Hypercube

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. A unit hypercube's longest diagonal in n dimensions is equal to ${\sqrt {n}}$ .

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope (originally from Elte, 1912) 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 (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 2n points in Rn with each coordinate equal to 0 or 1 is called the unit 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 $\left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},\cdots ,\pm {\frac {1}{2}}\right)$ . 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.

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 endpoints; 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

$E_{m,n}=2^{n-m}{n \choose m}$ ,     where ${n \choose m}={\frac {n!}{m!\,(n-m)!}}$  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 ${n \choose m}$  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 by 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: $2ns^{n-1}$ .

These numbers can also be generated by the linear recurrence relation

$E_{m,n}=2E_{m,n-1}+E_{m-1,n-1}\!$ ,     with $E_{0,0}=1\!$ ,  and undefined elements (where $n , $n<0$ , 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.

Hypercube elements $E_{m,n}\!$  (sequence A038207 in the OEIS)
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 4 1
3 3-cube Cube
Hexahedron
{4,3}

8 12 6 1
4 4-cube Tesseract
Octachoron
{4,3,3}

16 32 24 8 1
5 5-cube Penteract
Deca-5-tope
{4,3,3,3}

32 80 80 40 10 1
6 6-cube Hexeract
Dodeca-6-tope
{4,3,3,3,3}

64 192 240 160 60 12 1
7 7-cube Hepteract
{4,3,3,3,3,3}

128 448 672 560 280 84 14 1
8 8-cube Octeract
{4,3,3,3,3,3,3}

256 1024 1792 1792 1120 448 112 16 1
9 9-cube Enneract
{4,3,3,3,3,3,3,3}

512 2304 4608 5376 4032 2016 672 144 18 1
10 10-cube Dekeract
Icosa-10-tope
{4,3,3,3,3,3,3,3,3}

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.

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 γn. The other two are the hypercube dual family, the cross-polytopes, labeled as βn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn.

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 n.

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

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
= γn = 2{4}2{3}...2{3}2 = {4,3,..,3}. For p>2, they exist in $\mathbb {C} ^{n}$ . 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: $p^{n-m}{n \choose m}$ . This is pn vertices and pn facets.

Generalized hypercubes
p=2 p=3 p=4 p=5 p=6 p=7 p=8
$\mathbb {R} ^{2}$
γ2
2
= {4} =
4 vertices
$\mathbb {C} ^{2}$
γ3
2
=
9 vertices

γ4
2
=
16 vertices

γ5
2
=
25 vertices

γ6
2
=
36 vertices

γ7
2
=
49 vertices

γ8
2
=
64 vertices
$\mathbb {R} ^{3}$
γ2
3
= {4,3} =
8 vertices
$\mathbb {C} ^{3}$
γ3
3
=
27 vertices

γ4
3
=
64 vertices

γ5
3
=
125 vertices

γ6
3
=
216 vertices

γ7
3
=
343 vertices

γ8
3
=
512 vertices
$\mathbb {R} ^{4}$
γ2
4
= {4,3,3}
=
16 vertices
$\mathbb {C} ^{4}$
γ3
4
=
81 vertices

γ4
4
=
256 vertices

γ5
4
=
625 vertices

γ6
4
=
1296 vertices

γ7
4
=
2401 vertices

γ8
4
=
4096 vertices
$\mathbb {R} ^{5}$
γ2
5
= {4,3,3,3}
=
32 vertices
$\mathbb {C} ^{5}$
γ3
5
=
243 vertices

γ4
5
=
1024 vertices

γ5
5
=
3125 vertices

γ6
5
=
7776 vertices
γ7
5
=
16,807 vertices
γ8
5
=
32,768 vertices
$\mathbb {R} ^{6}$
γ2
6
= {4,3,3,3,3}
=
64 vertices
$\mathbb {C} ^{6}$
γ3
6
=
729 vertices

γ4
6
=
4096 vertices

γ5
6
=
15,625 vertices
γ6
6
=
46,656 vertices
γ7
6
=
117,649 vertices
γ8
6
=
262,144 vertices
$\mathbb {R} ^{7}$
γ2
7
= {4,3,3,3,3,3}
=
128 vertices
$\mathbb {C} ^{7}$
γ3
7
=
2187 vertices
γ4
7
=
16,384 vertices
γ5
7
=
78,125 vertices
γ6
7
=
279,936 vertices
γ7
7
=
823,543 vertices
γ8
7
=
2,097,152 vertices
$\mathbb {R} ^{8}$
γ2
8
= {4,3,3,3,3,3,3}
=
256 vertices
$\mathbb {C} ^{8}$
γ3
8
=
6561 vertices
γ4
8
=
65,536 vertices
γ5
8
=
390,625 vertices
γ6
8
=
1,679,616 vertices
γ7
8
=
5,764,801 vertices
γ8
8
=
16,777,216 vertices