Hypercube

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This article is about the mathematical concept. For the film, see Cube 2: Hypercube. For the computer architecture, see Connection Machine.
Perspective projections
Hexahedron.svg Hypercube.svg
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. A unit hypercube's longest diagonal in n-dimensions is equal to .

An n-dimensional hypercube is also called an n-cube or an n-dimensional cube. 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polytope" is also used, notably in the work of H. S. M. Coxeter (originally from Elte, 1912),[1] but it has now been superseded.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 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 coordinates equal to 0 or 1 is called "the" unit hypercube.

Construction[edit]

A diagram showing how to create a tesseract from a point.
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.

Coordinates[edit]

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.

Elements[edit]

Every n-cube of n > 0 is composed of elements, or n-cubes of a lower dimension, on the (n-1)-dimensional surface on the parent hypercube. A side is any element of (n-1) dimension of the parent hypercube. 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 (a cube has vertices, for instance).

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

,[2]     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 vertices defines a vertex in a -dimensional boundary. There are ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted times since it has that many vertices, we need to divide with this number.

This identity can also be used to generate the formula for the n-dimensional cube surface area. The surface area of a hypercube is: .

These numbers can also be generated by the linear recurrence relation

,     with ,     and undefined elements (where , , or ) = 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 (sequence A013609 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
( )
CDel node.png
1
1 1-cube Line segment
Ditel
{}
CDel node 1.png
2 1
2 2-cube Square
Tetragon
{4}
CDel node 1.pngCDel 4.pngCDel node.png
4 4 1
3 3-cube Cube
Hexahedron
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
8 12 6 1
4 4-cube Tesseract
Octachoron
{4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16 32 24 8 1
5 5-cube Penteract
Deca-5-tope
{4,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
32 80 80 40 10 1
6 6-cube Hexeract
Dodeca-6-tope
{4,3,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
64 192 240 160 60 12 1
7 7-cube Hepteract
Tetradeca-7-tope
{4,3,3,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
128 448 672 560 280 84 14 1
8 8-cube Octeract
Hexadeca-8-tope
{4,3,3,3,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
256 1024 1792 1792 1120 448 112 16 1
9 9-cube Enneract
Octadeca-9-tope
{4,3,3,3,3,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
1024 5120 11520 15360 13440 8064 3360 960 180 20 1

Graphs[edit]

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 12-cube.

Petrie polygon Orthographic projections
1-simplex t0.svg
Line segment
2-cube.svg
Square
3-cube graph.svg
Cube
4-cube graph.svg
4-cube (tesseract)
5-cube graph.svg
5-cube (penteract)
6-cube graph.svg
6-cube (hexeract)
7-cube graph.svg
7-cube (hepteract)
8-cube.svg
8-cube (octeract)
9-cube.svg
9-cube (enneract)
10-cube.svg
10-cube (dekeract)
11-cube.svg
11-cube (hendekeract)
12-cube.svg
12-cube (dodekeract)
Projection of a rotating tesseract.

Related families of polytopes[edit]

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 n.

Relation to n-simplices[edit]

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[edit]

Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes, γp
n
= p{4}2{3}...2{3}2, or CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 3.pngCDel node.png. Real solutions exist with p=2, i.e. γ2
n
= γn = 2{4}2{3}...2{3}2 = {4,3,..,3}. For p>2, they exist in . 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: . This is pn vertices and pn facets.[3]

Generalized hypercubes
p=2 p=3 p=4 p=5 p=6 p=7 p=8
2-generalized-2-cube.svg
γ2
2
= {4} = CDel node 1.pngCDel 4.pngCDel node.png
4 vertices
3-generalized-2-cube skew.svg
γ3
2
= CDel 3node 1.pngCDel 4.pngCDel node.png
9 vertices
4-generalized-2-cube.svg
γ4
2
= CDel 4node 1.pngCDel 4.pngCDel node.png
16 vertices
5-generalized-2-cube skew.svg
γ5
2
= CDel 5node 1.pngCDel 4.pngCDel node.png
25 vertices
6-generalized-2-cube.svg
γ6
2
= CDel 6node 1.pngCDel 4.pngCDel node.png
36 vertices
7-generalized-2-cube skew.svg
γ7
2
= CDel 7node 1.pngCDel 4.pngCDel node.png
49 vertices
8-generalized-2-cube.svg
γ8
2
= CDel 8node 1.pngCDel 4.pngCDel node.png
64 vertices
2-generalized-3-cube.svg
γ2
3
= {4,3} = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
8 vertices
3-generalized-3-cube.svg
γ3
3
= CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
27 vertices
4-generalized-3-cube.svg
γ4
3
= CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
64 vertices
5-generalized-3-cube.svg
γ5
3
= CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
125 vertices
6-generalized-3-cube.svg
γ6
3
= CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
216 vertices
7-generalized-3-cube.svg
γ7
3
= CDel 7node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
343 vertices
8-generalized-3-cube.svg
γ8
3
= CDel 8node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
512 vertices
2-generalized-4-cube.svg
γ2
4
= {4,3,3}
= CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16 vertices
3-generalized-4-cube.svg
γ3
4
= CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
81 vertices
4-generalized-4-cube.svg
γ4
4
= CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
256 vertices
5-generalized-4-cube.svg
γ5
4
= CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
625 vertices
6-generalized-4-cube.svg
γ6
4
= CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
1296 vertices
7-generalized-4-cube.svg
γ7
4
= CDel 7node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2401 vertices
8-generalized-4-cube.svg
γ8
4
= CDel 8node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4096 vertices
2-generalized-5-cube.svg
γ2
5
= {4,3,3,3}
= CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
32 vertices
3-generalized-5-cube.svg
γ3
5
= CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
243 vertices
4-generalized-5-cube.svg
γ4
5
= CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
1024 vertices
5-generalized-5-cube.svg
γ5
5
= CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3125 vertices
6-generalized-5-cube.svg
γ6
5
= CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7776 vertices
γ7
5
= CDel 7node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16,807 vertices
γ8
5
= CDel 8node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
32,768 vertices
2-generalized-6-cube.svg
γ2
6
= {4,3,3,3,3}
= CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
64 vertices
3-generalized-6-cube.svg
γ3
6
= CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
729 vertices
4-generalized-6-cube.svg
γ4
6
= CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4096 vertices
5-generalized-6-cube.svg
γ5
6
= CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
15,625 vertices
γ6
6
= CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
46,656 vertices
γ7
6
= CDel 7node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
117,649 vertices
γ8
6
= CDel 8node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
262,144 vertices
2-generalized-7-cube.svg
γ2
7
= {4,3,3,3,3,3}
= CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3-generalized-7-cube.svg
γ3
7
= CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2187 vertices
γ4
7
= CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16,384 vertices
γ5
7
= CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
78,125 vertices
γ6
7
= CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
279,936 vertices
γ7
7
= CDel 7node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
823,543 vertices
γ8
7
= CDel 8node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2,097,152 vertices
2-generalized-8-cube.svg
γ2
8
= {4,3,3,3,3,3,3}
= CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
256 vertices
3-generalized-8-cube.svg
γ3
8
= CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6561 vertices
γ4
8
= CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
65,536 vertices
γ5
8
= CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
390,625 vertices
γ6
8
= CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
1,679,616 vertices
γ7
8
= CDel 7node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5,764,801 vertices
γ8
8
= CDel 8node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16,777,216 vertices

See also[edit]

Notes[edit]

  1. ^ Elte, E. L. (1912). "The Semiregular Polytopes of the Hyperspaces". Groningen: University of Groningen.  Chapter IV, five dimensional semiregular polytope [1]
  2. ^ Coxeter, Regular polytopes, p.120
  3. ^ Coxeter, H. S. M. (1974), Regular complex polytopes, London & New York: Cambridge University Press, p. 180, MR 0370328 .

References[edit]

  • Bowen, J. P. (April 1982). "Hypercubes". Practical Computing. 5 (4): 97–99. Archived from the original on June 30, 2008. 
  • Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). Dover. p. 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.

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds