| Truncated octahedron | |
| Faces: | 14 |
| Edges: | 36 |
| Vertices: | 24 |
| Symmetry: | octahedral symmetry Oh |
| Net: | Polyhedron truncated 8 net.svg |
| Vertex Figure: | Polyhedron truncated 8 vertfig.svg |
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 and .
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
3a
a
V
a
Oh
4 ⋅ 62
The dihedral angle of a truncated octahedron between square-to-hexagon is , and that between adjacent hexagonal faces is .
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
(1,2,3,4)
x+y+z+w=10
(1,2,3,4)
S4
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. 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. More generally, every permutohedron and parallelohedron is zonohedron, a polyhedron that is centrally symmetric that can be defined by using Minkowski sum.
The truncated octahedron is a Goldberg polyhedron, a polyhedron with either hexagonal or pentagonal faces.
In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a faujasite-type of zeolite crystals.
In solid-state physics, the first Brillouin zone of the face-centered cubic lattice is a truncated octahedron.
The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.[2]
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.[3]
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.
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 | |
| Vertices: | 24 |
| Edges: | 36 |
| Automorphisms: | 48 |
| Chromatic Number: | 2 |
| Properties: | Cubic, Hamiltonian, regular, zero-symmetric |
| Book Thickness: | 3 |
| Queue Number: | 2 |
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. It has book thickness 3 and queue number 2.[4]
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].