Truncated octahedron Information
Truncated octahedron  

(Click here for rotating model)  
Type 
Archimedean solid Uniform polyhedron 
Elements  F = 14, E = 36, V = 24 (χ = 2) 
Faces by sides  6{4}+8{6} 
Conway notation  tO bT 
Schläfli symbols  t{3,4} tr{3,3} or 
t_{0,1}{3,4} or t_{0,1,2}{3,3}  
Wythoff symbol  2 4  3 3 3 2  
Coxeter diagram  
Symmetry group 
O_{h}, B_{3}, [4,3], (*432), order 48 T_{h}, [3,3] and (*332), order 24 
Rotation group  O, [4,3]^{+}, (432), order 24 
Dihedral angle  46: arccos(−1/√3) = 125°15′51″ 66: arccos(−1/3) = 109°28′16″ 
References  U_{08}, C_{20}, W_{7} 
Properties  Semiregular
convex
parallelohedron permutohedron zonohedron 
Colored faces 
4.6.6 ( Vertex figure) 
Tetrakis hexahedron ( dual polyhedron) 
Net 
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron G_{IV}(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3dimensional space, as a permutohedron.
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 cube has edge lengths 9/8√2 and 3/2√2.
Construction
A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point. These pyramids have both base side length (a) and lateral side length (e) of a, to form equilateral triangles. The base area is then a^{2}. Note that this shape is exactly similar to half an octahedron or Johnson solid J_{1}.
From the properties of square pyramids, we can now find the slant height, s, and the height, h, of the pyramid:
The volume, V, of the pyramid is given by:
Because six pyramids are removed by truncation, there is a total lost volume of √2a^{3}.
Orthogonal projections
The truncated octahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B_{2} and A_{2} Coxeter planes.
Centered by  Vertex  Edge 46 
Edge 66 
Face Square 
Face Hexagon 

Solid  
Wireframe  
Dual  
Projective symmetry 
[2]  [2]  [2]  [4]  [6] 
Spherical tiling
The truncated octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
squarecentered 
hexagoncentered  
Orthographic projection  Stereographic projections 

Coordinates
Orthogonal projection in
bounding box (±2,±2,±2) 
Truncated octahedron with hexagons replaced by 6 coplanar triangles. There are 8 new vertices at: (±1,±1,±1).  Truncated octahedron subdivided into as a topological rhombic triacontahedron 
All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.
The edge vectors have Cartesian coordinates (0, ±1, ±1) and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are (0, 0, ±1), (0, ±1, 0) and (±1, 0, 0). The face normals of the 8 hexagonal faces are (±1/√3, ±1/√3, ±1/√3). The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. The dihedral angle is approximately 1.910633 radians (109.471° OEIS: A156546) at edges shared by two hexagons or 2.186276 radians (125.263° OEIS: A195698) at edges shared by a hexagon and a square.
Dissection
The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupola on each face, and 6 square pyramids above the vertices.^{ [2]}
Removing the central octahedron and 2 or 4 triangular cupola creates two Stewart toroids, with dihedral and tetrahedral symmetry:
Genus 2  Genus 3 

D_{3d}, [2^{+},6], (2*3), order 12  T_{d}, [3,3], (*332), order 24 
Permutohedron
The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1, 2, 3, 4) form the vertices of a truncated octahedron in the threedimensional subspace x + y + z + w = 10. Therefore, the truncated octahedron is the permutohedron of order 4: each vertex corresponds to a permutation of (1, 2, 3, 4) and each edge represents a single pairwise swap of two elements.
Area and volume
The area A and the volume V of a truncated octahedron of edge length a are:
Uniform colorings
There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry, and two 2uniform coloring with dihedral symmetry as a truncated triangular antiprism. The construcational names are given for each. Their Conway polyhedron notation is given in parentheses.
1uniform  2uniform  

O_{h}, [4,3], (*432) Order 48 
T_{d}, [3,3], (*332) Order 24 
D_{4h}, [4,2], (*422) Order 16 
D_{3d}, [2^{+},6], (2*3) Order 12 
122 coloring 
123 coloring 
122 & 322 colorings 
122 & 123 colorings 
Truncated octahedron (tO) 
Bevelled tetrahedron (bT) 
Truncated square bipyramid (tdP4) 
Truncated triangular antiprism (tA3) 
Chemistry
The truncated octahedron exists in the structure of the faujasite crystals.
Data Hiding
The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.^{ [3]}
Related polyhedra
The truncated octahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2)  
{4,3}  t{4,3} 
r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3} 
h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  




 
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
It also exists as the omnitruncate of the tetrahedron family:
Family of uniform tetrahedral polyhedra  

Symmetry: [3,3], (*332)  [3,3]^{+}, (332)  
{3,3}  t{3,3}  r{3,3}  t{3,3}  {3,3}  rr{3,3}  tr{3,3}  sr{3,3} 
Duals to uniform polyhedra  
V3.3.3  V3.6.6  V3.3.3.3  V3.6.6  V3.3.3  V3.4.3.4  V4.6.6  V3.3.3.3.3 
Symmetry mutations
*n32 symmetry mutations of omnitruncated tilings: 4.6.2n  

Sym. *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3] 
*∞32 [∞,3] 
[12i,3] 
[9i,3] 
[6i,3] 
[3i,3]  
Figures  
Config.  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞  4.6.24i  4.6.18i  4.6.12i  4.6.6i 
Duals  
Config.  V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞  V4.6.24i  V4.6.18i  V4.6.12i  V4.6.6i 
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n  

Symmetry *nn2 [n,n] 
Spherical  Euclidean  Compact hyperbolic  Paracomp.  
*222 [2,2] 
*332 [3,3] 
*442 [4,4] 
*552 [5,5] 
*662 [6,6] 
*772 [7,7] 
*882 [8,8]... 
*∞∞2 [∞,∞]  
Figure  
Config.  4.4.4  4.6.6  4.8.8  4.10.10  4.12.12  4.14.14  4.16.16  4.∞.∞  
Dual  
Config.  V4.4.4  V4.6.6  V4.8.8  V4.10.10  V4.12.12  V4.14.14  V4.16.16  V4.∞.∞ 
This polyhedron is a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter–Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra ( zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures n.6.6, extending into the hyperbolic plane:
*n32 symmetry mutation of truncated tilings: n.6.6  

Sym. *n42 [n,3] 
Spherical  Euclid.  Compact  Parac.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 
[12i,3]  [9i,3]  [6i,3]  
Truncated figures 

Config.  2.6.6  3.6.6  4.6.6  5.6.6  6.6.6  7.6.6  8.6.6  ∞.6.6  12i.6.6  9i.6.6  6i.6.6  
nkis figures 

Config.  V2.6.6  V3.6.6  V4.6.6  V5.6.6  V6.6.6  V7.6.6  V8.6.6  V∞.6.6  V12i.6.6  V9i.6.6  V6i.6.6 
The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
*n42 symmetry mutation of truncated tilings: 4.2n.2n  

Symmetry *n42 [n,4] 
Spherical  Euclidean  Compact hyperbolic  Paracomp.  
*242 [2,4] 
*342 [3,4] 
*442 [4,4] 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4]  
Truncated figures 

Config.  4.4.4  4.6.6  4.8.8  4.10.10  4.12.12  4.14.14  4.16.16  4.∞.∞  
nkis figures 

Config.  V4.4.4  V4.6.6  V4.8.8  V4.10.10  V4.12.12  V4.14.14  V4.16.16  V4.∞.∞ 
Related polytopes
The truncated octahedron ( bitruncated cube), is first in a sequence of bitruncated hypercubes:
Image  ...  

Name  Bitruncated cube  Bitruncated tesseract  Bitruncated 5cube  Bitruncated 6cube  Bitruncated 7cube  Bitruncated 8cube  
Coxeter  
Vertex figure 
( )v{ } 
{ }v{ } 
{ }v{3} 
{ }v{3,3} 
{ }v{3,3,3}  { }v{3,3,3,3} 
Tessellations
The truncated octahedron exists in three different convex uniform honeycombs ( spacefilling tessellations):
Bitruncated cubic  Cantitruncated cubic  Truncated alternated cubic 

The celltransitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the bodycentered cubic lattice. The truncated octahedron is one of five threedimensional primary parallelohedra.
Objects
sculpture in Bonn
Rubik's Cube variant
model made with Polydron construction set
Pyrite crystal
Truncated octahedral graph
Truncated octahedral graph  

3fold symmetric
Schlegel diagram  
Vertices  24 
Edges  36 
Automorphisms  48 
Chromatic number  2 
Book thickness  3 
Queue number  2 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
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, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.^{ [4]} It has book thickness 3 and queue number 2.^{ [5]}
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].^{ [6]}
References
 ^ "Truncated Octahedron". Wolfram Mathworld.
 ^ Doskey, Alex. "Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1". www.doskey.com.
 ^ PerezGonzalez, F.; Balado, F.; Martin, J.R.H. (2003). "Performance analysis of existing and new methods for data hiding with knownhost information in additive channels". IEEE Transactions on Signal Processing. 51 (4): 960–980. doi: 10.1109/TSP.2003.809368.
 ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
 ^ Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
 ^ 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 048623729X. (Section 3–9)
 Freitas, Robert A. Jr.
"Uniform spacefilling using only truncated octahedra". Figure 5.5 of
Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999. Retrieved 20060908. External link in
publisher=
( help)  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 20060908. External link in
publisher=
( help)  Mäder, Roman. "The Uniform Polyhedra: Truncated Octahedron". Retrieved 20060908.
 Alexandrov, A.D. (1958). Konvexe Polyeder. Berlin: Springer. p. 539. ISBN 3540231587.
 Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0521554322.
External links
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