Cube

The cube or hexahedron is one of the five Platonic solids. It has 6 square faces, joining 3 to a triangular vertex. It is the 3-dimensional hypercube.

It is the only Platonic solid that can tile 3-dimensional space. This results in the cubic honeycomb.

It is also the uniform square prism.

Vertex coordinates
Coordinates for the vertices of a cube of edge length 1, centered at the origin, are:
 * $$(±\frac{1}{2},\,±\frac{1}{2},\,±\frac{1}{2}).$$

Representations
A cube can be represented by the following Coxeter diagrams:


 * x4o3o (regular)
 * x x4o (generally a square prism)
 * x x x (generally a cuboid)
 * s2s4x (generally a rectangular trapezoprism)
 * x2s4s
 * x2s4x
 * xx4oo&#x (BC2 axial, generally a square frustum)
 * xx xx&#x (A1×A1 axisl, rectangular frustum)
 * oqoo3ooqo&#xt (A2 axial, generally a trigonal trapezohedron
 * xx oqo&#xt (A1×A1, edge-first)
 * oqooqo&#xt (A1 only)
 * xxxx&#xr (A1 only, edge first)
 * qo3oo3oq&#zx (A3 subsymmetry, hull of two tetrahedra)
 * xx qo oq&#zx (rhombi prism)

Variations
A cube can be considered as the prism product of three mutually orthogonal dyads with the same length. By adjusting the sizes of these edges, we can create variations with different symmetry. Further notable variations of the cube arise from taking subgroups of the full BC3 symmetry.

All of these double as colorings of the cube, when their symmetry is transferred to the regular cube.

Square prism
A square prism is a variant of the cube constructed as the prism of a square. The two bases are squares, while the 4 lateral sides are rectangles. It can be represented as x4o y.

Cuboid
A cuboid, or rectangular prism, is a prism based on a rectangle. It has three different edge types and 6 rectangles as faces, in 3 parallel pairs. It can be represented as x y z.

Trigonal trapezohedron
A trigonal trapezohedron is a trapezohedron built from a triangle. It has a single face type, which is a rhombus.

Rectangular trapezoprism
A rectangular trapezoprism is made out of two perpendicular rectangles in parallel planes, connected by 4 isosceles trapezoids.

Parallelepiped
The parallelepiped is the 3-dimensional parallelotope, having three pairs of parallel parallelogram faces.

Related polyhedra
The cube can be augmented with a square pyramid to form the elongated square pyramid, a Johnson solid. If the opposite face is also agumented with a square pyramid, the result is the elongated square bipyramid.

A number of uniform polyhedron compounds are composed of cubes:


 * Rhombihedron (5)
 * Rhombihexahedron (3)
 * Rhombisnub dishexahedron (6, with rotational freedom)