HYPERCUBE

:''This article refers to the mathematical concept. For the movie of the same name, see .''



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, at right angles to each other.
An ''n''-dimensional hypercube is also called an 'n-cube'. The term "measure polytope" (which is apparently due to Coxeter; see Coxeter 1973) is also used but it is rare.
The hypercube is the special case of a hyperrectangle.
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 ''Rⁿ'' with coordinates equal to 0 or 1 is called '"the" unit hypercube'.

A point is a hypercube of dimension zero. If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one. If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a two-dimensional square. If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a three-dimensional cube. This can be generalized to any number of dimensions. For example, if one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).
The 1-skeleton of a hypercube is a hypercube graph.

Contents

Related families of polytopes

Elements

''n''-cube rotation

References

External links

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' family is the first of three regular polytope families, labeled by Coxeter as ''γ_n'', the other two being the hypercube dual family, the 'cross-polytopes', labeled as ''β_n'', and the 'simplices', labeled as ''α_n''. A fourth family, the infinite tessellation 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 2''n'' "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 2^''n'' (a cube has 2³ 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
:$2^\{n-m\}\{n\; choose\; m\}.$
For example, the boundary of a 4-cube contains 8 cubes, 24 squares, 32 lines and 16 vertices.
{| class="prettytable"
|+
Hypercube elements
|-
! n
! γ_n
! n-cube
! Graph
! Names
Schläfli symbol
Coxeter-Dynkin
! Vertices
! Edges
! Faces
! Cells
! ''4''-faces
! ''5''-faces
! ''6''-faces
! ''7''-faces
! ''8''-faces
|-
! 0-polytope
! γ₀
| 0-cube
|
| Point
-
| 1
|  
|  
|  
|  
|  
|  
|  
|  
|-
! 1
! γ₁
| 1-cube
|
| Line segment
{}

| 2
| 1
|  
|  
|  
|  
|  
|  
|  
|-
! 2
! γ₂
| 2-cube
|
| Square
'Tetragon'
{4}

| 4
| 4
| 1
|  
|  
|  
|  
|  
|  
|-
! 3
! γ₃
| 3-cube
|
| Cube
'Hexahedron'
{4,3}

| 8
| 12
| 6
| 1
|  
|  
|  
|  
|  
|-
! 4
! γ₄
| 4-cube
|
| Tesseract
'Octachoron'
{4,3,3}

| 16
| 32
| 24
| 8
| 1
|  
|  
|  
|  
|-
! 5
! γ₅
| 5-cube
|
| Penteract
'Decateron'
{4,3,3,3}

| 32
| 80
| 80
| 40
| 10
| 1
|  
|  
|  
|-
! 6
! γ₆
| 6-cube
|
| Hexeract
'Dodecapeton'
{4,3,3,3,3}

| 64
| 192
| 240
| 160
| 60
| 12
| 1
|  
|  
|-
! 7
! γ₇
| 7-cube
|
| Hepteract
'Tetradeca-7-tope'
{4,3,3,3,3,3}

| 128
| 448
| 672
| 560
| 280
| 84
| 14
| 1
|  
|-
! 8
! γ₈
| 8-cube
|
| Octeract
'Hexadeca-8-tope'
{4,3,3,3,3,3,3}

| 256
| 1024
| 1792
| 1792
| 1120
| 448
| 112
| 16
| 1
|-
! 9
! γ₉
| 9-cube
|
| Enneract
'Octadeca-9-tope'
{4,3,3,3,3,3,3,3}

| 512
| 2304
| 4608
| 5376
| 4032
| 2016
| 672
| 144
| 18
|}

''n''-cube rotation

Based on observations of how 1-, 2-, and 3-dimensional hypercubes can be rotated, it is possible to hypothesize how objects with ''n'' dimensions can be rotated. A 3-dimensional hypercube can be rotated about 3 axes in 2 different ways: rotation by edge or rotation by vertex. Rotation by edge involves changing the position of every vertex but the two vertices on that particular edge. Rotation by vertex involves changing the positions of all vertices but the point of rotation. A 3-dimensional hypercube can be rotated by edge and by vertex, a 2-dimensional hypercube can only be rotated by vertex, and so on. If these series of observations are extended to higher dimensions, a 4-dimensional hypercube can be rotated about a whole face, and a 5-cube can be rotated about a whole cube.

References

★ Bowen, J. P., Hypercubes, ''Practical Computing'', 5(4):97–99, April 1982.

★ Coxeter, H. S. M., ''Regular Polytopes''. 3rd edition, Dover, 1973, p. 123. ISBN 0-486-61480-8. p.296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5)

External links

★

★

★ Hypercube images (2D–15D)

★ Animation of a Hypercube