Truncated octahedron

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 hexagons and 6 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_IV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron.

Truncated octahedron
Type	Archimedean solid, Parallelohedron, Permutohedron, Plesiohedron, Zonohedron
Faces	14
Edges	36
Vertices	24
Symmetry group	octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$
Dual polyhedron	tetrakis hexahedron
Vertex figure
Net

The truncated octahedron was called the "mecon" by Buckminster Fuller.^[1]

Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths ⁠9/8⁠√2 and ⁠3/2⁠√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 pyramids. Considering that each length of the regular octahedron is ${\displaystyle 3a}$, and the edge length of a square pyramid is ${\displaystyle a}$ (the square pyramid is an equilateral, the first Johnson solid). From the equilateral square pyramid's property, its volume is ${\textstyle {\tfrac {\sqrt {2}}{6}}a^{3}}$. Because six equilateral square pyramids are removed by truncation, the volume of a truncated octahedron ${\displaystyle V}$ is obtained by subtracting the volume of a regular octahedron from those six:^[2] $${\displaystyle V={\frac {\sqrt {2}}{3}}(3a)^{3}-6\cdot {\frac {\sqrt {2}}{6}}a^{3}=8a^{3}{\sqrt {2}}\approx 11.3137.}$$ The surface area of a truncated octahedron can be obtained by summing all polygonals' area, six squares and eight hexagons. Considering the edge length ${\displaystyle a}$, this is:^[2] $${\displaystyle (6+12{\sqrt {3}})a^{2}\approx 26.7846a^{2}.}$$

3D model of a truncated octahedron

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.^[3] 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 ${\displaystyle \mathrm {O} _{\mathrm {h} }}$.^[4] A square and two hexagons surround each of its vertex, denoting its vertex figure as ${\displaystyle 4\cdot 6^{2}}$.^[5]

The dihedral angle of a truncated octahedron between square-to-hexagon is ${\textstyle \arccos(-1/{\sqrt {3}})\approx 125.26^{\circ }}$, and that between adjacent hexagonal faces is ${\textstyle \arccos(-1/3)\approx 109.47^{\circ }}$.^[6]

As a tilling space polyhedron

Truncated octahedron as a permutahedron of order 4

Truncated octahedron in tilling space

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 ${\displaystyle (1,2,3,4)}$ form the vertices of a truncated octahedron in the three-dimensional subspace ${\displaystyle x+y+z+w=10}$.^[7] Therefore, each vertex corresponds to a permutation of ${\displaystyle (1,2,3,4)}$ and each edge represents a single pairwise swap of two elements. It has the symmetric group ${\displaystyle S_{4}}$.^[8]

The truncated octahedron can be used as a tilling space. It is classified as plesiohedron, meaning it can be defined as the Voronoi cell of a symmetric Delone set.^[9] The plesiohedron includes the parallelohedron, a polyhedron can be translated without rotating and tilling space so that it fills the entire face. There are five three-dimensional primary parallelohedrons, one of which is the truncated octahedron.^[10] More generally, every permutohedron and parallelohedron is zonohedron, a polyhedron that is centrally symmetric that can be defined by using Minkowski sum.^[11]

Applications

The structure of the faujasite framework

First Brillouin zone of FCC lattice, showing symmetry labels for high symmetry lines and points.

In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a faujasite-type of zeolite crystals.^[12]

In solid-state physics, the first Brillouin zone of the face-centered cubic lattice is a truncated octahedron.^[13]

The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.^[14]

Dissection

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.^[15]

Removing the central octahedron and 2 or 4 triangular cupolae creates two Stewart toroids, with dihedral and tetrahedral symmetry:

Genus 2	Genus 3
D3d, [2+,6], (2*3), order 12	Td, [3,3], (*332), order 24

It is possible to slice a tesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.^[16]

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.

Objects

Jungle gym nets often include truncated octahedra.

• 
  ancient Chinese die
• 
  sculpture in Bonn
• 
  Rubik's Cube variant
• 
  model made with Polydron construction set
• 
  Pyrite crystal
• 
  Boleite crystal

Truncated octahedral graph

Truncated octahedral graph
3-fold symmetric Schlegel diagram
Vertices	24
Edges	36
Automorphisms	48
Chromatic number	2
Book thickness	3
Queue number	2
Properties	Cubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

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.^[17] It has book thickness 3 and queue number 2.^[18]

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].^[19]

Three different Hamiltonian cycles described by the three different LCF notations for the truncated octahedral graph

References

1. "Truncated Octahedron". Wolfram Mathworld.
2. Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
3. Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
4. Koca, M.; Koca, N. O. (2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010. World Scientific. p. 48.
5. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 78. ISBN 978-0-486-23729-9.
6. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.
7. Johnson, Tom; Jedrzejewski, Franck (2014). Looking at Numbers. Springer. p. 15. doi:10.1007/978-3-0348-0554-4. ISBN 978-3-0348-0554-4.
8. Crisman, Karl-Dieter (2011). "The Symmetry Group of the Permutahedron". The College Mathematics Journal. 42 (2): 135–139. doi:10.4169/college.math.j.42.2.135. JSTOR college.math.j.42.2.135.
9. Erdahl, R. M. (1999). "Zonotopes, dicings, and Voronoi's conjecture on parallelohedra". European Journal of Combinatorics. 20 (6): 527–549. doi:10.1006/eujc.1999.0294. MR 1703597.. 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 Grünbaum, Branko; Shephard, G. C. (1980). "Tilings with congruent tiles". Bulletin of the American Mathematical Society. New Series. 3 (3): 951–973. doi:10.1090/S0273-0979-1980-14827-2. MR 0585178.
10. Alexandrov, A. D. (2005). "8.1 Parallelohedra". Convex Polyhedra. Springer. pp. 349–359.
11. Jensen, Patrick M.; Trinderup, Camilia H.; Dahl, Anders B.; Dahl, Vedrana A. (2019). "Zonohedral Approximation of Spherical Structuring Element for Volumetric Morphology". In Felsberg, Michael; Forssén, Per-Erik; Sintorn, Ida-Maria; Unger, Jonas (eds.). Image Analysis: 21st Scandinavian Conference, SCIA 2019, Norrköping, Sweden, June 11–13, 2019, Proceedings. Springer. p. 131–132. doi:10.1007/978-3-030-20205-7. ISBN 978-3-030-20205-7.
12. Yen, Teh F. (2007). Chemical Processes for Environmental Engineering. Imperial College Press. p. 338. ISBN 978-1-86094-759-9.
13. Mizutani, Uichiro (2001). Introduction to the Electron Theory of Metals. Cambridge University Press. p. 112. ISBN 978-0-521-58709-9.
14. Perez-Gonzalez, F.; Balado, F.; Martin, J.R.H. (2003). "Performance analysis of existing and new methods for data hiding with known-host information in additive channels". IEEE Transactions on Signal Processing. 51 (4): 960–980. Bibcode:2003ITSP...51..960P. doi:10.1109/TSP.2003.809368.
15. Doskey, Alex. "Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1". www.doskey.com.
16. Borovik, Alexandre V.; Borovik, Anna (2010), "Exercise 14.4", Mirrors and Reflections, Universitext, New York: Springer, p. 109, doi:10.1007/978-0-387-79066-4, ISBN 978-0-387-79065-7, MR 2561378
17. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
18. Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
19. Weisstein, Eric W. "Truncated octahedral graph". MathWorld.

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3–9)
• Freitas, Robert A. Jr (1999). "Figure 5.5: Uniform space-filling using only truncated octahedra". Nanomedicine, Volume I: Basic Capabilities. Georgetown, Texas: Landes Bioscience. Retrieved 2006-09-08.
• Gaiha, P. & Guha, S.K. (1977). "Adjacent vertices on a permutohedron". SIAM Journal on Applied Mathematics. 32 (2): 323–327. doi:10.1137/0132025.
• Hart, George W. "VRML model of truncated octahedron". Virtual Polyhedra: The Encyclopedia of Polyhedra. Retrieved 2006-09-08.
• Mäder, Roman. "The Uniform Polyhedra: Truncated Octahedron". Retrieved 2006-09-08.
• Alexandrov, A.D. (1958). Konvexe Polyeder. Berlin: Springer. p. 539. ISBN 3-540-23158-7.
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.

External links

• Weisstein, Eric W., "Truncated octahedron" ("Archimedean solid") at MathWorld.
  • Weisstein, Eric W. "Truncated octahedral graph". MathWorld.
• Weisstein, Eric W. "Permutohedron". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x4o - toe".
• Editable printable net of a truncated octahedron with interactive 3D view