Hypercube

In geometry, a hypercube is an "n"-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in "n" dimension is equal to formula_1.
An "n"-dimensional hypercube is also called an n"-cube or an n"-dimensional cube. The term "measure polytope" is also used, notably in the work of H. S. M. Coxeter (originally from Elte, 1912), but it has now been superseded.
The hypercube is the special case of a hyperrectangle (also called an "n-orthotope").
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or "vertices") are the 2"n" points in R"n" with each coordinate equal to 0 or 1 is called "the" unit hypercube.
Construction.
A hypercube can be defined by increasing the numbers of dimensions of a shape:
This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the "d"-dimensional hypercube is the Minkowski sum of "d" mutually perpendicular unit-length line segments, and is therefore an example of a zonotope.
The 1-skeleton of a hypercube is a hypercube graph.