Truncated octahedron

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Truncated octahedron
Truncatedoctahedron.jpg
(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}
t0,1{3,4} or t0,1,2{3,3}
Wythoff symbol 2 4 | 3
3 3 2 |
Coxeter diagram 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 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:arccos(-1/sqrt(3))=125°15'51"
6-6:cos(-1/3)=109°28'16"
References U08, C20, W7
Properties Semiregular convex parallelohedron
permutohedron
Truncated octahedron.png
Colored faces
Truncated octahedron vertfig.png
4.6.6
(Vertex figure)
Tetrakishexahedron.jpg
Tetrakis hexahedron
(dual polyhedron)
Truncated Octahedron Net.svg
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. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron.

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[edit]

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 \scriptstyle {\sqrt{2}a^3}.

Orthogonal projections[edit]

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
Truncated
octahedron
Cube t12 v.png Cube t12 e46.png Cube t12 e66.png 3-cube t12 B2.svg 3-cube t12.svg
Hexakis
hexahedron
Dual cube t12 v.png Dual cube t12 e46.png Dual cube t12 e66.png Dual cube t12 B2.png Dual cube t12.png
Projective
symmetry
[2] [2] [2] [4] [6]

Spherical tiling[edit]

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.

Uniform tiling 432-t12.png Truncated octahedron stereographic projection square.png
square-centered
Truncated octahedron stereographic projection hexagon.png
hexagon-centered
Orthographic projection Stereographic projections

Coordinates[edit]

Truncated octahedron in unit cube.png Triangulated truncated octahedron.png
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).

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 ° OEISA156546) at edges shared by two hexagons or 2.186276 rad (125.263 ° OEISA195698) at edges shared by a hexagon and a square.

Dissection[edit]

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.[1]

Removing the central octahedron and 2 or 4 triangular cupola 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
Excavated truncated octahedron1.png Excavated truncated octahedron2.png

Permutohedron[edit]

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.

Permutohedron.svg

Area and volume[edit]

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

A = \left(6+12\sqrt{3}\right) a^2 \approx 26.7846097a^2
V = 8\sqrt{2} a^3 \approx 11.3137085a^3.

Uniform colorings[edit]

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
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
Uniform polyhedron-43-t12.svg
122 coloring
Uniform polyhedron-33-t012.png
123 coloring
Truncated square bipyramid.png
122 & 322 colorings
Truncated octahedron prismatic symmetry.png
122 & 123 colorings
Truncated octahedron
(tO)
Bevelled tetrahedron
(bT)
Truncated square bipyramid
(tdP4)
Truncated triangular antiprism
(tA3)

Related polyhedra[edit]

The truncated octahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

It also exists as the omnitruncate of the tetrahedron family:

Symmetry mutations[edit]

This polyhedron is a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. 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:

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:

Related polytopes[edit]

The truncated octahedron (bitruncated cube), is first in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
3-cube t12.svgTruncated octahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t12.svg5-cube t12 A3.svg 6-cube t12.svg6-cube t12 A5.svg 7-cube t12.svg7-cube t12 A5.svg 8-cube t12.svg8-cube t12 A7.svg ...
Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Tessellations[edit]

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-centered cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedra.

Truncated octahedral graph[edit]

Truncated octahedral graph
Truncated octahedral graph2.png
3-fold symmetric schlegel diagram
Vertices 24
Edges 36
Automorphisms 48
Chromatic number 2
Properties Cubic, Hamiltonian, regular, zero-symmetric

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.[2]

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].[3]

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

Truncated octahedral graph.neato.svg

References[edit]

  1. ^ http://www.doskey.com/polyhedra/Stewart05.html
  2. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 
  3. ^ Weisstein, Eric W., "Truncated octahedral graph", MathWorld.

External links[edit]