# Hypercube

An n-cube can be projected inside a regular 2n-gonal polygon by a skew orthogonal projection, shown here from the 2-cube to the 10-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, at right angles 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 The term "measure polytope" is also used, notably in the work of H.S.M. Coxeter, but it has now been superseded.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.

## Construction

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). 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

A unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates $(\pm 1/2, \pm 1/2, \cdots, \pm 1/2)$. 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 $(\pm 1, \pm 1, \cdots, \pm 1)$. 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 the first 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 tessellation 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.

## Elements

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: $2n^{2}-2n$

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

$E_{m,n} = 2^{n-m}{n \choose m}$,     where ${n \choose m}=\frac{n!}{m!\,(n-m)!}$ 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 $2^n$ vertices defines a vertex in a $m$-dimensional boundary. There are ${n \choose m}$ ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted $2^m$ 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

$E_{m,n} = 2E_{m,n-1} + E_{m-1,n-1} \!$,     with $E_{0,0} = 1 \!$,     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 $E_{1,3} \!$ = 12 lines in total.

Hypercube elements $E_{m,n} \!$
m 0 1 2 3 4 5 6 7 8 9 10
n γn n-cube Petrie
polygon

projection
Names
Schläfli symbol
Coxeter-Dynkin
Vertices Edges Faces Cells 4-faces 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
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
{4,3,3,3,3,3}
128 448 672 560 280 84 14 1
8 γ8 8-cube Octeract
{4,3,3,3,3,3,3}
256 1024 1792 1792 1120 448 112 16 1
9 γ9 9-cube Enneract
{4,3,3,3,3,3,3,3}
512 2304 4608 5376 4032 2016 672 144 18 1
10 γ10 10-cube Dekeract
icosa-10-tope
{4,3,3,3,3,3,3,3,3}
1024 5120 11520 15360 13440 8064 3360 960 180 20 1

## n-cube rotation

Hypercube rotation.

In general, rotation is a planar phenomenon requiring two dimensions to operate. Any additional dimensions in the space that the rotating object is embedded in manifests itself as a stationary set.

No rotation is possible in 1 dimension. An object in 1 dimension cannot rotate without leaving that 1-dimensional space.

In 2 dimensions, both dimensions are used for rotation, leaving a 0-dimensional stationary point. Hence, an object in 2 dimensions rotate about a point. The rotational axis commonly associated with 2-dimensional rotation actually lies outside of the 2-dimensional space itself, and thus is merely an artifact of anthropocentric bias toward 3-dimensional space. Hence, rotation in 2 dimensions is more properly understood as rotation about a center of rotation. Rotations in 2 dimensions are uniquely identified by the center of rotation and the rate of rotation.

In 3 dimensions, objects rotate about an axis, a stationary line, since there is one dimension "left over" as the other two participate in the rotation. The rotational axis is peculiar to odd-numbered dimensions. Rotations in 3 dimensions are uniquely identified by the axis of rotation and the rate of rotation.

Rotation in 4 dimensions are of two kinds: plane rotations and composite rotations. A plane rotation has a stationary plane which an object may rotate "around". This is because two dimensions participate in the rotation while the other two are stationary. Objects in 4 dimensions can also rotate independently in these two leftover dimensions, resulting in a composite rotation composed of two plane rotations at two independent rates of rotation. These composite rotations have a stationary point, just as in 2 dimensions. Hence, rotation in 4 dimensions are identified by a center of rotation, and two rates of rotation (planar rotation being a special case where one of the rates is zero).

In 5 dimensions, rotations have either a rotational axis or a rotational 3-space. With the former, there are two independent rates of rotation, just as in 4 dimensions. With the latter, there is 1 rate of rotation (2 dimensions participating in the rotation, and 3 dimensions forming the stationary 3-space).

In general, in n dimensions, if n is odd then rotations have an axis, and there are (n-1)/2 possible simultaneous plane rotations around that axis. If n is even, then rotations have stationary points (rotational centers), with n/2 possible simultaneous plane rotations. Each possible plane rotation has its own rate of rotation.

## 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.

## References

• Frederick J. Hill and Gerald R. Peterson, Introduction to Switching Theory and Logical Design: Second Edition, John Wiley & Sons, NY, 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.