Hypercube

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In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). 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.

An n-dimensional hypercube is also called an n-cube. 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polytope" is also used, notably in the work of H.S.M. Coxeter (originally from Elte, 1912[1]), but it has now been superseded.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 2n points in Rn 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 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.

Coordinates

A unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates (\pm 1/2, \pm 1/2, \cdots, \pm 1/2). It has an edge length of 1 and an n-dimensional volume of 1.

An n-dimensional hypercube is also often regarded as the convex hull of all sign permutations of the coordinates (\pm 1, \pm 1, \cdots, \pm 1). This form is often chosen due to ease of writing out the coordinates. Its edge length is 2, and its n-dimensional volume is 2n.

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