In geometry, a hypercube is an ndimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure whose 1skeleton 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.
An ndimensional hypercube is also called an ncube. The term "measure polytope" is also used, notably in the work of H.S.M. Coxeter (originally from Elte, 1912^{[1]}), but it has now been superseded.
The hypercube is the special case of a hyperrectangle (also called an 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 coordinates equal to 0 or 1 is called "the" unit hypercube.
Contents
Construction
This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the ddimensional hypercube is the Minkowski sum of d mutually perpendicular unitlength line segments, and is therefore an example of a zonotope.
The 1skeleton of a hypercube is a hypercube graph.
Coordinates
A unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates . It has an edge length of 1 and an ndimensional volume of 1.
An ndimensional hypercube is also often regarded as the convex hull of all sign permutations of the coordinates . This form is often chosen due to ease of writing out the coordinates. Its edge length is 2, and its ndimensional volume is 2^{n}.
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