Truncated octahedron

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Truncated octahedron

The truncated octahedron is an Archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
. It has 8 regular hexagon

Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces, 6 square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron

Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
.

Construction

A truncated octahedron is constructed from a regular octahedron

Octahedron

Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
, one from each point.

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Encyclopedia

The truncated octahedron is an Archimedean solid

Archimedean solid

Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces, 6 square

Square (geometry)

Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
.

Construction

A truncated octahedron is constructed from a regular octahedron

Octahedron

Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
, one from each point. These pyramids have both base side length, a, and lateral side length, e, of a. The base area is then a². Note that this shape is exactly similar to half an octahedron or Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
J₁.

From the properties of square pyramids, we can now find the slant height, s, and the height, h of the pyramid:

The volume, V, of the pyramid is given by:

Because six pyramids are removed by truncation, there is a total lost volume of v2 a³.

Coordinates and permutations

All permutation

Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
s of (0, ±1, ±2) are Cartesian coordinates of the vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
of a truncated

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
octahedron

Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.

The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1,2,3,4) form the vertices of a truncated octahedron in the three-dimensional subspace x + y + z + w = 10. Therefore, the truncated octahedron is the permutohedron

Permutohedron

In mathematics, the permutohedron of order n is an -dimensional polytope embedded in an n-dimensional space, the vertices of which are formed by Permutation the coordinates of the vector ....
of order 4.

Area and volume

The area A and the volume

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
V of a truncated octahedron of edge length a are:

Uniform colourings

There are two uniform colourings, with tetrahedral symmetry

Tetrahedral symmetry

A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
and octahedral symmetry

Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
:

Truncated octahedron	Omnitruncated tetrahedron
122 colouring O_h symmetry Wythoff Wythoff symbol In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane.... : 2 4 \| 3	123 colouring T_h symmetry Wythoff: 3 3 2 \|

Related polyhedra

The truncated octahedron exists within the set of truncated forms between a cube

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
and octahedron

Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
:

Cube

Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....

cuboctahedron

Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....

Truncated octahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....

Tessellations

The truncated octahedron exists in three different convex uniform honeycomb

Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
s (space-filling tessellations

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
):

Bitruncated cubic Bitruncated cubic honeycomb The Bitruncation cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedron.It is one of 28 Convex uniform honeycomb....	Cantitruncated cubic Cantitruncated cubic honeycomb The cantitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedron, truncated octahedron, and cubes in a ratio of 1:1:3....	Truncated alternated cubic Truncated alternated cubic honeycomb The truncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated octahedron, cuboctahedron and truncated tetrahedron in a ratio of 1:1:2....

The cell-transitive bitruncated cubic honeycomb

Bitruncated cubic honeycomb

The Bitruncation cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedron.It is one of 28 Convex uniform honeycomb....
can also be seen as the Voronoi tessellation of the body-centred cubic lattice

Crystal structure

In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a particular way, and a lattice....
.

Truncated octahedron

Construction

Coordinates and permutations

Area and volume

Uniform colourings

Related polyhedra

Tessellations

External links