Truncated octahedron

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Truncated octahedron
Truncatedoctahedron.jpg
Type Archimedean solid,
Parallelohedron,
Permutohedron,
Plesiohedron,
Zonohedron
Faces 14
Edges 36
Vertices 24
Symmetry group octahedral symmetry
Dual polyhedron tetrakis hexahedron
Vertex figure
Polyhedron truncated 8 vertfig.svg
Net
Polyhedron truncated 8 net.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.

Contents

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 9/82 and 3/22.

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 Truncated octahedron.stl
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

Symmetric group 4; permutohedron 3D; transpositions (1-based).png
Truncated octahedron as a permutahedron of order 4
Truncated octahedra.jpg
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

Structural features of the faujasite zeolite framework (FAU).svg
The structure of the faujasite framework
Brillouin Zone (1st, FCC).svg
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]

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]

Dissection

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:

Genus 2Genus 3
D3d, [2+,6], (2*3), order 12 Td, [3,3], (*332), order 24
Excavated truncated octahedron1.png Excavated truncated octahedron2.png

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.

Objects

43840 Salou, Tarragona, Spain - panoramio (21), crop.jpg
Bundek climbing frame 20150307 DSC 0109, crop.jpg
Jungle gym nets often include truncated octahedra.

Truncated octahedral graph

Truncated octahedral graph
Truncated octahedral graph2.png
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]

Three different Hamiltonian cycles described by the three different LCF notations for the truncated octahedral graph Truncated octahedral Hamiltonicity.svg
Three different Hamiltonian cycles described by the three different LCF notations for the truncated octahedral graph

Related Research Articles

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces. It has twelve edges and eight vertices. It can be represented as a rectangular cuboid with six square faces, or a parallelepiped with equal edges. It is an example of many type of solids: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.

<span class="mw-page-title-main">Rhombicuboctahedron</span> Archimedean solid with 26 faces

In geometry, rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It is named by Johannes Kepler in his 1618 Harmonices Mundi, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a rhombic dodecahedron.

<span class="mw-page-title-main">Truncated icosahedron</span> A soccerball-shaped like polyhedron

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs that are typically patterned with white hexagons and black pentagons. It can be found in the application of geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.

<span class="mw-page-title-main">Snub cube</span> Archimedean solid with 38 faces

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol , and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol .

<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron.

<span class="mw-page-title-main">Truncated cube</span> Archimedean solid with 14 regular faces

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

<span class="mw-page-title-main">Truncated cuboctahedron</span> Archimedean solid in geometry

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

<span class="mw-page-title-main">Snub dodecahedron</span> Archimedean solid with 92 faces

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

<span class="mw-page-title-main">Triakis octahedron</span> Catalan solid with 24 faces

In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.

<span class="mw-page-title-main">Tetrakis hexahedron</span> Catalan solid with 24 faces

In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.

<span class="mw-page-title-main">Disdyakis dodecahedron</span> Geometric shape with 48 faces

In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.

<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

<span class="mw-page-title-main">Chamfered dodecahedron</span> Goldberg polyhedron with 42 faces

In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

<span class="mw-page-title-main">Chamfer (geometry)</span> Geometric operation which truncates the edges of polyhedra

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

References

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