Cube (geometry)

Three dimensions

Cube
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Type	Platonic
Face polygon	square
Faces	6
Edges	12
Vertices	8
Faces per vertex	3
Vertices per face	4
Symmetry group	octahedral (Oh)
Dual polyhedron	octahedron
Properties	regular, convex, zonohedron

A cube (or hexahedron) is a 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 triangular trapezohedron, and is dual to the octahedron. Canonical coordinates for the vertices of a cube centered at the origin are (±1,±1,±1), while the interior of the same consists of all points (x₀, x₁, x₂) with -1 < x_i < 1.

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

<math>A=6a^2<math>

<math>V=a^3<math>

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. The compound of two tetrahedra is made from the cube in like fashion. The cube is unique among the Platonic solids for being able to tile space regularly, and finds many uses because of this. For instance, sugar is frequently pressed into cubes containing a convenient amount to sweeten beverages, and the familiar six-sided die is cube shaped.

Four dimensions

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

Arbitrary dimensions

In an n-dimensional space the analog of the figure is called n-dimensional cube, or simply cube, if it doesn't lead to a confusion.

See also

• Unit cube