Hypercube

Summary

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

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

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.

Vertex coordinates

A unit hypercube of dimension is the convex hull of all the points whose Cartesian coordinates are each equal to either or . This hypercube is also the cartesian product of copies of the unit interval . 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 points whose vectors of Cartesian coordinates are

Here the symbol means that each coordinate is either equal to or to . This unit hypercube is also the cartesian product . Any unit hypercube has an edge length of and an -dimensional volume of .

The -dimensional hypercube obtained as the convex hull of the points with coordinates or, equivalently as the Cartesian product is also often considered due to the simpler form of its vertex coordinates. Its edge length is , and its -dimensional volume is .

Faces

Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension admits facets, or faces of dimension : a (-dimensional) line segment has endpoints; a (-dimensional) square has sides or edges; a -dimensional cube has square faces; a (-dimensional) tesseract has three-dimensional cube as its facets. The number of vertices of a hypercube of dimension is (a usual, -dimensional cube has vertices, for instance).

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

,[3]     where and denotes the factorial of .

For example, the boundary of a -cube () contains cubes (-cubes), squares (-cubes), line segments (-cubes) and vertices (-cubes). This identity can be proven by a simple combinatorial argument: for each of the vertices of the hypercube, there are ways to choose a collection of edges incident to that vertex. Each of these collections defines one of the -dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the -dimensional faces of the hypercube is counted times since it has that many vertices, and we need to divide by this number.

The number of facets of the hypercube can be used to compute the -dimensional volume of its boundary: that volume is times the volume of a -dimensional hypercube; that is, where is the length of the edges of the hypercube.

These numbers can also be generated by the linear recurrence relation

,     with , and when , , or .

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 line segments.

Number of -dimensional faces of a -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
( )
CDel node.png
1
1 1-cube Line segment
Dion[4]
{}
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

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 15-cube.

Petrie polygon Orthographic projections
0-point t0.svg
Point
1-simplex t0.svg
Line segment
2-cube.svg
Square
3-cube graph.svg
Cube
4-cube graph.svg
Tesseract
5-cube graph.svg
5-cube
6-cube graph.svg
6-cube
7-cube graph.svg
7-cube
8-cube.svg
8-cube
9-cube.svg
9-cube
10-cube.svg
10-cube
11-cube.svg
11-cube
12-cube.svg
12-cube
13-cube.svg
13-cube
14-cube.svg
14-cube
15-cube.svg
15-cube
16-cube t0 A15.svg
16-cube
Projection of a rotating tesseract.

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 n.

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

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, 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.[5]

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
128 vertices
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

Notes

  1. ^ Elte, E. L. (1912). "IV, Five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Netherlands: University of Groningen. ISBN 141817968X.
  2. ^ Coxeter 1973, pp. 122–123, §7.2 see illustration Fig 7.2C.
  3. ^ Coxeter 1973, p. 122, §7·25.
  4. ^ Johnson, Norman W.; Geometries and Transformations, Cambridge University Press, 2018, p.224.
  5. ^ 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). Regular Polytopes (3rd ed.). §7.2. see illustration Fig. 7-2C: Dover. pp. 122-123. ISBN 0-486-61480-8.CS1 maint: location (link) 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

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