Hypercube
| Cube | Tesseract |
In geometry, a hypercube is an n-dimensional analogue of a square and a cube. 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 is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.
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 are the 2n points in Rn 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: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
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
Every n-cube of n > 0 is composed of elements, or n-cubes of a lower dimension, on the -dimensional surface on the parent hypercube. A side is any element of -dimension of the parent hypercube. A hypercube of dimension n has 2n sides. The number of vertices of a hypercube is .The number of m-dimensional hypercubes on the boundary of an n-cube is
For example, the boundary of a 4-cube contains 8 cubes, 24 squares, 32 lines and 16 vertices.
This identity can be proved by combinatorial arguments; each of the vertices defines a vertex in
a m-dimensional boundary. There are ways of choosing which lines 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 by 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
For example, extending a square via its 4 vertices adds one extra line per vertex, and also adds the final second square, to form a cube, giving = 12 lines in total.
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.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 |
.
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 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 hγn.
n-cubes can be combined with their duals to form compound polytopes:
- In two dimensions, we obtain the octagrammic star figure,
- 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
This relation may be used to generate the face lattice of an -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 = p2...22, or... Real solutions exist with p=2, i.e. γ = γn = 22...22 =. For p>2, they exist in. The facets are generalized -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 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.
| p=2 | p=3 | p=4 | p=5 | p=6 | p=7 | p=8 | ||
γ = square| = 4 vertices | γ = 9 vertices | γ = 16 vertices | γ = 25 vertices | γ = 36 vertices | γ = 49 vertices | γ = 64 vertices | ||
γ = cube| = 8 vertices | γ = 27 vertices | γ = 64 vertices | γ = 125 vertices | γ = 216 vertices | γ = 343 vertices | γ = 512 vertices | ||
γ = tesseract| = 16 vertices | γ = 81 vertices | γ = 256 vertices | γ = 625 vertices | γ = 1296 vertices | γ = 2401 vertices | γ = 4096 vertices | ||
γ = 5-cube| = 32 vertices | γ = 243 vertices | γ = 1024 vertices | γ = 3125 vertices | γ = 7776 vertices | γ = 16,807 vertices | γ = 32,768 vertices | ||
γ = 6-cube| = 64 vertices | γ = 729 vertices | γ = 4096 vertices | γ = 15,625 vertices | γ = 46,656 vertices | γ = 117,649 vertices | γ = 262,144 vertices | ||
γ = 7-cube| = 128 vertices | γ = 2187 vertices | γ = 16,384 vertices | γ = 78,125 vertices | γ = 279,936 vertices | γ = 823,543 vertices | γ = 2,097,152 vertices | ||
γ = 8-cube| = 256 vertices | γ = 6561 vertices | γ = 65,536 vertices | γ = 390,625 vertices | γ = 1,679,616 vertices | γ = 5,764,801 vertices | γ = 16,777,216 vertices |