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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 In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
() 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 r ...
(). 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 British ...
,
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
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 of IBM ...
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s aligned in each of the space's
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 can ...
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 all of ...
(also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
. 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 can ...
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 In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
). This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a
Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...
: the ''d''-dimensional hypercube is the Minkowski sum of ''d'' mutually perpendicular unit-length line segments, and is therefore an example of a
zonotope In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
. 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 In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, ...
.


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 t ...
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\ti ...
,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 (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
. 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 In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
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 In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
: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 polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
s that are represented in any number of dimensions. The hypercube (offset) family is one of three
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
families, labeled by
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 to ...
as ''γn''. The other two are the hypercube dual family, the
cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
s, labeled as ''βn,'' and the
simplices 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. ...
, labeled as ''αn''. A fourth family, the infinite tessellations of hypercubes, he labeled as ''δn''. Another related family of semiregular and
uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
s is the
demihypercube In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-polytopes constructed from alternation of an ''n''-hypercube, labeled as ''hγn'' for being ''half'' of the hype ...
s, 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-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
s) to form compound polytopes: * In two dimensions, we obtain the
octagram In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line". Detai ...
mic star figure , * In three dimensions we obtain the
compound of cube and octahedron The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction The 14 Cartesian coordinates of the vertices of the compound are. : 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2) : ...
, * In four dimensions we obtain the
compound of tesseract and 16-cell In 4-dimensional geometry, the tesseract 16-cell compound is a polytope compound composed of a regular tesseract and its dual, the regular 16-cell. Its convex hull is the regular 24-cell, which is self-dual. A '' compound polytope'' is a figure t ...
.


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 is ...
to the
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ea ...
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 A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
. 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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
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 In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
perimeter seen in these orthogonal projections is called a
petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
. 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 The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...
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 (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
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 In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a paramete ...
, the symmetry group of the hypercube *
Hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...
*
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) ''Crucifixion (Corpus Hypercubus)'' is a 1954 oil-on-canvas painting by Salvador Dalí. A nontraditional, surrealist portrayal of the Crucifixion, it depicts Christ on a polyhedron net of a tesseract (hypercube). It is one of his best-known p ...
'' (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 representati ...
) 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 ''logical ...
or
Karnaugh map 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 ...
.


External links

* *
www.4d-screen.de
(Rotation of 4D – 7D-Cube) *
Rotating a Hypercube
' by Enrique Zeleny,
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
Stereoscopic Animated Hypercube



A001787    Number of edges in an n-dimensional hypercube.
at
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
{{Polytopes Multi-dimensional geometry Cubes