Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 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 ${\sqrt {n}}$ .

In the following perspective projections, cube is 3-cube and tesseract is 4-cube.

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube.^[1]^[2] The term measure polytope (originally from Elte, 1912)^[3] is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.^[4]

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

By the number of dimensions

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

Projection of a rotating tesseract.

A unit hypercube of dimension $n$ is the convex hull of all the $2^{n}$ points whose $n$ Cartesian coordinates are each equal to either $0$ or $1$ . These points are its vertices. The hypercube with these coordinates is also the cartesian product $[0,1]^{n}$ of $n$ copies of the unit interval $[0,1]$ . 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 $2^{n}$ points whose vectors of Cartesian coordinates are

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

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

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

Faces

Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension $n$ admits $2n$ facets, or faces of dimension $n-1$ : a ( $1$ -dimensional) line segment has $2$ endpoints; a ( $2$ -dimensional) square has $4$ sides or edges; a $3$ -dimensional cube has $6$ square faces; a ( $4$ -dimensional) tesseract has $8$ three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension $n$ is $2^{n}$ (a usual, $3$ -dimensional cube has $2^{3}=8$ vertices, for instance).^[5]

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

E_{m,n}=2^{n-m}{n \choose m}

,^[6] 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), $32$ line segments ( $1$ -cubes) and $16$ vertices ( $0$ -cubes). This identity can be proven by a simple combinatorial argument: for each of the $2^{n}$ vertices of the hypercube, there are ${\tbinom {n}{m}}$ ways to choose a collection of $m$ edges incident to that vertex. Each of these collections defines one of the $m$ -dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the $m$ -dimensional faces of the hypercube is counted $2^{m}$ times since it has that many vertices, and we need to divide $2^{n}{\tbinom {n}{m}}$ by this number.

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

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

E_{m,n}=0

when

n<m

,

n<0

, or

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 $E_{1,3}=12$ line segments.

The extended f-vector for an n-cube can also be computed by expanding $(2x+1)^{n}$ (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 $E_{m,n}$ of $m$ -dimensional faces of a $n$ -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

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 16-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

Related families of polytopes

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

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, 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.

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, 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 $\mathbb {C} ^{n}$ . 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.^[9]

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

Relation to exponentiation

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.

Notes

^ Paul Dooren; Luc Ridder. "An adaptive algorithm for numerical integration over an n-dimensional cube".
^ Xiaofan Yang; Yuan Tang. "A (4n − 9)/3 diagnosis algorithm on n-dimensional cube network".
^ Elte, E. L. (1912). "IV, Five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Netherlands: University of Groningen. ISBN 141817968X.
^ Coxeter 1973, pp. 122–123, §7.2 see illustration Fig 7.2C.
^ Miroslav Vořechovský; Jan Mašek; Jan Eliáš (November 2019). "Distance-based optimal sampling in a hypercube: Analogies to N-body systems". Advances in Engineering Software. 137. 102709. doi:10.1016/j.advengsoft.2019.102709. ISSN 0965-9978.
^ Coxeter 1973, p. 122, §7·25.
^ Johnson, Norman W.; Geometries and Transformations, Cambridge University Press, 2018, p.224.
^ Noga Alon. "Transmitting in the n-dimensional cube".
^ Coxeter, H. S. M. (1974), Regular complex polytopes, London & New York: Cambridge University Press, p. 180, MR 0370328.

References

Bowen, J. P. (April 1982). "Hypercube". Practical Computing. 5 (4): 97–99. Archived from the original on 2008-06-30. Retrieved June 30, 2008.
Coxeter, H. S. M. (1973). "§7.2. see illustration Fig. 7-2c". Regular Polytopes (3rd ed.). Dover. pp. 122-123. ISBN 0-486-61480-8. p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
Hill, Frederick J.; Gerald R. Peterson (1974). Introduction to Switching Theory and Logical Design: Second Edition. New York: John Wiley & Sons. ISBN 0-471-39882-9. Cf Chapter 7.1 "Cubical Representation of Boolean Functions" wherein the notion of "hypercube" is introduced as a means of demonstrating a distance-1 code (Gray code) as the vertices of a hypercube, and then the hypercube with its vertices so labelled is squashed into two dimensions to form either a Veitch diagram or Karnaugh map.

External links

Weisstein, Eric W. "Hypercube". MathWorld.
Weisstein, Eric W. "Hypercube graphs". MathWorld.
Rotating a Hypercube by Enrique Zeleny, Wolfram Demonstrations Project.
Rudy Rucker and Farideh Dormishian's Hypercube Downloads
A001787 Number of edges in an n-dimensional hypercube. at OEIS

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Paul Dooren; Luc Ridder. "An adaptive algorithm for numerical integration over an n-dimensional cube".

[2] Xiaofan Yang; Yuan Tang. "A (4n − 9)/3 diagnosis algorithm on n-dimensional cube network".

[3] Elte, E. L. (1912). "IV, Five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Netherlands: University of Groningen. ISBN 141817968X.

[FOOTNOTECoxeter1973122–123§7.2_see_illustration_Fig_7.2<small>C</small>-4] Coxeter 1973, pp. 122–123, §7.2 see illustration Fig 7.2C.

[5] Miroslav Vořechovský; Jan Mašek; Jan Eliáš (November 2019). "Distance-based optimal sampling in a hypercube: Analogies to N-body systems". Advances in Engineering Software. 137. 102709. doi:10.1016/j.advengsoft.2019.102709. ISSN 0965-9978.

[FOOTNOTECoxeter1973122§7·25-6] Coxeter 1973, p. 122, §7·25.

[7] Johnson, Norman W.; Geometries and Transformations, Cambridge University Press, 2018, p.224.

[8] Noga Alon. "Transmitting in the n-dimensional cube".

[9] Coxeter, H. S. M. (1974), Regular complex polytopes, London & New York: Cambridge University Press, p. 180, MR 0370328.

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