# Hypercube

An *n*-dimensional hypercube is more commonly referred to as an ** n-cube** or sometimes as an

**. The term**

*n*-dimensional cube**measure polytope**(originally from Elte, 1912)

^{[1]}is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.

^{[2]}

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.

A hypercube can be defined by increasing the numbers of dimensions of a shape:

**1**– If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.

**3**– If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.

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 number of *m*-dimensional hypercubes (just referred to as *m*-cube from here on) on the boundary of an *n*-cube is

For example, the boundary of a 4-cube (n=4) contains 8 cubes (3-cubes), 24 squares (2-cubes), 32 lines (1-cubes) and 16 vertices (0-cubes).

An ** n-cube** can be projected inside a regular 2

*n*-gonal polygon by a skew orthogonal projection, shown here from the line segment to the 15-cube.

The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions.

The **hypercube (offset)** family is one of three regular polytope families, labeled by Coxeter as *γ _{n}*. The other two are the hypercube dual family, the

**cross-polytopes**, labeled as

*β*and the

_{n,}**simplices**, labeled as

*α*. A fourth family, the infinite tessellations of hypercubes, he labeled as

_{n}*δ*.

_{n}Another related family of semiregular and uniform polytopes is the **demihypercubes**, which are constructed from hypercubes with alternate vertices deleted and simplex facets added in the gaps, labeled as *hγ _{n}*.

*n*-cubes can be combined with their duals (the cross-polytopes) to form compound polytopes:

The graph of the *n*-hypercube's edges is isomorphic to the Hasse diagram of the (*n*−1)-simplex's face lattice. This can be seen by orienting the *n*-hypercube so that two opposite vertices lie vertically, corresponding to the (*n*-1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (*n*-1)-simplex's facets (*n*-2 faces), and each vertex connected to those vertices maps to one of the simplex's *n*-3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices.

This relation may be used to generate the face lattice of an (*n-1*)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.

The regular polygon perimeter seen in these orthogonal projections is called a petrie polygon. The generalized squares (n=2) are shown with edges outlined as red and blue alternating color *p*-edges, while the higher n-cubes are drawn with black outlined *p*-edges.