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}
Schläfli symbol	t0,1{3,4} t0,1,2{3,3}
Wythoff symbol	2 4 | 3 3 3 2 |
Coxeter diagram
Symmetry group	Oh, BC3, [4,3], (*432), order 48 Th, [3,3] and (*332), order 24
Rotation group	O, [4,3]+, (432), order 24
Dihedral Angle	4-6:cos(-1/sqrt(3))=125°15'51" 6-6:cos(-1/3)=109°28'16"
References	U08, C20, W7
Properties	Semiregular convex zonohedron permutohedron
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.

If the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths $\tfrac{9}{8}\scriptstyle {\sqrt{2}}$ and $\tfrac{3}{2}\scriptstyle{\sqrt{2}}$.

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:

$h = \sqrt{e^2-\frac{1}{2}a^2}=\frac{\sqrt{2}}{2}a$

$s = \sqrt{h^2 + \frac{1}{4}a^2} = \sqrt{\frac{1}{2}a^2 + \frac{1}{4}a^2}=\frac{\sqrt{3}}{2}a$

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

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

Because six pyramids are removed by truncation, there is a total lost volume of $\scriptstyle {\sqrt{2}a^3}$.

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 B₂ and A₂ Coxeter planes.

Orthogonal projections

Centered by	Vertex	Edge 4-6	Edge 6-6	Face Square	Face Hexagon
Image
Projective symmetry	[2]	[2]	[2]	[4]	[6]

Coordinates and permutohedron

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 rad (109.471 °  A156546) at edges shared by two hexagons or 2.186276 rad (125.263 °  A195698) at edges shared by a hexagon and a square.

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.

Area and volume

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

$A = (6+12\sqrt{3}) a^2 \approx 26.7846097a^2$

$V = 8\sqrt{2} a^3 \approx 11.3137085a^3.$

Uniform colorings

There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry:

Octahedral symmetry	Tetrahedral symmetry (Omnitruncated tetrahedron)
122 coloring Wythoff: 2 4 | 3	123 coloring Wythoff: 3 3 2 |

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], (*332)	[4,3+], (3*2)
{4,3}	t0,1{4,3}	t1{4,3}	t1,2{4,3}	{3,4}	t0,2{4,3}	t0,1,2{4,3}	s{4,3}	h{4,3}	h1,2{4,3}
Duals to uniform polyhedra
V4.4.4	V3.8.8	V3.4.3.4	V4.6.6	V3.3.3.3	V3.4.4.4	V4.6.8	V3.3.3.3.4	V3.3.3	V3.3.3.3.3

It also exists as the omnitruncate of the tetrahedron family:

Family of uniform tetrahedral polyhedra

Symmetry: [3,3], (*332)							[3,3]+, (332)
{3,3}	t0,1{3,3}	t1{3,3}	t1,2{3,3}	t2{3,3}	t0,2{3,3}	t0,1,2{3,3}	s{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

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n

Symmetry *n32 [n,3]	Spherical				Euclidean	Hyperbolic
	*232 [2,3] D3h	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]	*∞32 [∞,3]
Coxeter Schläfli	t0,1,2{2,3}	t0,1,2{3,3}	t0,1,2{4,3}	t0,1,2{5,3}	t0,1,2{6,3}	t0,1,2{7,3}	t0,1,2{8,3}	t0,1,2{∞,3}
Omnitruncated figure
Vertex figure	4.6.4	4.6.6	4.6.8	4.6.10	4.6.12	4.6.14	4.6.16	4.6.∞
Dual figures
Coxeter
Omnitruncated duals
Face configuration	V4.6.4	V4.6.6	V4.6.8	V4.6.10	V4.6.12	V4.6.14	V4.6.16	V4.6.∞

This polyhedron can be considered 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:

Dimensional family of truncated polyhedra and tilings: n.6.6

Symmetry *n42 [n,3]	Spherical				Euclidean	Hyperbolic...
	*232 [2,3] D3h	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Order	12	24	48	120	∞
Truncated figures	2.6.6	3.6.6	4.6.6	5.6.6	6.6.6	7.6.6	8.6.6	∞.6.6
Coxeter Schläfli	t0,1{3,2}	t0,1{3,3}	t0,1{3,4}	t0,1{3,5}	t0,1{3,6}	t0,1{3,7}	t0,1{3,8}	t0,1{3,∞}
Uniform dual figures
n-kis figures	V2.6.6	V3.6.6	V4.6.6	V5.6.6	V6.6.6	V7.6.6	V8.6.6	V∞.6.6
Coxeter

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:

Dimensional family of truncated polyhedra and tilings: 4.2n.2n

Symmetry *n42 [n,4]	Spherical		Euclidean	Hyperbolic...
	*242 [2,4] D4h	*342 [3,4] Oh	*442 [4,4] P4m	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]
Truncated figures	4.4.4	4.6.6	4.8.8	4.10.10	4.12.12	4.14.14	4.16.16	4.∞.∞
Coxeter Schläfli	t1,2{4,2}	t1,2{4,3}	t1,2{4,4}	t1,2{4,5}	t1,2{4,6}	t1,2{4,7}	t1,2{4,8}	t1,2{4,∞}
Uniform dual figures
n-kis figures	V4.4.4	V4.6.6	V4.8.8	V4.10.10	V4.12.12	V4.14.14	V4.16.16	V4.∞.∞
Coxeter

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.

References

• 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. 
• Gaiha, P., and 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). Convex polyhedra. Berlin: Springer. p. 539. ISBN 3-540-23158-7. 

External links

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