Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube ; the special case for Four-dimensional space| is known as a tesseract. 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 dimensions is equal to.
An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.
The hypercube is the special case of a hyperrectangle.
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners are the 2ⁿ points in Rⁿ with each coordinate equal to 0 or 1 is called the unit hypercube.

Construction

By the number of dimensions

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.

Vertex coordinates

A unit hypercube of dimension is the convex hull of all the points whose Cartesian coordinates are each equal to either or. These points are its vertices. The hypercube with these coordinates is also the cartesian product of copies of the unit interval. Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a translation. It is the convex hull of the points whose vectors of Cartesian coordinates are
Here the symbol means that each coordinate is either equal to or to. This unit hypercube is also the cartesian product. Any unit hypercube has an edge length of and an -dimensional volume of.
The -dimensional hypercube obtained as the convex hull of the points with coordinates or, equivalently as the Cartesian product is also often considered due to the simpler form of its vertex coordinates. Its edge length is, and its -dimensional volume is.

Faces

Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension admits facets, or faces of dimension : a line segment has endpoints; a square has sides or edges; a -dimensional cube has square faces; a tesseract has three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension is .
The number of the -dimensional hypercubes contained in the boundary of an -cube is
For example, the boundary of a -cube contains cubes, squares, line segments and vertices. This identity can be proven by a simple combinatorial argument: for each of the vertices of the hypercube, there are ways to choose a collection of edges incident to that vertex. Each of these collections defines one of the -dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the -dimensional faces of the hypercube is counted times since it has that many vertices, and we need to divide by this number.
The number of facets of the hypercube can be used to compute the -dimensional volume of its boundary: that volume is times the volume of a -dimensional hypercube; that is, where is the length of the edges of the hypercube.
These numbers can also be generated by the linear recurrence relation.
For example, extending a square via its 4 vertices adds one extra line segment per vertex. Adding the opposite square to form a cube provides line segments.
The extended f-vector for an n-cube can also be computed by expanding , and reading off the coefficients of the resulting polynomial. For example, the elements of a tesseract is ⁴ = ² =.

			m	0	1	2	3	4	5	6	7	8	9	10
n	n-cube	Names	Schläfli Coxeter	Vertex 0-face	Edge 1-face	Face 2-face	Cell 3-face	4-face	5-face	6-face	7-face	8-face	9-face	10-face
0	0-cube	Point Monon		1
1	1-cube	Line segment Dion		2	1
2	2-cube	Square Tetragon		4	4	1
3	3-cube	Cube Hexahedron		8	12	6	1
4	4-cube	Tesseract Octachoron		16	32	24	8	1
5	5-cube	Penteract Deca-5-tope		32	80	80	40	10	1
6	6-cube	Hexeract Dodeca-6-tope		64	192	240	160	60	12	1
7	7-cube	Hepteract Tetradeca-7-tope		128	448	672	560	280	84	14	1
8	8-cube	Octeract Hexadeca-8-tope		256	1024	1792	1792	1120	448	112	16	1
9	9-cube	Enneract Octadeca-9-tope		512	2304	4608	5376	4032	2016	672	144	18	1
10	10-cube	Dekeract Icosa-10-tope		1024	5120	11520	15360	13440	8064	3360	960	180	20	1

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 15-cube.

Line segment	Square	Cube	Tesseract
5-cube	6-cube	7-cube	8-cube
9-cube	10-cube	11-cube	12-cube
13-cube	14-cube	15-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 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, is 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.
n-cubes can be combined with their duals to form compound polytopes:

In two dimensions, we obtain the octagrammic star figure,
In three dimensions we obtain the compound of cube and octahedron,
In four dimensions we obtain the compound of tesseract and 16-cell.
Relation to (''n''−1)-simplices

The graph of the n-hypercube's edges is isomorphic to the Hasse diagram of the -simplex's face lattice. This can be seen by orienting the n-hypercube so that two opposite vertices lie vertically, corresponding to the -simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the -simplex's facets, 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 -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, γ = _p₂...₂₂, or... Real solutions exist with p = 2, i.e. γ = γ_n = ₂₂...₂₂ =. For p > 2, they exist in. The facets are generalized -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 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:. This is pⁿ vertices and pn facets.

p=2	p=3	p=4	p=5	p=6	p=7	p=8
γ = square\| = 4 vertices	γ = 9 vertices	γ = 16 vertices	γ = 25 vertices	γ = 36 vertices	γ = 49 vertices	γ = 64 vertices
γ = cube\| = 8 vertices	γ = 27 vertices	γ = 64 vertices	γ = 125 vertices	γ = 216 vertices	γ = 343 vertices	γ = 512 vertices
γ = tesseract\| = 16 vertices	γ = 81 vertices	γ = 256 vertices	γ = 625 vertices	γ = 1296 vertices	γ = 2401 vertices	γ = 4096 vertices
γ = 5-cube\| = 32 vertices	γ = 243 vertices	γ = 1024 vertices	γ = 3125 vertices	γ = 7776 vertices	γ = 16,807 vertices	γ = 32,768 vertices
γ = 6-cube\| = 64 vertices	γ = 729 vertices	γ = 4096 vertices	γ = 15,625 vertices	γ = 46,656 vertices	γ = 117,649 vertices	γ = 262,144 vertices
γ = 7-cube\| = 128 vertices	γ = 2187 vertices	γ = 16,384 vertices	γ = 78,125 vertices	γ = 279,936 vertices	γ = 823,543 vertices	γ = 2,097,152 vertices
γ = 8-cube\| = 256 vertices	γ = 6561 vertices	γ = 65,536 vertices	γ = 390,625 vertices	γ = 1,679,616 vertices	γ = 5,764,801 vertices	γ = 16,777,216 vertices