Ingeometry, thetruncated octahedron is theArchimedean solid that arises from a regularoctahedron by removing sixequilateral square pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regularhexagons and 6squares), 36 edges, and 24 vertices. Since each of its faces haspoint symmetry, the truncated octahedron is a 6-zonohedron. It is also theGoldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as apermutohedron.
The truncated octahedron was called the "mecon" byBuckminster Fuller.[1]
Itsdual polyhedron is thetetrakis hexahedron. If the original truncated octahedron has unit edge length, then its dual tetrakis hexahedron has edge lengths9/8√2 and3/2√2.
As suggested by its name, a truncated octahedron is constructed from aregular octahedron by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving out sixsquare pyramids.
TheCartesian coordinates of the vertices of a truncated octahedron with edge length 1 are all permutations of[2]The truncated octahedron is the result of the regular octahedron's edges being expanded. The edges are separated and pushed away in the direction of two-fold rotational symmetry axes passing through the midpoint of opposite edges in theoctahedral symmetry. With the distance between the closest parallel edges that is the regular octahedron's edge-length, connecting the endpoints of each edge to form hexagons and triangles, the highlight shows that the thirdJohnson solid, thetriangular cupola, coincides with the hexagonal faces.[3]
Loeb (1986) constructs a truncated octahedron by attaching eight cubes to each face.[4] In this hinge, eight segments on each half-cube form hexagons, and the hexagons bisect the cubes. It thus can fold inward and outward, forming a truncated octahedron and a cube, respectively.[5]
Setting the edge length of the regular octahedron equal to, it follows that the length of each edge of a square pyramid (to be removed) is (the square pyramid has fourequilateral triangles as faces, the firstJohnson solid). The volume of each of these equilateral square pyramids is. Because six of them are removed by truncation, the volume of the truncated octahedron is given by[6]The surface area of this truncated octahedron can be obtained by summing all polygons' area, six squares and eight hexagons:[6]
The truncated octahedron is one of the thirteenArchimedean solids. In other words, it is a highly symmetric, semi-regular polyhedron with two or more different regular polygonal faces meeting at a vertex.[7] Thedual polyhedron of a truncated octahedron is thetetrakis hexahedron. They both have the same three-dimensional symmetry group as the regular octahedron does, theoctahedral symmetry.[8] Each vertex is surrounded by a square and two hexagons, so itsvertex figure is.[9]
A truncated octahedron has two differentdihedral angles, angles between two polygonal faces. The angle between square and hexagonal faces is, and the angle between two adjacent hexagonal faces is.[10]
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The truncated octahedron can be described as apermutohedron of order 4, or4-permutohedron, meaning it can be represented with even more symmetric coordinates in four dimensions: all permutations of form the vertices of a truncated octahedron in the three-dimensional subspace.[11] Therefore, each vertex corresponds to a permutation of and each edge represents a single pairwise swap of two elements.[12] With this labeling, the swaps are of elements whose values differ by one. If, instead, the truncated octahedron is labeled by the inverse permutations, then the edges correspond to swaps of elements whose positions differ by one. With this alternative labeling, the edges and vertices of the truncated octahedron form theCayley graph of thesymmetric group, the group of four-element permutations, as generated by swaps of consecutive positions.[13]
The truncated octahedron is aspace-filling polyhedron; that is, it can tile space by translating its copies face-to-face in order to form ahoneycomb. It is classified asplesiohedron, meaning it can be defined as theVoronoi cell of a symmetricDelone set.[14] Plesiohedra,translated without rotating, can be repeated to fill space. The truncated octahedron is one of the five three-dimensional primaryparallelohedra; the other four are thecube, thehexagonal prism, therhombic dodecahedron, and theelongated dodecahedron. The truncated octahedron is generated from six line segments with four triples of coplanar segments. In the most symmetric form, it is generated from six line segments parallel to the face diagonals of a cube.[15] The truncated octahedron tile a honeycomb known as abitruncated cubic honeycomb.[16] More generally, every permutohedron and parallelohedron is azonohedron, a polyhedron that iscentrally symmetric and can be defined by aMinkowski sum.[17]
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The truncated octahedron has been used as a fourteen-sideddie, dating back toChina inWarring States period, although a cube was also used as well.[18]
The truncated octahedron appears in the structure in the framework of afaujasite-type ofzeolite crystals. The structure consists of sodalite cages that resemble truncated octahedra, connecting each other by hexagonal prisms[19]
Insolid-state physics, the firstBrillouin zone of theface-centered cubic lattice is a truncated octahedron.[20]
The shape of a truncated octahedron appears as theWigner–Seitz cell of asodium. The background for this discovery dates back toEugene Wigner andFrederick Seitz's proposal on the Wigner–Seitz cell's application tocondensed matter physics on solving theSchrödinger equation for free electrons in sodium.[21][22]
The truncated octahedron (in fact, the generalized truncated octahedron) appears in theerror analysis ofquantization index modulation (QIM) in conjunction with repetition coding.[23]
The truncated octahedron can be dissected into a centraloctahedron, surrounded by 8triangular cupolae on each face, and 6square pyramids above the vertices.[24] Removing the central octahedron and 2 or 4 triangular cupolae creates twoStewart toroids, with dihedral and tetrahedral symmetry.
It is possible to slice atesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.[25]
Thecell-transitivebitruncated cubic honeycomb can also be seen as theVoronoi tessellation of thebody-centered cubic lattice. The truncated octahedron is one of five three-dimensional primaryparallelohedra.
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| Truncated octahedral graph | |
|---|---|
 3-fold symmetricSchlegel 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 themathematical field ofgraph theory, atruncated octahedral graph is thegraph of vertices and edges of the truncated octahedron. It has 24vertices and 36 edges, and is acubicArchimedean graph.[26] It hasbook thickness 3 andqueue number 2.[27]
As aHamiltoniancubic graph, it can be represented byLCF 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].[28]
| LCF [3, −7, 7, −3]6 | LCF [5, −7, 7, −5]6 | Configuration | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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