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

**. The term**

*n*-dimensional cube**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 2^{n} points in **R**^{n} with each coordinate equal to 0 or 1 is called *the* unit hypercube.

## Construction

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 vector 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.

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 | PointMonon |
( ) |
1 | ||||||||||

1 | 1-cube | Line segmentDion^{[4]} |
{} |
2 | 1 | |||||||||

2 | 2-cube | SquareTetragon |
{4} |
4 | 4 | 1 | ||||||||

3 | 3-cube | CubeHexahedron |
{4,3} |
8 | 12 | 6 | 1 | |||||||

4 | 4-cube | TesseractOctachoron |
{4,3,3} |
16 | 32 | 24 | 8 | 1 | ||||||

5 | 5-cube | PenteractDeca-5-tope |
{4,3,3,3} |
32 | 80 | 80 | 40 | 10 | 1 | |||||

6 | 6-cube | HexeractDodeca-6-tope |
{4,3,3,3,3} |
64 | 192 | 240 | 160 | 60 | 12 | 1 | ||||

7 | 7-cube | HepteractTetradeca-7-tope |
{4,3,3,3,3,3} |
128 | 448 | 672 | 560 | 280 | 84 | 14 | 1 | |||

8 | 8-cube | OcteractHexadeca-8-tope |
{4,3,3,3,3,3,3} |
256 | 1024 | 1792 | 1792 | 1120 | 448 | 112 | 16 | 1 | ||

9 | 9-cube | EnneractOctadeca-9-tope |
{4,3,3,3,3,3,3,3} |
512 | 2304 | 4608 | 5376 | 4032 | 2016 | 672 | 144 | 18 | 1 | |

10 | 10-cube | DekeractIcosa-10-tope |
{4,3,3,3,3,3,3,3,3} |
1024 | 5120 | 11520 | 15360 | 13440 | 8064 | 3360 | 960 | 180 | 20 | 1 |

### Graphs

An ** n-cube** can be projected inside a regular 2

*n*-gonal polygon by a skew orthogonal projection, shown here from the line segment to the 15-cube.

Point |
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 |
16-cube |

## 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

**cross-polytopes**, labeled as

*β*and the

_{n,}**simplices**, labeled as

*α*. A fourth family, the infinite tessellations of hypercubes, he labeled as

_{n}*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}*.

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

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

## 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*, γ^{p}_{n} = _{p}{4}_{2}{3}..._{2}{3}_{2}, or ... Real solutions exist with *p*=2, i.e. γ^{2}_{n} = γ_{n} = _{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 *p*^{n} vertices and *pn* facets.^{[5]}

p=2 |
p=3 |
p=4 |
p=5 |
p=6 |
p=7 |
p=8
| ||
---|---|---|---|---|---|---|---|---|

γ ^{2}_{2} = {4} = 4 vertices |
γ ^{3}_{2} = 9 vertices |
γ ^{4}_{2} = 16 vertices |
γ ^{5}_{2} = 25 vertices |
γ ^{6}_{2} = 36 vertices |
γ ^{7}_{2} = 49 vertices |
γ ^{8}_{2} = 64 vertices | ||

γ ^{2}_{3} = {4,3} = 8 vertices |
γ ^{3}_{3} = 27 vertices |
γ ^{4}_{3} = 64 vertices |
γ ^{5}_{3} = 125 vertices |
γ ^{6}_{3} = 216 vertices |
γ ^{7}_{3} = 343 vertices |
γ ^{8}_{3} = 512 vertices | ||

γ ^{2}_{4} = {4,3,3}= 16 vertices |
γ ^{3}_{4} = 81 vertices |
γ ^{4}_{4} = 256 vertices |
γ ^{5}_{4} = 625 vertices |
γ ^{6}_{4} = 1296 vertices |
γ ^{7}_{4} = 2401 vertices |
γ ^{8}_{4} = 4096 vertices | ||

γ ^{2}_{5} = {4,3,3,3}= 32 vertices |
γ ^{3}_{5} = 243 vertices |
γ ^{4}_{5} = 1024 vertices |
γ ^{5}_{5} = 3125 vertices |
γ ^{6}_{5} = 7776 vertices |
γ^{7}_{5} = 16,807 vertices |
γ^{8}_{5} = 32,768 vertices | ||

γ ^{2}_{6} = {4,3,3,3,3}= 64 vertices |
γ ^{3}_{6} = 729 vertices |
γ ^{4}_{6} = 4096 vertices |
γ ^{5}_{6} = 15,625 vertices |
γ^{6}_{6} = 46,656 vertices |
γ^{7}_{6} = 117,649 vertices |
γ^{8}_{6} = 262,144 vertices | ||

γ ^{2}_{7} = {4,3,3,3,3,3}= 128 vertices |
γ ^{3}_{7} = 2187 vertices |
γ^{4}_{7} = 16,384 vertices |
γ^{5}_{7} = 78,125 vertices |
γ^{6}_{7} = 279,936 vertices |
γ^{7}_{7} = 823,543 vertices |
γ^{8}_{7} = 2,097,152 vertices | ||

γ ^{2}_{8} = {4,3,3,3,3,3,3}= 256 vertices |
γ ^{3}_{8} = 6561 vertices |
γ^{4}_{8} = 65,536 vertices |
γ^{5}_{8} = 390,625 vertices |
γ^{6}_{8} = 1,679,616 vertices |
γ^{7}_{8} = 5,764,801 vertices |
γ^{8}_{8} = 16,777,216 vertices |

## See also

- Hypercube interconnection network of computer architecture
- Hyperoctahedral group, the symmetry group of the hypercube
- Hypersphere
- Simplex
*Crucifixion (Corpus Hypercubus)*(famous artwork)

## Notes

**^**Elte, E. L. (1912). "IV, Five dimensional semiregular polytope".*The Semiregular Polytopes of the Hyperspaces*. Netherlands: University of Groningen. ISBN 141817968X.**^**Coxeter 1973, pp. 122-123, §7.2 see illustration Fig 7.2C.**^**Coxeter 1973, p. 122, §7·25.**^**Johnson, Norman W.;*Geometries and Transformations*, Cambridge University Press, 2018, p.224.**^**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

Wikimedia Commons has media related to Hypercubes. |

- Weisstein, Eric W. "Hypercube".
*MathWorld*. - Weisstein, Eric W. "Hypercube graphs".
*MathWorld*. - www.4d-screen.de (Rotation of 4D – 7D-Cube)
*Rotating a Hypercube*by Enrique Zeleny, Wolfram Demonstrations Project.- Stereoscopic Animated Hypercube
- Rudy Rucker and Farideh Dormishian's Hypercube Downloads
- A001787 Number of edges in an n-dimensional hypercube. at OEIS