Truncated octahedron

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 ${\displaystyle t{\begin{Bmatrix}3\\3\end{Bmatrix}}}$
	t0,1{3,4} or t0,1,2{3,3}
Wythoff symbol	2 4 | 3 3 3 2 |
Coxeter diagram
Symmetry group	Oh, B3, [4,3], (*432), order 48 Th, [3,3] and (*332), order 24
Rotation group	O, [4,3]+, (432), order 24
Dihedral angle	${\displaystyle {\begin{aligned}{\text{4-6:}}&\ \arccos {\tfrac {-1}{\sqrt {3}}}=125^{\circ }15'51''\\{\text{6-6:}}&\ \arccos {\tfrac {-1}{3}}=109^{\circ }28'16''\end{aligned}}}$
References	U08, C20, W7
Properties	Semiregular convex parallelohedron permutohedron zonohedron
Colored faces	4.6.6 (Vertex figure)
Tetrakis hexahedron (dual polyhedron)	Net

In

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.

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 a². Note that this shape is exactly similar to half an octahedron or Johnson solid J₁.

From the properties of square pyramids, we can now find the slant height, s, and the height, h, of the pyramid:

${\displaystyle {\begin{aligned}h&={\sqrt {e^{2}-{\tfrac {1}{2}}a^{2}}}&&={\tfrac {1}{\sqrt {2}}}a\\s&={\sqrt {h^{2}+{\tfrac {1}{4}}a^{2}}}&&={\sqrt {{\tfrac {1}{2}}a^{2}+{\tfrac {1}{4}}a^{2}}}&&={\tfrac {\sqrt {3}}{2}}a\end{aligned}}}$

The volume, V, of the pyramid is given by:

${\displaystyle V={\tfrac {1}{3}}a^{2}h={\tfrac {\sqrt {2}}{6}}a^{3}}$

Because six pyramids are removed by truncation, there is a total lost volume of √2a³.

Orthogonal projections

The truncated octahedron has five special

Orthogonal projections

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

, 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

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

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.^[2]

Removing the central octahedron and 2 or 4 triangular cupolae creates two

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

Area and volume

The surface area S and the volume V of a truncated octahedron of edge length a are:

${\displaystyle {\begin{aligned}S&=\left(6+12{\sqrt {3}}\right)a^{2}&&\approx 26.784\,6097a^{2}\\V&=8{\sqrt {2}}a^{3}&&\approx 11.313\,7085a^{3}.\end{aligned}}}$

Uniform colorings

There are two

is given in parentheses.

1-uniform		2-uniform
Oh, [4,3], (*432) Order 48	Td, [3,3], (*332) Order 24	D4h, [4,2], (*422) Order 16	D3d, [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{31,1}	t{3,4} t{31,1}	{3,4} {31,1}	rr{4,3} s2{3,4}	tr{4,3}	sr{4,3}	h{4,3} {3,3}	h2{4,3} t{3,3}	s{3,4} s{31,1}
		=	=	=				= or	= or	=
Duals to uniform polyhedra
V43	V3.82	V(3.4)2	V4.62	V34	V3.43	V4.6.8	V34.4	V33	V3.62	V35

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 mutation of omnitruncated tilings: 4.6.2n v t e
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 v t e
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

.

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 v t e
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

4.2n.2n, extending into the hyperbolic plane:

*n42 symmetry mutation of truncated tilings: 4.2n.2n v t e
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:

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}

It is possible to slice a tesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.^[4]

Tessellations

The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations):

Bitruncated cubic	Cantitruncated cubic	Truncated alternated cubic

The

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

In the

References

• 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).
  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.

• 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