Truncated octahedron

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.

Truncated octahedron

(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 14, E = 36, V = 24 (χ = 2)
Faces by sides6{4}+8{6}
Conway notationtO
bT
Schläfli symbolst{3,4}
tr{3,3} or
t0,1{3,4} or t0,1,2{3,3}
Wythoff symbol3
3 3 2 |
Coxeter diagram
Symmetry groupOh, B3, [4,3], (*432), order 48
Th, [3,3] and (*332), order 24
Rotation groupO, [4,3]+, (432), order 24
Dihedral angle4-6: arccos(−1/√3) = 125°15′51″
6-6: arccos(−1/3) = 109°28′16″
ReferencesU08, C20, W7
PropertiesSemiregular convex parallelohedron
permutohedron
zonohedron

Colored faces
4.6.6
(Vertex figure)

Tetrakis hexahedron
(dual polyhedron)
Net
3D model of a truncated octahedron

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 cube 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:

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

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, 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]

Spherical tiling

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

Coordinates
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.

Dissection

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.[2]

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

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

Uniform colorings

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

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]

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

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:

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:

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}

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.

Objects
Jungle gym nets often include truncated octahedra.

Truncated octahedral graph
Truncated octahedral graph
3-fold symmetric Schlegel diagram
Vertices24
Edges36
Automorphisms48
Chromatic number2
Book thickness3
Queue number2
PropertiesCubic, 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, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[4] It has book thickness 3 and queue number 2.[5]

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].[6]

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

References
  1. "Truncated Octahedron". Wolfram Mathworld.
  2. Doskey, Alex. "Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1". www.doskey.com.
  3. Perez-Gonzalez, F.; Balado, F.; Martin, J.R.H. (2003). "Performance analysis of existing and new methods for data hiding with known-host information in additive channels". IEEE Transactions on Signal Processing. 51 (4): 960–980. doi:10.1109/TSP.2003.809368.
  4. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
  5. Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
  6. Weisstein, Eric W. "Truncated octahedral graph". MathWorld.
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