{{r|jj}} Therefore, each vertex corresponds to a permutation of <math> (1, 2, 3, 4) </math> and each edge represents a single pairwise swap of two elements.{{r|crisman}} With this labeling, the swaps are of elements whose values differ by one. If, instead, the truncated octahedron is labeled by the inverse permutations, the edges correspond to swaps of elements whose positions differ by one. With this alternative labeling, the edges and vertices of the truncated octahedron form the [[Cayley graph]] of the [[symmetric group]] <math> S_4 </math>, the group of four-element permtutations, as generated by swaps of consecutive positions.{{r|budden}}

The truncated octahedron can tile space. It is classified as [[plesiohedron]], meaning it can be defined as the [[Voronoi cell]] of a symmetric [[Delone set]].{{r|erdahl}} Plesiohedra, [[Translation (geometry)|translated]] without rotating, can be repeated to fill space. There are five three-dimensional primary parallelohedrons, one of which is the truncated octahedron.{{r|alexandrov}} More generally, every permutohedron and parallelohedron is a [[zonohedron]], a polyhedron that is [[centrally symmetric]] and can be defined by a [[Minkowski sum]].{{r|jtdd}}

== Applications ==
{{multiple image
 | image1 = Structural features of the faujasite zeolite framework (FAU).svg
 | caption1 = The structure of the faujasite framework
 | image2 = Brillouin Zone (1st, FCC).svg
 | caption2 = First Brillouin zone of [[Cubic crystal system|FCC lattice]], showing symmetry labels for high symmetry lines and points.
 | total_width = 500
}} 
In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a [[faujasite]]-type of [[zeolite]] crystals.{{r|yen}}

In [[solid-state physics]], the first [[Brillouin zone]] of the [[face-centered cubic]] lattice is a truncated octahedron.{{r|mizutani}}

The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.<ref>{{cite journal
|author1=Perez-Gonzalez, F.| author2= Balado, F. | author3= Martin, J.R.H. 
|year=2003
|journal=IEEE Transactions on Signal Processing
|title=Performance analysis of existing and new methods for data hiding with known-host information in additive channels 
|volume=51
|number=4
|pages=960–980
|doi=10.1109/TSP.2003.809368
| bibcode= 2003ITSP...51..960P }}</ref>

== Dissection ==
The truncated octahedron can be dissected into a central [[octahedron]], surrounded by 8 [[triangular cupola]]e on each face, and 6 [[square pyramid]]s above the vertices.<ref>{{cite web|url=http://www.doskey.com/polyhedra/Stewart05.html|title=Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1|first=Alex|last=Doskey|website=www.doskey.com}}</ref>

{{multiple image
 | image1 = Excavated truncated octahedron1.png
 | image2 = Excavated truncated octahedron2.png
 | total_width = 400
 | footer = Second and third  genus toroids
}}
Removing the central octahedron and 2 or 4 triangular cupolae creates two [[Stewart toroid]]s, with dihedral and tetrahedral symmetry:

It is possible to slice a [[tesseract]] by a hyperplane so that its sliced cross-section is a truncated octahedron.<ref>{{citation
 | last1 = Borovik | first1 = Alexandre V.
 | last2 = Borovik | first2 = Anna
 | contribution = Exercise 14.4
 | contribution-url = https://books.google.com/books?id=bVdzDL8KW3wC&pg=PA109
 | doi = 10.1007/978-0-387-79066-4
 | isbn = 978-0-387-79065-7
 | mr = 2561378
 | page = 109
 | publisher = Springer | location = New York
 | series = Universitext
 | title = Mirrors and Reflections
 | title-link = Mirrors and Reflections
 | year = 2010}}</ref>

The [[cell-transitive]] [[bitruncated cubic honeycomb]] can also be seen as the [[Voronoi tessellation]] of the [[Crystal structure|body-centered cubic lattice]]. The truncated octahedron is one of five three-dimensional primary [[Parallelohedron#Zonohedra that tile space|parallelohedra]].

==Objects==

{{multiple image
 | align = right  | total_width = 300
 | image1 = 43840 Salou, Tarragona, Spain - panoramio (21), crop.jpg
 | image2 = Bundek climbing frame 20150307 DSC 0109, crop.jpg
 | footer = [[Jungle gym]] nets often include truncated octahedra.
}}

<gallery>
14-sided Chinese dice from warring states period.jpg | ancient Chinese die
CvO 2.jpg | sculpture in [[Bonn]]
DaYan Gem solved cubemeister com.jpg | [[Rubik's Cube]] variant
Polydron 1170197.jpg | model made with Polydron [[construction set]]
Pyrite-249304.jpg | [[Pyrite]] crystal
File:Boleite-rare-09-45da.jpg | [[Boleite]] crystal
</gallery>

== Truncated octahedral graph ==
{{Infobox graph
 | name = Truncated octahedral graph
 | image = [[File:Truncated octahedral graph2.png|240px]]
 | image_caption = 3-fold symmetric [[Schlegel diagram]]
 | namesake = 
 | vertices = 24
 | edges = 36
 | automorphisms = 48
 | radius = 
 | diameter = 
 | girth = 
 | chromatic_number = 2
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = [[Cubic graph|Cubic]], [[hamiltonian graph|Hamiltonian]], [[regular graph|regular]], [[Zero-symmetric graph|zero-symmetric]]
|book thickness=3|queue number=2}}
In the [[mathematics|mathematical]] field of [[graph theory]], a '''truncated octahedral graph''' is the [[1-skeleton|graph of vertices and edges]] of the truncated octahedron. It has 24 [[Vertex (graph theory)|vertices]] and 36 edges, and is a [[cubic graph|cubic]] [[Archimedean graph]].<ref>{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=[[Oxford University Press]]|year= 1998|page=269}}</ref> It has [[book thickness]] 3 and [[queue number]] 2.<ref>Wolz, Jessica; ''Engineering Linear Layouts with SAT.'' Master Thesis, University of Tübingen, 2018</ref>

As a [[Hamiltonian graph|Hamiltonian]] [[cubic graph]], it can be represented by [[LCF notation]] in multiple ways: [3, −7, 7, −3]<sup>6</sup>, [5, −11, 11, 7, 5, −5, −7, −11, 11, −5, −7, 7]<sup>2</sup>, and [−11, 5, −3, −7, −9, 3, −5, 5, −3, 9, 7, 3, −5, 11, −3, 7, 5, −7, −9, 9, 7, −5, −7, 3].<ref>{{MathWorld |urlname=TruncatedOctahedralGraph |title=Truncated octahedral graph}}</ref>

[[File:Truncated octahedral Hamiltonicity.svg|600px|thumb|left|Three different Hamiltonian cycles described by the three different [[LCF notation]]s for the truncated octahedral graph]]

{{Clear}}

==References==
{{Reflist|refs=
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<ref name=berman>{{cite journal
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<ref name=budden>{{citation
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<ref name=crisman>{{cite journal
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<ref name=erdahl>{{cite journal
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 }}. Voronoi conjectured that all tilings of higher dimensional spaces by translates of a single [[convex polytope]] are combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case of [[zonotope]]s. But as he writes (p.&nbsp;429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see {{cite journal
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 | last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard
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 | title = Tilings with congruent tiles
 | volume = 3
 | year = 1980| doi-access = free
}}</ref>

<ref name=jj>{{cite book
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 | isbn = 978-3-0348-0554-4
}}</ref>

<ref name=johnson>{{cite journal
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 | s2cid = 122006114
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}}</ref>

<ref name=jtdd>{{cite book
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 | editor-last4 = Unger | editor-first4 = Jonas
 | contribution = Zonohedral Approximation of Spherical Structuring Element for Volumetric Morphology
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 | page = 131&ndash;132
 | title = Image Analysis: 21st Scandinavian Conference, SCIA 2019, Norrköping, Sweden, June 11–13, 2019, Proceedings
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<ref name=kocakoca>{{cite book
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 | page = 48
 | publisher = World Scientific
}}</ref>

<ref name=mizutani>{{cite book
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| isbn = 978-0-521-58709-9
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<ref name=williams>{{cite book
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}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3–9)
*{{cite book
|author=Freitas, Robert A. Jr
|contribution=Figure 5.5: Uniform space-filling using only truncated octahedra
|url=http://www.nanomedicine.com/NMI.htm|title= Nanomedicine, Volume I: Basic Capabilities|publisher= Landes Bioscience|location= Georgetown, Texas|year= 1999
|contribution-url=http://www.nanomedicine.com/NMI/Figures/5.5.jpg
|access-date=2006-09-08}}
*{{cite journal
|author1=Gaiha, P. |author2=Guha, S.K.
 |name-list-style=amp |year=1977
|title=Adjacent vertices on a permutohedron
|journal=SIAM Journal on Applied Mathematics
|volume=32
|issue=2
|pages=323–327
|doi=10.1137/0132025}}
*{{cite encyclopedia
|author=Hart, George W
|contribution=VRML model of truncated octahedron
|url=http://www.georgehart.com/virtual-polyhedra/vp.html |title=Virtual Polyhedra: The Encyclopedia of Polyhedra
|contribution-url=http://www.georgehart.com/virtual-polyhedra/vrml/truncated_octahedron.wrl
|access-date=2006-09-08}}
*{{cite web 
|author=Mäder, Roman
|title=The Uniform Polyhedra: Truncated Octahedron
|url=http://www.mathconsult.ch/showroom/unipoly/08.html
|access-date=2006-09-08}}
*{{cite book
|author=Alexandrov, A.D.
|year=1958
|title=Konvexe Polyeder
|title-link=Convex Polyhedra (book)
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*{{cite book
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}}

==External links==
{{Commons category|Truncated octahedron}}
*{{mathworld2 |urlname=TruncatedOctahedron |title=Truncated octahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
**{{MathWorld |urlname=TruncatedOctahedralGraph |title=Truncated octahedral graph}}
*{{MathWorld |urlname=Permutohedron |title=Permutohedron}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x4o - toe}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=3UtM7vifCnAn5PXmSbX99Eu3LJZAs0nAWn3JyT7et98rnPxTGYml4FXjuQ2tE4viYN0KMgAstBRd0otTWLThQWl9BNNC4uigRoZQUQOQibYqtCuLQw9Ui3OofXtQPEsqQ7#applet Editable printable net of a truncated octahedron with interactive 3D view]

{{Archimedean solids}}
{{Polyhedron navigator}}

[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]
[[Category:Space-filling polyhedra]]
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