Cube

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This article is about the geometric shape. For other meanings of the word "cube", see cube (disambiguation).

Template:Reg polyhedra db A cube <ref>English cube from Old French < Latin cubus < Greek kubos, "a cube, a die, vertebra". In turn from PIE *keu(b)-, "to bend, turn".</ref> (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. The cube is a special kind of square prism, of rectangular parallelepiped and of 3-sided trapezohedron, and is dual to the octahedron. Thus it has octahedral symmetry.

1 Cartesian coordinates
2 Area and volume
3 Geometric relations
4 Higher dimensions
5 Related polyhedra
6 Combinatorial cubes
7 Note
8 See also
9 External links

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Cartesian coordinates

Cartesian coordinates for the vertices of a cube centered at the origin and edge length 2 are

(±1,±1,±1)

while the interior of the same consists of all points (x₀, x₁, x₂) with -1 < x_i < 1.

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Area and volume

The area A and volume V of a cube of edge length a are:

A cube construction has the largest volume among cuboids (rectangular boxes) with a given surface area (e.g., paper, cardboard, sheet metal, etc.). Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).

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Geometric relations

The cube is unique among the Platonic solids for being able to tile space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry).

Image:Hexahedron flat.png Image:Dice.jpg

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Higher dimensions

Image:Expo 67 cubes in a room.jpg

In the four-dimensional Euclidean space, the analogue of a cube has a special name — a tesseract or hypercube.

The analog of the cube in the n-dimensional Euclidean space is called n-dimensional cube, or simply n-cube, if it does not lead to confusion. The name measure polytope is also used.

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Related polyhedra

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.

One such regular tetrahedron has a volume of 1/3 of that of the cube. The remaining space consists of four equal irregular polyhedra with a volume of 1/6 of that of the cube, each.

The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with 6 octagonal faces and 8 triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount.

A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes. Image:Stella octangula.png Image:Cuboctahedron.jpg Image:Truncatedhexahedron.jpg Image:Rhombicuboctahedron.jpg
The figures shown have the same symmetries as the cube (see octahedral symmetry).

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Combinatorial cubes

As noted above, an n-dimensional cube is often called an n-cube.

A further extension of the cube concept is the k-ary n-cube of combinatorics and computer science. This may be viewed as an n-dimensional torus that is a k × ... × k (with k repeated n times) cube of grids with wrap-around edges, according to "On k-ary n-cubes: Theory and Applications," by Weizhen Mao and David M. Nicol.

Cubes of this sort occur in the theory of parallel processing in computers. For some purely mathematical properties of one such structure, the 4-ary 3-cube, see geometry of the 4×4×4 cube.

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