Cube

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In geometry, a cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry).

A cube is the three-dimensional case of the more general concept of a hypercube.

It has 11 nets.[2] If one were to colour the cube so that no two adjacent faces had the same colour, one would need 3 colours.

If the original cube has edge length 1, its dual octahedron has edge length \sqrt{2}.

Contents

Cartesian coordinates

For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are

while the interior consists of all points (x0x1x2) with −1 < x i < 1.

Formulae

For a cube of edge length a,

As the volume of a cube is the third power of its sides a×a×a, third powers are called cubes, by analogy with squares and second powers.

A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).

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