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Archimedean solid From Wikipedia, the free encyclopedia
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-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.
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 hexahedron has edge lengths 9/8√2 and 3/2√2.
A truncated octahedron is constructed from a regular octahedron by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving out six square pyramids. Considering that each length of the regular octahedron is , and the edge length of a square pyramid is (the square pyramid is an equilateral, the first Johnson solid). From the equilateral square pyramid's property, its volume is . Because six equilateral square pyramids are removed by truncation, the volume of a truncated octahedron is obtained by subtracting the volume of a regular octahedron from those six:[2] The surface area of a truncated octahedron can be obtained by summing all polygonals' area, six squares and eight hexagons. Considering the edge length , this is:[2]
The truncated octahedron is one of the thirteen Archimedean solids. In other words, it has a highly symmetric and semi-regular polyhedron with two or more different regular polygonal faces that meet in a vertex.[3] The dual polyhedron of a truncated octahedron is the tetrakis hexahedron. They both have the same three-dimensional symmetry group as the regular octahedron does, the octahedral symmetry .[4] A square and two hexagons surround each of its vertex, denoting its vertex figure as .[5]
The dihedral angle of a truncated octahedron between square-to-hexagon is , and that between adjacent hexagonal faces is .[6]
The truncated octahedron can be described as a permutohedron of order 4 or 4-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 .[7] Therefore, each vertex corresponds to a permutation of and each edge represents a single pairwise swap of two elements. It has the symmetric group .[8]
The truncated octahedron can be used as a tilling space. It is classified as plesiohedron, meaning it can be defined as the Voronoi cell of a symmetric Delone set.[9] The plesiohedron includes the parallelohedron, a polyhedron can be translated without rotating and tilling space so that it fills the entire face. There are five three-dimensional primary parallelohedrons, one of which is the truncated octahedron.[10] More generally, every permutohedron and parallelohedron is zonohedron, a polyhedron that is centrally symmetric that can be defined by using Minkowski sum.[11]
The truncated octahedron is a Goldberg polyhedron, a polyhedron with either hexagonal or pentagonal faces.[12]
In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a faujasite-type of zeolite crystals.[13]
In solid-state physics, the first Brillouin zone of the face-centered cubic lattice is a truncated octahedron.[14]
The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.[15]
The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupolae on each face, and 6 square pyramids above the vertices.[16]
Removing the central octahedron and 2 or 4 triangular cupolae creates two Stewart toroids, with dihedral and tetrahedral symmetry:
It is possible to slice a tesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.[17]
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 | |
---|---|
![]() 3-fold symmetric Schlegel 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 the mathematical field of graph theory, a truncated octahedral graph is the graph of vertices and edges of the truncated octahedron. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[18] It has book thickness 3 and queue number 2.[19]
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].[20]
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