Truncated octahedron#Truncated octahedral graph

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 pyramids. Setting the edge length of the regular octahedron equal to $3a$, it follows that the length of each edge of a square pyramid (to be removed) is $a$ (the square pyramid has four equilateral triangles as faces, the first Johnson solid). From the equilateral square pyramid's property, its volume is $\backslash tfrac\{\backslash sqrt\{2\}\}\{6\}a^3$. Because six equilateral square pyramids are removed by truncation, the volume of a truncated octahedron $V$ is obtained by subtracting the volume of those six from that of a regular octahedron:{{r|berman}}

$$V\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{3\}\; (3a)^3\; -\; 6\; \backslash cdot\; \backslash frac\{\backslash sqrt\{2\}\}\{6\}\; a^3\; =\; 8\backslash sqrt\{2\}a^3\; \backslash approx\; 11.3137\; a^3.$$

The surface area of a truncated octahedron $A$ can be obtained by summing all polygonals' area, six squares and eight hexagons. Considering the edge length $a$, this is:{{r|berman}}

$$A\; =\; (6\; +\; 12\backslash sqrt\{3\})a^2\; \backslash approx\; 26.7846a^2.$$

File:Truncated octahedron.stl

The truncated octahedron is one of the thirteen Archimedean solids. 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 $\backslash mathrm\{O\}\_\backslash mathrm\{h\}$.{{r|kocakoca}} A square and two hexagons surround each of its vertex, denoting its vertex figure as $4\; \backslash cdot\; 6^2$.{{r|williams}}

The dihedral angle of a truncated octahedron between square-to-hexagon is $\backslash arccos(-1/\backslash sqrt\{3\})\; \backslash approx\; 125.26^\backslash circ$, and that between adjacent hexagonal faces is $\backslash arccos\; (-1/3)\; \backslash approx\; 109.47^\backslash circ$.{{r|johnson}}

The Cartesian coordinates of the vertices of a truncated octahedron with edge length 1 are all permutations of{{cite journal

| last = Chieh | first = C.

| bibcode = 1979AcCrA..35..946C

| date = November 1979

| doi = 10.1107/s0567739479002114

| issue = 6

| journal = Acta Crystallographica Section A

| pages = 946–952

| publisher = International Union of Crystallography (IUCr)

| title = The Archimedean truncated octahedron, and packing of geometric units in cubic crystal structures

| url = https://scholar.archive.org/work/lfzy6gofjvc55nygiq36od3cwi

| volume = 35}}

$$$$

\bigl(\pm\sqrt{2}, \pm\tfrac{\sqrt{2}}{2}, 0\bigr).

{{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 $(1,\; 2,\; 3,\; 4)$ form the vertices of a truncated octahedron in the three-dimensional subspace $x\; +\; y\; +\; z\; +\; w\; =\; 10$.{{r|jj}} Therefore, each vertex corresponds to a permutation of $(1,\; 2,\; 3,\; 4)$ 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 $S\_4$, the group of four-element permutations, 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, translated without rotating, can be repeated to fill space. There are five three-dimensional primary parallelohedrons, one of which is the truncated octahedron. This polyhedron is generated from six line segments with four triples of coplanar segments, with the most symmetric form being generated from six line segments parallel to the face diagonals of a cube;{{r|alexandrov}} an example of the honeycomb is the bitruncated cubic honeycomb.{{r|tz}} 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}}

{{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 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.{{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 }}

The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupolae on each face, and 6 square pyramids above the vertices.{{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}}

{{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 toroids, 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.{{cite book

| 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}}

The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centered cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedra.

{{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.

}}

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

{{Infobox graph

| name = Truncated octahedral graph

| image = 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, Hamiltonian, regular, zero-symmetric

|book thickness=3|queue number=2}}

In the mathematical field of graph theory, a truncated octahedral graph is the graph of vertices and edges of the truncated octahedron. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.{{cite book|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=Oxford University Press|year= 1998|page=269}} It has book thickness 3 and queue number 2.Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018

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

File:Truncated octahedral Hamiltonicity.svgs for the truncated octahedral graph]]

{{Clear}}

{{Reflist|refs=

{{cite book

| last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov

| contribution = 8.1 Parallelohedra

| contribution-url = https://books.google.com/books?id=R9vPatr5aqYC&pg=PA349

| pages = 349–359

| publisher = Springer

| title = Convex Polyhedra | title-link = Convex Polyhedra (book)

| year = 2005

}}

{{cite journal

| last = Berman | first = Martin

| year = 1971

| title = Regular-faced convex polyhedra

| journal = Journal of the Franklin Institute

| volume = 291

| issue = 5

| pages = 329–352

| doi = 10.1016/0016-0032(71)90071-8

| mr = 290245

}}

{{citation

| last = Budden | first = Frank

| date = December 1985

| doi = 10.2307/3617571

| issue = 450

| journal = The Mathematical Gazette

| jstor = 3617571

| pages = 271–278

| publisher = JSTOR

| title = Cayley graphs for some well-known groups

| volume = 69}}

{{cite journal

| last = Crisman | first = Karl-Dieter

| year = 2011

| title = The Symmetry Group of the Permutahedron

| journal = The College Mathematics Journal

| volume = 42 | issue = 2 | pages = 135–139

| doi = 10.4169/college.math.j.42.2.135

| jstor = college.math.j.42.2.135

}}

{{cite book

| last = Diudea | first = M. V.

| year = 2018

| title = Multi-shell Polyhedral Clusters

| series = Carbon Materials: Chemistry and Physics

| volume = 10

| publisher = Springer

| isbn = 978-3-319-64123-2

| doi = 10.1007/978-3-319-64123-2

| url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39

| page = 39

}}

{{cite journal

| last = Erdahl | first = R. M.

| doi = 10.1006/eujc.1999.0294

| issue = 6

| journal = European Journal of Combinatorics

| mr = 1703597

| pages = 527–549

| title = Zonotopes, dicings, and Voronoi's conjecture on parallelohedra

| volume = 20

| year = 1999| doi-access = free

}}. 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 zonotopes. But as he writes (p. 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

| last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum

| last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard

| doi = 10.1090/S0273-0979-1980-14827-2

| issue = 3

| journal = Bulletin of the American Mathematical Society

| mr = 585178

| pages = 951–973

| series = New Series

| title = Tilings with congruent tiles

| volume = 3

| year = 1980| doi-access = free

}}

{{cite book

| last1 = Johnson | first1 = Tom

| last2 = Jedrzejewski | first2 = Franck

| year = 2014

| title = Looking at Numbers

| publisher = Springer

| url = https://books.google.com/books?id=xtE-AgAAQBAJ&pg=PA15

| page = 15

| doi = 10.1007/978-3-0348-0554-4

| isbn = 978-3-0348-0554-4

}}

{{cite journal

| last = Johnson | first = Norman W. | authorlink = Norman W. Johnson

| year = 1966

| title = Convex polyhedra with regular faces

| journal = Canadian Journal of Mathematics

| volume = 18

| pages = 169–200

| doi = 10.4153/cjm-1966-021-8

| mr = 0185507

| s2cid = 122006114

| zbl = 0132.14603| doi-access = free

}}

{{cite book

| last1 = Jensen | first1 = Patrick M.

| last2 = Trinderup | first2 = Camilia H.

| last3 = Dahl | first3 = Anders B.

| last4 = Dahl | first4 = Vedrana A.

| year = 2019

| editor-last1 = Felsberg | editor-first1 = Michael

| editor-last2 = Forssén | editor-first2 = Per-Erik

| editor-last3 = Sintorn | editor-first3 = Ida-Maria

| editor-last4 = Unger | editor-first4 = Jonas

| contribution = Zonohedral Approximation of Spherical Structuring Element for Volumetric Morphology

| contribution-url = https://books.google.com/books?id=uLWZDwAAQBAJ&pg=PA131

| page = 131–132

| title = Image Analysis: 21st Scandinavian Conference, SCIA 2019, Norrköping, Sweden, June 11–13, 2019, Proceedings

| publisher = Springer

| doi = 10.1007/978-3-030-20205-7

| isbn = 978-3-030-20205-7

}}

{{cite book

| last1 = Koca | first1 = M.

| last2 = Koca | first2 = N. O.

| year = 2013

| title = Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010

| contribution = Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes

| contribution-url = https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48

| page = 48

| publisher = World Scientific

}}

{{cite book

| last = Mizutani | first = Uichiro

| year = 2001

| title = Introduction to the Electron Theory of Metals

| url = https://books.google.com/books?id=zY5z_UGqAcwC&pg=PA112

| page = 112

| publisher = Cambridge University Press

| isbn = 978-0-521-58709-9

}}

{{cite journal

| last1 = Thuswaldner | first1 = Jörg

| last2 = Zhang | first2 = Shu-qin

| arxiv = 1811.06718

| doi = 10.1090/tran/7930

| issue = 1

| journal = Transactions of the American Mathematical Society

| mr = 4042883

| pages = 491–527

| title = On self-affine tiles whose boundary is a sphere

| volume = 373

| year = 2020

}}

{{cite book

| last = Williams | first = Robert | authorlink = Robert Williams (geometer)

| year = 1979

| title = The Geometrical Foundation of Natural Structure: A Source Book of Design

| publisher = Dover Publications, Inc.

| url = https://archive.org/details/geometricalfound00will/page/78

| page = 78

| isbn = 978-0-486-23729-9 }}

{{cite book

| last = Yen | first = Teh F.

| year = 2007

| title = Chemical Processes for Environmental Engineering

| publisher = Imperial College Press

| url = https://books.google.com/books?id=KHGelcDf1qQC&pg=PA388

| page = 338

| isbn = 978-1-86094-759-9

}}

}}

• {{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 book

|author=Cromwell, P.

|year=1997

|title=Polyhedra

|location=United Kingdom

|publisher=Cambridge

|pages=79–86 Archimedean solids

|isbn=0-521-55432-2

}}

{{Commons category|Truncated octahedron}}

• {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x4o - toe}}
• {{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}}

• [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

Category:Truncated tilings

Category:Zonohedra