 Hypercube

This article is about the mathematical concept. For the film, see Cube 2: Hypercube.
Perspective projections Cube (3cube) Tesseract (4cube) In geometry, a hypercube is an ndimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure whose 1skeleton 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.
An ndimensional hypercube is also called an ncube. The term "measure polytope" is also used, notably in the work of H.S.M. Coxeter (originally from Elte, 1912^{[1]}), but it has now been superseded.
The hypercube is the special case of a hyperrectangle (also called an orthotope).
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2^{n} points in R^{n} with coordinates equal to 0 or 1 is called "the" unit hypercube.
Contents
Construction
 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 2dimensional square.
 3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3dimensional cube.
 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4dimensional 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 ddimensional hypercube is the Minkowski sum of d mutually perpendicular unitlength line segments, and is therefore an example of a zonotope.
The 1skeleton 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 . It has an edge length of 1 and an ndimensional volume of 1.
An ndimensional hypercube is also often regarded as the convex hull of all sign permutations of the coordinates . This form is often chosen due to ease of writing out the coordinates. Its edge length is 2, and its ndimensional volume is 2^{n}.
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 γ_{n}. The other two are the hypercube dual family, the crosspolytopes, labeled as β_{n}, and the simplices, labeled as α_{n}. A fourth family, the infinite tessellations of hypercubes, he 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}.
Elements
A hypercube of dimension n has 2n "sides" (a 1dimensional line has 2 end points; a 2dimensional square has 4 sides or edges; a 3dimensional cube has 6 2dimensional faces; a 4dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is 2^{n} (a cube has 2^{3} vertices, for instance).
A simple formula to calculate the number of "n2"faces in an ndimensional hypercube is: 2n^{2} − 2n
The number of mdimensional hypercubes (just referred to as mcube from here on) on the boundary of an ncube is
 , where and n! denotes the factorial of n.
For example, the boundary of a 4cube (n=4) contains 8 cubes (3cubes), 24 squares (2cubes), 32 lines (1cubes) and 16 vertices (0cubes).
This identity can be proved by combinatorial arguments; each of the 2^{n} vertices defines a vertex in a mdimensional boundary. There are 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 with this number. Hence the identity above.
These numbers can also be generated by the linear recurrence relation
 , with , and undefined elements = 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 = 12 lines in total.
Hypercube elements m 0 1 2 3 4 5 6 7 8 9 10 n γ_{n} ncube Names
Schläfli symbol
CoxeterDynkinVertices Edges Faces Cells (3faces) 4faces (Hypercells) 5faces 6faces 7faces 8faces 9faces 10faces 0 γ_{0} 0cube Point
1 1 γ_{1} 1cube Line segment
{}
2 1 2 γ_{2} 2cube Square
Tetragon
{4}
4 4 1 3 γ_{3} 3cube Cube
Hexahedron
{4,3}
8 12 6 1 4 γ_{4} 4cube Tesseract
Octachoron
{4,3,3}
16 32 24 8 1 5 γ_{5} 5cube Penteract
Decateron
{4,3,3,3}
32 80 80 40 10 1 6 γ_{6} 6cube Hexeract
Dodecapeton
{4,3,3,3,3}
64 192 240 160 60 12 1 7 γ_{7} 7cube Hepteract
Tetradeca7tope
{4,3,3,3,3,3}
128 448 672 560 280 84 14 1 8 γ_{8} 8cube Octeract
Hexadeca8tope
{4,3,3,3,3,3,3}
256 1024 1792 1792 1120 448 112 16 1 9 γ_{9} 9cube Enneract
Octadeca9tope
{4,3,3,3,3,3,3,3}
512 2304 4608 5376 4032 2016 672 144 18 1 10 γ_{10} 10cube Dekeract
icosa10tope
{4,3,3,3,3,3,3,3,3}
1024 5120 11520 15360 13440 8064 3360 960 180 20 1 Graphs
An ncube can be projected inside a regular 2ngonal polygon by a skew orthogonal projection, shown here from the line segment to the dodekeract.
Petrie polygon Orthographic projections
Line segment
Square
Cube
4cube (tesseract)
5cube (penteract)
6cube (hexeract)
7cube (hepteract)
8cube (octeract)
9cube (enneract)
10cube (dekeract)
11cube (hendekeract)
12cube (dodekeract)Relation to nsimplices
The graph of the nhypercube's edges is isomorphic to the Hasse diagram of the (n1)simplex's face lattice. This can be seen by orienting the nhypercube so that two opposite vertices lie vertically, corresponding to the (n1)simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (n1)simplex's facets (n2 faces), and each vertex connected to those vertices maps to one of the simplex's n3 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 (n1)simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.
See also
 Hyperoctahedral group, the symmetry group of the hypercube
 Hypersphere
 Simplex
 Hypercube interconnection network of computer architecture
Notes
References
 Bowen, J. P. (April 1982). "Hypercubes". Practical Computing 5 (4): 97–99. http://www.jpbowen.com/publications/ndcubes.html.
 Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). Dover. pp. 123. ISBN 0486614808. p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
 Hill, Frederick J.; Gerald R. Peterson. Introduction to Switching Theory and Logical Design: Second Edition. NY: John Wiley & Sons. ISBN 0471398829. Cf Chapter 7.1 "Cubical Representation of Boolean Functions" wherein the notion of "hypercube" is introduced as a means of demonstrating a distance1 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" from MathWorld.
 Weisstein, Eric W., "Hypercube graphs" from MathWorld.
 Olshevsky, George, Measure polytope at Glossary for Hyperspace.
 www.4dscreen.de (Rotation of 4D – 7DCube)
 Rotating a Hypercube by Enrique Zeleny, Wolfram Demonstrations Project.
 Stereoscopic Animated Hypercube
Dimension Dimensional spaces One · Two · Three · Four · Five · Six · Seven · Eight · ndimensions · Spacetime · Projective space · HyperplanePolytopes and Shapes Concepts and mathematics Cartesian coordinates · Linear algebra · Geometric algebra · Conformal geometry · Reflection · Rotation · Plane of rotation · Space · Fractal dimension · MultiverseFundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: Multidimensional geometry
 Cubes
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Look at other dictionaries:
hypercube — n. A mathematical object existing in more than three dimensions, analogous to the cube in that each two dimensional facet of the surface is a square; a generalization of a cube in more than three dimensions. [PJC] … The Collaborative International Dictionary of English
Hypercube — Pour les articles homonymes, voir Hypercube (homonymie). Une projection d un hypercube (dans une image bidimensionnelle) Un hypercube est, en géométrie, un analogue … Wikipédia en Français
Hypercube — der Name Hypercube bezeichnet: ein geometrisches Gebilde, auch Hyperkubus genannt, siehe Hyperwürfel einen Film, siehe Cube 2: Hypercube Hypercube Inc., einen US amerikanischen Hersteller für Molecular Modelling Software, siehe Hypercube… … Deutsch Wikipedia
hypercube — noun Date: 1909 1. a geometric figure (as a tesseract) in Euclidean space of n dimensions that is analogous to a cube in three dimensions 2. a computer architecture in which each processor is connected to n others based on analogy to a hypercube… … New Collegiate Dictionary
hypercube — ˌ noun Etymology: hyper (herein) + cube (I) 1. : a geometric figure in Euclidean space of n dimensions that is analogous to a cube in three dimensions in having 2n vertices each of which is connected to n other vertices by mutu … Useful english dictionary
hypercube — noun /ˈhaɪ.pə.kjuːb,ˈhaɪ.pɚ.kjuːb/ a) A geometric figure in four or more dimensions, which is analogous to a cube in three dimensions. Specifically, the n dimensional equivalent of a cube for any non negative integer n. b) A … Wiktionary
hypercube — ● n. m. ►ARCHI ordinateur multiprocesseur, ceux ci étant aux sommets de cubes imbriqués et reliés entre eux selon un réseau suivant les arêtes de ces cubes … Dictionnaire d'informatique francophone
hypercube — noun a geometrical figure in four or more dimensions which is analogous to a cube in three dimensions … English new terms dictionary
hypercube — hy·per·cube … English syllables
hypercube — /ˈhaɪpəkjub/ (say huypuhkyoohb) noun → tesseract … Australian English dictionary