Hypercube

Construction

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

0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one. 2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square. 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. 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

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 ( ± 1 2 , ± 1 2 , ⋯ , ± 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 ( ± 1 , ± 1 , ⋯ , ± 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.

Elements

Every n-cube of n > 0 is composed of elements, or n-cubes of a lower dimension, on the (n-1)-dimensional surface on the parent hypercube. A side is any element of (n-1)-dimension of the parent hypercube. A hypercube of dimension n has 2n sides (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is 2 n (a cube has 2 3 vertices, for instance).

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

E m , n = 2 n − m ( n m ) ,     where ( n m ) = n ! m ! ( n − m ) ! and n! denotes the factorial of n.

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).

This identity can be proved by combinatorial arguments; each of the 2 n vertices defines a vertex in a m-dimensional boundary. There are ( n m ) ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted 2 m times since it has that many vertices, we need to divide with this number.

This identity can also be used to generate the formula for the n-dimensional cube surface area. The surface area of a hypercube is: 2 n s n − 1 .

These numbers can also be generated by the linear recurrence relation

E m , n = 2 E m , n − 1 + E m − 1 , n − 1 ,     with E 0 , 0 = 1 ,  and undefined elements (where n < m , n < 0 , or m < 0 ) = 0 .

For example, extending a square via its 4 vertices adds one extra line (edge) per vertex, and also adds the final second square, to form a cube, giving E 1 , 3 = 12 lines in total.

Graphs

An n-cube can be projected inside a regular 2n-gonal polygon by a skew orthogonal projection, shown here from the line segment to the 12-cube.

Related families of polytopes

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_n, and the simplices, labeled as α_n. A fourth family, the infinite tessellations of hypercubes, he labeled as 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.

Relation to n-simplices

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.

Generalized hypercubes

Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes, γp
n = _p{4}₂{3}...₂{3}₂, or ... Real solutions exist with p=2, i.e. γ2
n = γ_n = ₂{4}₂{3}...₂{3}₂ = {4,3,..,3}. For p>2, they exist in C n . The facets are generalized (n-1)-cube and the vertex figure are regular simplexes.

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.

The number of m-face elements in a p-generalized n-cube are: p n − m ( n m ) . This is pⁿ vertices and pn facets.