Truncated octahedron

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.

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 B₂ and A₂ Coxeter planes.

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]

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

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.

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 Stewart toroids, with dihedral and tetrahedral symmetry:

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

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.

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

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

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 constructional names are given for each. Their Conway polyhedron notation 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)

The truncated octahedron exists in the structure of the faujasite crystals.

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]

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
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
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
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.∞.∞

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]

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.

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
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.[5] It has book thickness 3 and queue number 2.[6]

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].[7]

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

• 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. {{cite web}}: 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 2006-09-08. {{cite web}}: External link in |publisher= (help)
• 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.

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