Hypercube

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.

A unit hypercube of dimension ${\displaystyle n}$ is the convex hull of all the ${\displaystyle 2^{n}}$ points whose ${\displaystyle n}$ Cartesian coordinates are each equal to either ${\displaystyle 0}$ or ${\displaystyle 1}$. These points are its vertices. The hypercube with these coordinates 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 ${\displaystyle 2^{n}}$ 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 ${\displaystyle 2^{n}}$.

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 cubes 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).^[5]

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

${\displaystyle E_{m,n}=2^{n-m}{n \choose m}}$,^[6]     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 ${\displaystyle n<m}$, ${\displaystyle n<0}$, 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.

The extended f-vector for an n-cube can also be computed by expanding ${\displaystyle (2x+1)^{n}}$ (concisely, (2,1)ⁿ), and reading off the coefficients of the resulting polynomial. For example, the elements of a tesseract is (2,1)⁴ = (4,4,1)² = (16,32,24,8,1).

Number ${\displaystyle E_{m,n}}$ of ${\displaystyle m}$-dimensional faces of a ${\displaystyle n}$-dimensional hypercube (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[7]	{}	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

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.

Petrie polygon Orthographic projections

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

The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions.^[8]

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 (the cross-polytopes) to form compound polytopes:

• In two dimensions, we obtain the octagrammic star figure {8/2},
• In three dimensions we obtain the compound of cube and octahedron,
• In four dimensions we obtain the compound of tesseract and 16-cell.

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.

Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes, γ^p
_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 ${\displaystyle \mathbb {C} ^{n}}$. 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.^[9]

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

Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type of figurate number corresponding to an n-cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield a square number or "perfect square", which can be arranged into a square shape with a side length corresponding to that of the base. Similarly, the exponent 3 will yield a perfect cube, an integer which can be arranged into a cube shape with a side length of the base. As a result, the act of raising a number to 2 or 3 is more commonly referred to as "squaring" and "cubing", respectively. However, the names of higher-order hypercubes do not appear to be in common use for higher powers.

• Hypercube interconnection network of computer architecture
• Hyperoctahedral group, the symmetry group of the hypercube
• Hypersphere
• Simplex
• Parallelotope
• Crucifixion (Corpus Hypercubus), a painting by Salvador Dalí featuring an unfolded 4-cube