# Cube

Cube
Cube
Click on picture for large version.
Type Platonic
Face polygon square
Faces 6
Edges 12
Vertices 8
Faces per vertex 3
Vertices per face 4
Symmetry group octahedral (Oh)
of order 48
Dual polyhedron octahedron
Properties regular, convex, zonohedron
Cube
Vertex Figure

A cube (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.

## Canonical coordinates

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 < xi < 1.

## Area and volume

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

$\displaystyle A=6a^2$
$\displaystyle V=a^3$

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).

The cube is unique among the Platonic solids for being able to tile space regularly.

File:Dice.jpg
The familiar six-sided die is cube shaped

## Higher dimensions

File:Expo 67 cubes in a room.jpg
Room of cubes at Expo 67

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 cube, if it does not lead to a confusion. The name measure polytope is also used.

## 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.

File:Stella octangula.png
The tetrahedra in the cube (stella octangula)

Template:- The figures shown have the same symmetries as the cube (see octahedral symmetry).

## Trivia

If each edge of a cube is replaced by a one ohm resistor, the resistance between opposite vertices is 5/6 ohms, and that between adjacent vertices 7/12 ohms.