Truncated octahedron


Truncated octahedron
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-Dynkin CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry Oh, [4,3], (*432)
Th, [3,3] and (*332)
Dihedral Angle
References U08, C20, W7
Properties Semiregular convex zonohedron
permutohedron
Truncated octahedron color
Colored faces
Truncated octahedron
4.6.6
(Vertex figure)
Tetrakishexahedron.jpg
Tetrakis hexahedron
(dual polyhedron)
Truncated octahedron Net
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 \frac{9}{8}\sqrt{2} and \frac{3}{2}\sqrt{2}.

Contents


Construction

Truncated Octahedron with Construction.svg   Square Pyramid.svg

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 √2 a³.

Images

Orthographic projections
3-cube t12.svg 3-cube t12 B2.svg

Coordinates and permutohedron

All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.

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.

Permutohedron.svg

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)
Truncated Octahedron 122 Colouring.svg
122 coloring
Wythoff: 2 4 | 3
Truncated Octahedron 123 Colouring.svg
123 coloring
Wythoff: 3 3 2 |

Related polyhedra

The truncated octahedron exists within the set of truncated forms between a cube and octahedron:

Uniform polyhedron-43-t0.png
Cube
 
Uniform polyhedron-43-t01.png
Truncated cube
 
Uniform polyhedron-43-t1.png
cuboctahedron
 
Uniform polyhedron-43-t12.png
Truncated
octahedron
Uniform polyhedron-43-t2.png
Octahedron
 

It also exists as the omnitruncate of the tetrahedron family:

Uniform polyhedron-33-t0.png
Tetrahedron
 
Uniform polyhedron-33-t01.png
Truncated
tetrahedron
Uniform polyhedron-33-t1.png
Rectified
tetrahedron
Uniform polyhedron-33-t02.png
Cantellated
tetrahedron
Uniform polyhedron-33-t012.png
Omnitruncated
tetrahedron
Uniform polyhedron-33-s012.png
Snub
tetrahedron

Tessellations

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

Bitruncated cubic Cantitruncated cubic Truncated alternated cubic
Bitruncated Cubic Honeycomb.svg Cantitruncated Cubic Honeycomb.svg Truncated Alternated Cubic Honeycomb.svg

The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centred cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedron.

References

  • 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. 
  • Alexandrov, A. D. (1958). Convex polyhedra. Berlin : Springer, cop.. pp. 539. doi:3-540-23158-7. 

External links


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