Perspective projections Cube (3-cube) Tesseract (4-cube)
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). 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.
An n-dimensional hypercube is also called an n-cube. The term "measure polytope" is also used, notably in the work of H.S.M. Coxeter (originally from Elte, 1912), but it has now been superseded.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 2n points in Rn with coordinates equal to 0 or 1 is called "the" unit hypercube.
- 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 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 n-dimensional volume of 1.
An n-dimensional 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 n-dimensional volume is 2n.
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 γn. The other two are the hypercube dual family, the cross-polytopes, labeled as βn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn.
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 hγn.
A hypercube of dimension n has 2n "sides" (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is 2n (a cube has 23 vertices, for instance).
A simple formula to calculate the number of "n-2"-faces in an n-dimensional hypercube is: 2n2 − 2n
The number of m-dimensional hypercubes (just referred to as m-cube from here on) on the boundary of an n-cube is
- , where 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 lines (1-cubes) and 16 vertices (0-cubes).
This identity can be proved by combinatorial arguments; each of the 2n vertices defines a vertex in a m-dimensional boundary. There are ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted 2m 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 γn n-cube Names
Vertices Edges Faces Cells (3-faces) 4-faces (Hypercells) 5-faces 6-faces 7-faces 8-faces 9-faces 10-faces 0 γ0 0-cube Point
1 1 γ1 1-cube Line segment
2 1 2 γ2 2-cube Square
4 4 1 3 γ3 3-cube Cube
8 12 6 1 4 γ4 4-cube Tesseract
16 32 24 8 1 5 γ5 5-cube Penteract
32 80 80 40 10 1 6 γ6 6-cube Hexeract
64 192 240 160 60 12 1 7 γ7 7-cube Hepteract
128 448 672 560 280 84 14 1 8 γ8 8-cube Octeract
256 1024 1792 1792 1120 448 112 16 1 9 γ9 9-cube Enneract
512 2304 4608 5376 4032 2016 672 144 18 1 10 γ10 10-cube Dekeract
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 dodekeract.
Petrie polygon Orthographic projections
Relation to n-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.
- Hyperoctahedral group, the symmetry group of the hypercube
- Hypercube interconnection network of computer architecture
- 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 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. Introduction to Switching Theory and Logical Design: Second Edition. NY: 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.
- Weisstein, Eric W., "Hypercube" from MathWorld.
- Weisstein, Eric W., "Hypercube graphs" from MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- www.4d-screen.de (Rotation of 4D – 7D-Cube)
- Rotating a Hypercube by Enrique Zeleny, Wolfram Demonstrations Project.
- Stereoscopic Animated Hypercube
Dimension Dimensional spaces Polytopes and Shapes Concepts and mathematics Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An BCn Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5-cell 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 n-polytopes n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes
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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