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Hypercube

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From Wikipedia, the free encyclopedia

Convex polytope, the n-dimensional analogue of a square and a cube

For other uses, seeHypercube (disambiguation).

In the followingperspective projections,cube is 3-cube andtesseract is 4-cube.

Ingeometry, ahypercube is ann-dimensional analogue of asquare (n = 2) and acube (n = 3); the special case forn = 4 is known as atesseract. It is aclosed,compact,convex figure whose 1-skeleton consists of groups of oppositeparallel line segments aligned in each of the space'sdimensions,perpendicular to each other and of the same length. A unit hypercube's longest diagonal inn dimensions is equal to ${\sqrt {n}}$ .

Ann-dimensional hypercube is more commonly referred to as ann-cube or sometimes as ann-dimensional cube.^[1]^[2] The termmeasure polytope (originally from Elte, 1912)^[3] is also used, notably in the work ofH. S. M. Coxeter who also labels the hypercubes the γn polytopes.^[4]

The hypercube is the special case of ahyperrectangle (also called ann-orthotope).

Aunit hypercube is a hypercube whose side has length oneunit. Often, the hypercube whose corners (orvertices) are the 2ⁿ points inRⁿ with each coordinate equal to 0 or 1 is calledthe unit hypercube.

Construction

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By the number of dimensions

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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 aperpendicular 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 unittesseract).

This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as aMinkowski sum: thed-dimensional hypercube is the Minkowski sum ofd mutually perpendicular unit-length line segments, and is therefore an example of azonotope.

The1-skeleton of a hypercube is ahypercube graph.

Vertex coordinates

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A unit hypercube of dimension $n {\displaystyle n}$ is theconvex hull of all the $2^{n}$ points whose $n {\displaystyle n}$ Cartesian coordinates are each equal to either $0 {\displaystyle 0}$ or $1 {\displaystyle 1}$ . These points are itsvertices. The hypercube with these coordinates is also thecartesian product $[0,1]^{n}$ of $n {\displaystyle n}$ copies of the unitinterval $[0,1]$ . Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by atranslation. 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 $1 {\displaystyle 1}$ and an $n {\displaystyle n}$ -dimensional volume of $1 {\displaystyle 1}$ .

The $n {\displaystyle n}$ -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 $2 {\displaystyle 2}$ , and its $n {\displaystyle n}$ -dimensional volume is $2^{n}$ .

Faces

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

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

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

,^[6] where

{n \choose m}={\frac {n!}{m!\,(n-m)!}}

and

n! {\displaystyle n!}

denotes thefactorial of

n {\displaystyle n}

For example, the boundary of a $4 {\displaystyle 4}$ -cube ( $n=4$ ) contains $8 {\displaystyle 8}$ cubes ( $3 {\displaystyle 3}$ -cubes), $24 {\displaystyle 24}$ squares ( $2 {\displaystyle 2}$ -cubes), $32 {\displaystyle 32}$ line segments ( $1 {\displaystyle 1}$ -cubes) and $16 {\displaystyle 16}$ vertices ( $0 {\displaystyle 0}$ -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 $m {\displaystyle m}$ edges incident to that vertex. Each of these collections defines one of the $m {\displaystyle m}$ -dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the $m {\displaystyle m}$ -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 $s {\displaystyle s}$ is the length of the edges of the hypercube.

These numbers can also be generated by the linearrecurrence 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 extendedf-vector for ann-cube can also be computed by expanding $(2x+1)^{n}$ (concisely, (2,1)ⁿ), and reading off the coefficients of the resultingpolynomial. 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 $m {\displaystyle m}$ -dimensional faces of a $n {\displaystyle n}$ -dimensional hypercube (sequenceA038207 in theOEIS)
			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

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Ann-cube can be projected inside a regular 2n-gonal polygon by askew 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

Related families of polytopes

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

Thehypercube family is one of threeregular polytope families, labeled byCoxeter asγ_n. The other two are the hypercube dual family, thecross-polytopes, labeled asβ_n, and thesimplices, labeled asα_n. A fourth family, theinfinite tessellations of hypercubes, is labeled asδ_n.

Another related family of semiregular anduniform polytopes is thedemihypercubes, which are constructed from hypercubes with alternate vertices deleted andsimplex facets added in the gaps, labeled ashγ_n.

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

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

Relation to (n−1)-simplices

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The graph of then-hypercube's edges isisomorphic to theHasse diagram of the (n−1)-simplex'sface lattice. This can be seen by orienting then-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'sn−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

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Regularcomplex polytopes can be defined incomplex Hilbert space calledgeneralized hypercubes, γ^p
_n =_p{4}₂{3}...₂{3}₂, or... Real solutions exist withp = 2, i.e. γ²
_n = γ_n =₂{4}₂{3}...₂{3}₂ = {4,3,..,3}. Forp > 2, they exist in $\mathbb {C} ^{n}$ . The facets are generalized (n−1)-cube and thevertex figure are regularsimplexes.

Theregular polygon perimeter seen in these orthogonal projections is called aPetrie polygon. The generalized squares (n = 2) are shown with edges outlined as red and blue alternating colorp-edges, while the highern-cubes are drawn with black outlinedp-edges.

The number ofm-face elements in ap-generalizedn-cube are: $p^{n-m}{n \choose m}$ . This ispⁿ vertices andpn 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

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Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type offigurate number corresponding to ann-cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield asquare 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 yielda 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

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^Paul Dooren; Luc Ridder (1976)."An adaptive algorithm for numerical integration over an n-dimensional cube".Journal of Computational and Applied Mathematics.2 (3):207–217.doi:10.1016/0771-050X(76)90005-X.
^Xiaofan Yang; Yuan Tang (15 April 2007)."A (4n − 9)/3 diagnosis algorithm on n-dimensional cube network".Information Sciences.177 (8):1771–1781.doi:10.1016/j.ins.2006.10.002.
^Elte, E. L. (1912). "IV, Five dimensional semiregular polytope".The Semiregular Polytopes of the Hyperspaces. Netherlands:University of Groningen.ISBN 141817968X.{{cite book}}:ISBN / Date incompatibility (help)
^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 (1992)."Transmitting in the n-dimensional cube".Discrete Applied Mathematics.37–38:9–11.doi:10.1016/0166-218X(92)90121-P.
^Coxeter, H. S. M. (1974),Regular complex polytopes, London & New York:Cambridge University Press, p. 180,MR 0370328.

References

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Bowen, J. P. (April 1982)."Hypercube".Practical Computing.5 (4):97–99. Archived fromthe original on 2008-06-30. RetrievedJune 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 inn 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 aVeitch diagram orKarnaugh map.

External links

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Wikimedia Commons has media related toHypercubes.

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes andshapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

v t e Fundamental convexregular anduniform 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
Uniformn-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 •Polytope operations