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.
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.
Classifications
As an Archimedean solid
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]
3D model of a truncated octahedron
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]
As a tilling space polyhedron
Truncated octahedron as a permutahedron of order 4
Truncated octahedron in tilling space
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]
As a Goldberg polyhedron
The truncated octahedron is a
Goldberg polyhedron, a polyhedron with either hexagonal or pentagonal faces.[12]
Applications
The structure of the faujasite framework
First Brillouin zone of
FCC lattice, showing symmetry labels for high symmetry lines and points.
In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a
faujasite-type of
zeolite crystals.[13]
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]
^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.
Bibcode:
2003ITSP...51..960P.
doi:
10.1109/TSP.2003.809368.
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc.
ISBN0-486-23729-X. (Section 3–9)