Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract . 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 ${\displaystyle {\sqrt {n}}}${\displaystyle {\sqrt {n}}}.

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube.^{[1] [2]} The term measure polytope (originally from Elte, 1912)^[3] is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.^[4]

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.

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.

A unit hypercube of dimension ${\displaystyle n}${\displaystyle n} is the convex hull of all the ${\displaystyle 2^{n}}${\displaystyle 2^{n}} points whose ${\displaystyle n}${\displaystyle n} Cartesian coordinates are each equal to either ${\displaystyle 0}${\displaystyle 0} or ${\displaystyle 1}${\displaystyle 1}. These points are its vertices. The hypercube with these coordinates is also the cartesian product ${\displaystyle [0,1]^{n}}${\displaystyle [0,1]^{n}} of ${\displaystyle n}${\displaystyle n} copies of the unit interval ${\displaystyle [0,1]}${\displaystyle [0,1]}. 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 ${\displaystyle 2^{n}}${\displaystyle 2^{n}} points whose vectors of Cartesian coordinates are

${\displaystyle \left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},\cdots ,\pm {\frac {1}{2}}\right)\!\!.}${\displaystyle \left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},\cdots ,\pm {\frac {1}{2}}\right)\!\!.}

Here the symbol ${\displaystyle \pm }${\displaystyle \pm } means that each coordinate is either equal to ${\displaystyle 1/2}${\displaystyle 1/2} or to ${\displaystyle -1/2}${\displaystyle -1/2}. This unit hypercube is also the cartesian product ${\displaystyle [-1/2,1/2]^{n}}${\displaystyle [-1/2,1/2]^{n}}. Any unit hypercube has an edge length of ${\displaystyle 1}${\displaystyle 1} and an ${\displaystyle n}${\displaystyle n}-dimensional volume of ${\displaystyle 1}${\displaystyle 1}.

The ${\displaystyle n}${\displaystyle n}-dimensional hypercube obtained as the convex hull of the points with coordinates ${\displaystyle (\pm 1,\pm 1,\cdots ,\pm 1)}${\displaystyle (\pm 1,\pm 1,\cdots ,\pm 1)} or, equivalently as the Cartesian product ${\displaystyle [-1,1]^{n}}${\displaystyle [-1,1]^{n}} is also often considered due to the simpler form of its vertex coordinates. Its edge length is ${\displaystyle 2}${\displaystyle 2}, and its ${\displaystyle n}${\displaystyle n}-dimensional volume is ${\displaystyle 2^{n}}${\displaystyle 2^{n}}.

Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension ${\displaystyle n}${\displaystyle n} admits ${\displaystyle 2n}${\displaystyle 2n} facets, or faces of dimension ${\displaystyle n-1}${\displaystyle n-1}: a (${\displaystyle 1}${\displaystyle 1}-dimensional) line segment has ${\displaystyle 2}${\displaystyle 2} endpoints; a (${\displaystyle 2}${\displaystyle 2}-dimensional) square has ${\displaystyle 4}${\displaystyle 4} sides or edges; a ${\displaystyle 3}${\displaystyle 3}-dimensional cube has ${\displaystyle 6}${\displaystyle 6} square faces; a (${\displaystyle 4}${\displaystyle 4}-dimensional) tesseract has ${\displaystyle 8}${\displaystyle 8} three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension ${\displaystyle n}${\displaystyle n} is ${\displaystyle 2^{n}}${\displaystyle 2^{n}} (a usual, ${\displaystyle 3}${\displaystyle 3}-dimensional cube has ${\displaystyle 2^{3}=8}${\displaystyle 2^{3}=8} vertices, for instance).^[5]

The number of the ${\displaystyle m}${\displaystyle m}-dimensional hypercubes (just referred to as ${\displaystyle m}${\displaystyle m}-cubes from here on) contained in the boundary of an ${\displaystyle n}${\displaystyle n}-cube is

${\displaystyle E_{m,n}=2^{n-m}{n \choose m}}${\displaystyle E_{m,n}=2^{n-m}{n \choose m}},^[6]     where ${\displaystyle {n \choose m}={\frac {n!}{m!,円(n-m)!}}}${\displaystyle {n \choose m}={\frac {n!}{m!,円(n-m)!}}} and ${\displaystyle n!}${\displaystyle n!} denotes the factorial of ${\displaystyle n}${\displaystyle n}.

For example, the boundary of a ${\displaystyle 4}${\displaystyle 4}-cube (${\displaystyle n=4}${\displaystyle n=4}) contains ${\displaystyle 8}${\displaystyle 8} cubes (${\displaystyle 3}${\displaystyle 3}-cubes), ${\displaystyle 24}${\displaystyle 24} squares (${\displaystyle 2}${\displaystyle 2}-cubes), ${\displaystyle 32}${\displaystyle 32} line segments (${\displaystyle 1}${\displaystyle 1}-cubes) and ${\displaystyle 16}${\displaystyle 16} vertices (${\displaystyle 0}${\displaystyle 0}-cubes). This identity can be proven by a simple combinatorial argument: for each of the ${\displaystyle 2^{n}}${\displaystyle 2^{n}} vertices of the hypercube, there are ${\displaystyle {\tbinom {n}{m}}}${\displaystyle {\tbinom {n}{m}}} ways to choose a collection of ${\displaystyle m}${\displaystyle m} edges incident to that vertex. Each of these collections defines one of the ${\displaystyle m}${\displaystyle m}-dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the ${\displaystyle m}${\displaystyle m}-dimensional faces of the hypercube is counted ${\displaystyle 2^{m}}${\displaystyle 2^{m}} times since it has that many vertices, and we need to divide ${\displaystyle 2^{n}{\tbinom {n}{m}}}${\displaystyle 2^{n}{\tbinom {n}{m}}} by this number.

The number of facets of the hypercube can be used to compute the ${\displaystyle (n-1)}${\displaystyle (n-1)}-dimensional volume of its boundary: that volume is ${\displaystyle 2n}${\displaystyle 2n} times the volume of a ${\displaystyle (n-1)}${\displaystyle (n-1)}-dimensional hypercube; that is, ${\displaystyle 2ns^{n-1}}${\displaystyle 2ns^{n-1}} where ${\displaystyle s}${\displaystyle s} is the length of the edges of the hypercube.

These numbers can also be generated by the linear recurrence relation.

${\displaystyle E_{m,n}=2E_{m,n-1}+E_{m-1,n-1}\!}${\displaystyle E_{m,n}=2E_{m,n-1}+E_{m-1,n-1}\!},     with ${\displaystyle E_{0,0}=1}${\displaystyle E_{0,0}=1}, and ${\displaystyle E_{m,n}=0}${\displaystyle E_{m,n}=0} when ${\displaystyle n<m}${\displaystyle n<m}, ${\displaystyle n<0}${\displaystyle n<0}, or ${\displaystyle m<0}${\displaystyle m<0}.

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 ${\displaystyle E_{1,3}=12}${\displaystyle E_{1,3}=12} line segments.

The extended f-vector for an n-cube can also be computed by expanding ${\displaystyle (2x+1)^{n}}${\displaystyle (2x+1)^{n}} (concisely, (2,1)ⁿ), and reading off the coefficients of the resulting polynomial. For example, the elements of a tesseract is (2,1)⁴ = (4,4,1)² = (16,32,24,8,1).

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 16-cube.

The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions.^[8]

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_n, and the simplices , labeled as α_n. A fourth family, the infinite tessellations of hypercubes, is 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.

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.

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.

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. γ²
_n = γ_n = ₂{4}₂{3}...₂{3}₂ = {4,3,..,3}. For p > 2, they exist in ${\displaystyle \mathbb {C} ^{n}}${\displaystyle \mathbb {C} ^{n}}. 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: ${\displaystyle p^{n-m}{n \choose m}}${\displaystyle p^{n-m}{n \choose m}}. This is pⁿ vertices and pn facets.^[9]

Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type of figurate number corresponding to an n-cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield a square number or "perfect square", which can be arranged into a square shape with a side length corresponding to that of the base. Similarly, the exponent 3 will yield a perfect cube, an integer which can be arranged into a cube shape with a side length of the base. As a result, the act of raising a number to 2 or 3 is more commonly referred to as "squaring" and "cubing", respectively. However, the names of higher-order hypercubes do not appear to be in common use for higher powers.

• Hypercube interconnection network of computer architecture
• Hyperoctahedral group, the symmetry group of the hypercube
• Hypersphere
• Simplex
• Parallelotope
• Crucifixion (Corpus Hypercubus) , a painting by Salvador Dalí featuring an unfolded 4-cube

• Bowen, J. P. (April 1982). "Hypercube". Practical Computing . 5 (4): 97–99. Archived from the original on 2008年06月30日. Retrieved June 30, 2008.
• Coxeter, H. S. M. (1973). "§7.2. see illustration Fig. 7-2c". Regular Polytopes (3rd ed.). Dover. pp. 122-123. ISBN 0-486-61480-8. 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.

• Weisstein, Eric W. "Hypercube". MathWorld .
• Weisstein, Eric W. "Hypercube graphs". MathWorld .
• Rotating a Hypercube by Enrique Zeleny, Wolfram Demonstrations Project.
• Rudy Rucker and Farideh Dormishian's Hypercube Downloads
• A001787    Number of edges in an n-dimensional hypercube. at OEIS

• Regular polytopes
• Cubes

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