# Truncated octahedron

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") 3-dimensional space, as a permutohedron.

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 48T _{h}, [3,3] and (*332), order 24 |

Rotation group | O, [4,3]^{+}, (432), order 24 |

Dihedral angle | 4-6: arccos(−1/√3) = 125°15′51″ 6-6: arccos(−1/3) = 109°28′16″ |

References | U_{08}, C_{20}, W_{7} |

Properties | Semiregular convex parallelohedron permutohedron |

Colored faces |
4.6.6 (Vertex figure) |

Tetrakis hexahedron (dual polyhedron) |
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 cube has edge lengths 9/8√2 and 3/2√2.

## Construction

A truncated octahedron is constructed from a regular octahedron with side length 3*a* 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 √2*a*^{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 4-6 |
Edge 6-6 |
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.

square-centered |
hexagon-centered | |

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 three-dimensional 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.

## Uniform colorings

There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a *truncated triangular antiprism*. The construcational names are given for each. Their Conway polyhedron notation is given in parentheses.

1-uniform | 2-uniform | ||
---|---|---|---|

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) |

## 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}} |

= |
= |
= |
||||||||

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 (4.6.2*p*) and Coxeter–Dynkin diagram *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 | |

n-kis 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 4.2*n*.2*n*, 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.∞.∞ | |||

n-kis 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 5-cube | Bitruncated 6-cube | Bitruncated 7-cube | Bitruncated 8-cube | |

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 (space-filling tessellations):

Bitruncated cubic | Cantitruncated cubic | Truncated alternated cubic |
---|---|---|

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

## 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, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[3] It has book thickness 3 and queue number 2.[4]

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].[5]

## 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*. - 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 0-486-23729-X. (Section 3–9) - Freitas, Robert A., Jr. "Uniform space-filling using only truncated octahedra". Figure 5.5 of Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999. Retrieved 2006-09-08. External link in
`|publisher=`

(help)CS1 maint: multiple names: authors list (link) - 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. External link in
`|publisher=`

(help) - Mäder, Roman. "The Uniform Polyhedra: Truncated Octahedron". Retrieved 2006-09-08.
- Alexandrov, A.D. (1958).
*Convex polyhedra*. 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

Wikimedia Commons has media related to .Truncated octahedron |

- Eric W. Weisstein,
*Truncated octahedron*(*Archimedean solid*) at 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