Cube

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

It is the only Platonic solid that can tile 3-dimensional Euclidean space. This results in the cubic honeycomb. It also forms the cells of the 4D tesseract.

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:
 * $$\left(±\frac{1}{2},\,±\frac{1}{2},\,±\frac{1}{2}\right).$$

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 (B2 axial, generally a square frustum)
 * xx xx&#x (A1×A1 axial, 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 B3 symmetry.

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

The most notable variations incllude:


 * Square prism - two squares and 4 rectangles, identical vertices
 * Cuboid or rectangular prism - 3 pairs of rectangles, identical vertices
 * Rectangular trapezoprism - 2 rectangles and 4 isosceles trapezoids with digonal antiprism symmetry, identical vertices
 * Triangular antitegum - 6 identical rhombic faces, trigonal antiprism symmetry
 * Square frustum - 2 base squares of different sizes, 4 isosceles trapezoid sides
 * Triangular gyrotegum: 6 identical tetragonal faces, chiral triangular prismatic symmetry
 * Rectangular frustum - 2 different base rectangles, with 2 pairs of isosceles trapezoid sides, digonal pyramid symmetry

Related polyhedra
The cube can be augmented with a square pyramid to form the elongated square pyramid, which is a Johnson solid. If the opposite face is also augmented 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)
 * An infinite number of prismatic compounds that are the prisms of compounds of squares

It is nontrivial to find a realizable polychoron with an odd number of cells, all of which are combinatorial cubes. The first such polychoron to be found has 16,533 cells. See cubical polytope.