# Cube (geometry)

 Contents

## Three dimensions

Cube Click on picture for large version.
TypePlatonic
Face polygonsquare
Faces6
Edges12
Vertices8
Faces per vertex3
Vertices per face4
Symmetry groupoctahedral (Oh)
Dual polyhedronoctahedron
Propertiesregular, 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 < xi < 1.

Missing image
Hexahedron_flat.png
image:hexahedron flat.png

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

[itex]A=6a^2[itex]
[itex]V=a^3[itex]

Note that a cube construction will always create the largest volume possible per amount of material available (e.g. paper, cardboard, sheet metal, etc.) provided a flat six-sided face is a requirement. (The proof requires calculus, and assumes 2D squares can be created with no waste.) A similar object having a rectangular shape will always have a lesser volume than a cube for the same liner measurement (length + width + height).

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.

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.

## 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.

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