Hypercube
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Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube . 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 ${\displaystyle {\sqrt {n}}}$.

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)[1] is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the ?n 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 2n points in Rn with each coordinate equal to 0 or 1 is called the unit hypercube.

## Construction

A diagram showing how to create a tesseract from a point.
An animation showing how to create a tesseract from a point.

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.

## Vertex coordinates

A unit hypercube of dimension ${\displaystyle n}$ is the convex hull of all the points whose ${\displaystyle n}$ Cartesian coordinates are each equal to either ${\displaystyle 0}$ or ${\displaystyle 1}$. This hypercube is also the cartesian product ${\displaystyle [0,1]^{n}}$ of ${\displaystyle n}$ copies of the unit interval ${\displaystyle [0,1]}$. 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

${\displaystyle \left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},\cdots ,\pm {\frac {1}{2}}\right)\!\!.}$

Here the symbol ${\displaystyle \pm }$ means that each coordinate is either equal to ${\displaystyle 1/2}$ or to ${\displaystyle -1/2}$. This unit hypercube is also the cartesian product ${\displaystyle [-1/2,1/2]^{n}}$. Any unit hypercube has an edge length of ${\displaystyle 1}$ and an ${\displaystyle n}$-dimensional volume of ${\displaystyle 1}$.

The ${\displaystyle n}$-dimensional hypercube obtained as the convex hull of the points with coordinates ${\displaystyle (\pm 1,\pm 1,\cdots ,\pm 1)}$ or, equivalently as the Cartesian product ${\displaystyle [-1,1]^{n}}$ is also often considered due to the simpler form of its vertex coordinates. Its edge length is ${\displaystyle 2}$, and its ${\displaystyle n}$-dimensional volume is ${\displaystyle 2^{n}}$.

## Faces

Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension ${\displaystyle n}$ admits ${\displaystyle 2n}$ facets, or faces of dimension ${\displaystyle n-1}$: a (${\displaystyle 1}$-dimensional) line segment has ${\displaystyle 2}$ endpoints; a (${\displaystyle 2}$-dimensional) square has ${\displaystyle 4}$ sides or edges; a ${\displaystyle 3}$-dimensional cube has ${\displaystyle 6}$ square faces; a (${\displaystyle 4}$-dimensional) tesseract has ${\displaystyle 8}$ three-dimensional cube as its facets. The number of vertices of a hypercube of dimension ${\displaystyle n}$ is ${\displaystyle 2^{n}}$ (a usual, ${\displaystyle 3}$-dimensional cube has ${\displaystyle 2^{3}=8}$ vertices, for instance).

The number of the ${\displaystyle m}$-dimensional hypercubes (just referred to as ${\displaystyle m}$-cubes from here on) contained in the boundary of an ${\displaystyle n}$-cube is

${\displaystyle E_{m,n}=2^{n-m}{n \choose m}}$,[3]     where ${\displaystyle {n \choose m}={\frac {n!}{m!\,(n-m)!}}}$ and ${\displaystyle n!}$ denotes the factorial of ${\displaystyle n}$.

For example, the boundary of a ${\displaystyle 4}$-cube (${\displaystyle n=4}$) contains ${\displaystyle 8}$ cubes (${\displaystyle 3}$-cubes), ${\displaystyle 24}$ squares (${\displaystyle 2}$-cubes), ${\displaystyle 32}$ line segments (${\displaystyle 1}$-cubes) and ${\displaystyle 16}$ vertices (${\displaystyle 0}$-cubes). This identity can be proven by a simple combinatorial argument: for each of the ${\displaystyle 2^{n}}$ vertices of the hypercube, there are ${\displaystyle {\tbinom {n}{m}}}$ ways to choose a collection of ${\displaystyle m}$ edges incident to that vertex. Each of these collections defines one of the ${\displaystyle m}$-dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the ${\displaystyle m}$-dimensional faces of the hypercube is counted ${\displaystyle 2^{m}}$ times since it has that many vertices, and we need to divide ${\displaystyle 2^{n}{\tbinom {n}{m}}}$ by this number.

The number of facets of the hypercube can be used to compute the ${\displaystyle (n-1)}$-dimensional volume of its boundary: that volume is ${\displaystyle 2n}$ times the volume of a ${\displaystyle (n-1)}$-dimensional hypercube; that is, ${\displaystyle 2ns^{n-1}}$ where ${\displaystyle s}$ is the length of the edges of the hypercube.

These numbers can also be generated by the linear recurrence relation

${\displaystyle E_{m,n}=2E_{m,n-1}+E_{m-1,n-1}\!}$,     with ${\displaystyle E_{0,0}=1}$, and ${\displaystyle E_{m,n}=0}$ when ${\displaystyle n, ${\displaystyle n<0}$, or ${\displaystyle m<0}$.

For example, extending a square via its 4 vertices adds one extra line segment (edge) per vertex. Adding the opposite square to form a cube provides ${\displaystyle E_{1,3}=12}$ line segments.

Number ${\displaystyle E_{m,n}}$ of ${\displaystyle m}$-dimensional faces of a ${\displaystyle n}$-dimensional hypercube (sequence 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[4]
{}

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
Tetradeca-7-tope
{4,3,3,3,3,3}

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

256 1024 1792 1792 1120 448 112 16 1
9 9-cube Enneract
Octadeca-9-tope
{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.

 Point 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 16-cube
Projection of a rotating tesseract.

## 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 ?n. The other two are the hypercube dual family, the cross-polytopes, labeled as ?n, and the simplices, labeled as ?n. A fourth family, the infinite tessellations of hypercubes, he labeled as ?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 h?n.

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

## Relation to (n-1)-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 ${\displaystyle \mathbb {C} ^{n}}$. 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: ${\displaystyle p^{n-m}{n \choose m}}$. This is pn vertices and pn facets.[5]

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

γ4
2
=
16 vertices

γ5
2
=
25 vertices

γ6
2
=
36 vertices

γ7
2
=
49 vertices

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

γ4
3
=
64 vertices

γ5
3
=
125 vertices

γ6
3
=
216 vertices

γ7
3
=
343 vertices

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

γ4
4
=
256 vertices

γ5
4
=
625 vertices

γ6
4
=
1296 vertices

γ7
4
=
2401 vertices

γ8
4
=
4096 vertices
${\displaystyle \mathbb {R} ^{5}}$
γ2
5
= {4,3,3,3}
=
32 vertices
${\displaystyle \mathbb {C} ^{5}}$
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
${\displaystyle \mathbb {R} ^{6}}$
γ2
6
= {4,3,3,3,3}
=
64 vertices
${\displaystyle \mathbb {C} ^{6}}$
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
${\displaystyle \mathbb {R} ^{7}}$
γ2
7
= {4,3,3,3,3,3}
=
128 vertices
${\displaystyle \mathbb {C} ^{7}}$
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
${\displaystyle \mathbb {R} ^{8}}$
γ2
8
= {4,3,3,3,3,3,3}
=
256 vertices
${\displaystyle \mathbb {C} ^{8}}$
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

## Notes

1. ^ Elte, E. L. (1912). "IV, Five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Netherlands: University of Groningen. ISBN 141817968X.
2. ^ Coxeter 1973, pp. 122-123, §7.2 see illustration Fig 7.2C.
3. ^ Coxeter 1973, p. 122, §7·25.
4. ^ Johnson, Norman W.; Geometries and Transformations, Cambridge University Press, 2018, p.224.
5. ^ Coxeter, H. S. M. (1974), Regular complex polytopes, London & New York: Cambridge University Press, p. 180, MR 0370328.

## References

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