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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a hypercube is an ''n''-dimensional analogue of a square () and a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
(). It is a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
, convex figure whose 1-
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
consists of groups of opposite
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
line segments aligned in each of the space's
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
s,
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
who also labels the hypercubes the γn polytopes. The hypercube is the special case of a
hyperrectangle In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If al ...
(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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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 A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of a hypercube is a hypercube graph.


Vertex coordinates

A unit hypercube of dimension n is the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of all the points whose n
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
are each equal to either 0 or 1. This hypercube is also the
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
,1n of n copies of the unit interval ,1/math>. Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. It is the convex hull of the points whose vectors of Cartesian coordinates are : \left(\pm \frac, \pm \frac, \cdots, \pm \frac\right)\!\!. Here the symbol \pm means that each coordinate is either equal to 1/2 or to -1/2. This unit hypercube is also the cartesian product 1/2,1/2n. Any unit hypercube has an edge length of 1 and an n-dimensional volume of 1. The n-dimensional hypercube obtained as the convex hull of the points with coordinates (\pm 1, \pm 1, \cdots, \pm 1) or, equivalently as the Cartesian product 1,1n is also often considered due to the simpler form of its vertex coordinates. Its edge length is 2, and its n-dimensional volume is 2^n.


Faces

Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension n admits 2n facets, or faces of dimension n-1: a (1-dimensional) line segment has 2 endpoints; a (2-dimensional) square has 4 sides or edges; a 3-dimensional cube has 6 square faces; a (4-dimensional) tesseract has 8 three-dimensional cube as its facets. The number of vertices of a hypercube of dimension n is 2^n (a usual, 3-dimensional cube has 2^3=8 vertices, for instance). The number of the m-dimensional hypercubes (just referred to as m-cubes from here on) contained in the boundary of an n-cube is : E_ = 2^ ,     where =\frac 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 line segments (1-cubes) and 16 vertices (0-cubes). This identity can be proven by a simple combinatorial argument: for each of the 2^n vertices of the hypercube, there are \tbinom n m ways to choose a collection of m edges incident to that vertex. Each of these collections defines one of the m-dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the m-dimensional faces of the hypercube is counted 2^m times since it has that many vertices, and we need to divide 2^n\tbinom n m by this number. The number of facets of the hypercube can be used to compute the (n-1)-dimensional volume of its boundary: that volume is 2n times the volume of a (n-1)-dimensional hypercube; that is, 2ns^ where s is the length of the edges of the hypercube. These numbers can also be generated by the linear recurrence relation :E_ = 2E_ + E_ \!,     with E_= 1, and E_=0 when n < m, n < 0, or 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 E_=12 line segments.


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 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 ''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 In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
facets added in the gaps, labeled as ''hγn''. ''n''-cubes can be combined with their duals (the cross-polytopes) 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

The graph of the ''n''-hypercube's edges is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the Hasse diagram of the (''n''−1)-
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
'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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
called ''generalized hypercubes'', γ = ''p''2...22, or ... Real solutions exist with ''p'' = 2, i.e. γ = γ''n'' = 22...22 = . For ''p'' > 2, they exist in \mathbb^n. The facets are generalized (''n''−1)-cube and the
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
are regular
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es. 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: p^. This is ''p''''n'' vertices and ''pn'' facets..


Relation to exponentiation

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 In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
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.


See also

* Hypercube interconnection network of computer architecture * Hyperoctahedral group, the symmetry group of the hypercube * Hypersphere *
Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
* Parallelotope * '' Crucifixion (Corpus Hypercubus)'' (famous artwork)


Notes


References

* * p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5) * 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 The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representa ...
) 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 The Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 ''logica ...
or Karnaugh map.


External links

* *
www.4d-screen.de
(Rotation of 4D – 7D-Cube) *
Rotating a Hypercube
' by Enrique Zeleny, Wolfram Demonstrations Project.
Stereoscopic Animated Hypercube



A001787    Number of edges in an n-dimensional hypercube.
at OEIS {{Polytopes Multi-dimensional geometry Cubes