Hypercube

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Hypercube

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a hypercube is an n-dimensional analogue of a square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
(n = 2) and a cube

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
(n = 3). It is a closed

Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
, compact

Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
, convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
figure whose 1-skeleton consists of groups of opposite parallel

Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
s aligned in each of the space's dimension

Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
s, at right angle

Right angle

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Encyclopedia

In geometry

Geometry

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
(n = 2) and a cube

Cube

Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
, compact

Compact space

Convex set

Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
s aligned in each of the space's dimension

Dimension

Right angle

In geometry and trigonometry, a right angle is an angle of 90 degree s, corresponding to a quarter turn . It can be defined; as the angle such that twice that angle amounts to a half turn, or 180?....
s 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

Hyperrectangle

In geometry, an orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of interval s....
(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.

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

Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
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

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
). 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

Hypercube graph

In the mathematics field of graph theory, the hypercube graph Qn is a regular graph with 2n vertex , which correspond to the subsets of a Set with n elements....
.

Coordinates

A unit hypercube of n dimensions is the convex hull

Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
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.

Related families of polytopes

The hypercubes are one of the few families of regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
families, labeled by Coxeter as ?_n. The other two are the hypercube dual family, the cross-polytope
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
s, labeled as ß_n, and the simplices
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
, labeled as a_n. A fourth family, the infinite tessellation of hypercubes

Hypercubic honeycomb

In geometry, a hypercubic honeycomb is a family of List_of_regular_polytopes#Tessellations in n-dimensions with the Schl?fli symbols and containing the symmetry of Coxeter_diagram#Infinite_Coxeter_groups Rn for n>=3....
, he labeled as d_n.

Another related family of semiregular and uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
s is the demihypercubes which are constructed from hypercubes with alternate vertices deleted and simplex

Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
facets added in the gaps, labeled as h?_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 cell

Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
s). 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:

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

, where and n! denotes the factorial

Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to n....
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 vertices defines a vertex in a -dimensional boundrary. There are ways of choosing which lines ("sides") that defines the subspace that the boundrary is in. But, each side is counted 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

Recurrence relation

In mathematics, a recurrence relation is an equation that defines a sequence recursion: each term of the sequence is defined as a Function of the preceding terms....

, with , 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 = 12 lines in total.

Hypercube elements
				m	0	1	2	3	4	5	6	7	8	9
n Polytope In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....	?_n	n-cube	Petrie polygon Petrie polygon In geometry, a Petrie polygon is a skew polygon such that every two consecutive Edge belong to a Face of a regular polyhedron.This definition extends to higher regular polytopes.... projection	Names Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Vertices Vertex (geometry) In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....	Edges Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....	Faces Face (geometry) In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....	Cells Cell (geometry) In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....	Hypercells Hypercell In geometry, a hypercell is a descriptive term for an element of a polytope or tessellation, usually representing an element one dimension higher than a Cell ....	5-faces	6-faces	7-faces	8-faces	9-faces
0	?₀	0-cube		Point -	1
1	?₁	1-cube		Line segment Line segment In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....	2	1
2	?₂	2-cube		Square Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted .... Tetragon	4	4	1
3	?₃	3-cube		Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids.... Hexahedron	8	12	6	1
4	?₄	4-cube		Tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square .... Octachoron	16	32	24	8	1
5 5-polytope In geometry, a five-dimensional polytope, or 5-polytope, is a polytope in 5-dimensional space. Each polyhedron cell being shared by exactly two polychoron facets....	?₅	5-cube		Penteract Penteract In Fifth dimension geometry, a penteract is a name for a Fifth dimension hypercube with 32 Vertex , 80 Edge s, 80 square Face , 40 cubic Cell , and 10 tesseract hypercells.... Decateron	32	80	80	40	10	1
6 6-polytope In geometry, a six-dimensional polytope, or 6-polytope, is a polytope in 6-dimensional space. Each polychoron Ridge being shared by exactly two 5-polytope Facet ....	?₆	6-cube		Hexeract Hexeract A hexeract is a name for a six-dimensional hypercube with 64 Vertex , 192 Edge s, 240 square Face , 160 cubic Cell , 60 tesseract hypercell, and 12 penteract 5-faces.... Dodecapeton	64	192	240	160	60	12	1
7 7-polytope In geometry, a seven-dimensional polytope, or 7-polytope, is a polytope in 7-dimensional space. Each polyteron Ridge being shared by exactly two 6-polytope Facet ....	?₇	7-cube		Hepteract Hepteract A hepteract is a seven-dimensional hypercube with 128 Vertex , 448 Edge s, 672 square Face , 560 cubic Cell , 280 tesseract hypercell, 84 penteract 5-faces, and 14 hexeract 6-faces.... Tetradeca-7-tope	128	448	672	560	280	84	14	1
8 8-polytope In geometry, an eight-dimensional polytope, or 8-polytope, is a polytope in 8-dimensional space. Each 6-polytope Ridge being shared by exactly two 7-polytope Facet ....	?₈	8-cube		Octeract Octeract An octeract is an eight-dimensional hypercube with 256 Vertex , 1024 Edge s, 1792 square Face , 1792 cubic Cell , 1120 tesseract hypercell, 448 penteract 5-faces, 112 hexeract 6-faces, and 16 hepteract 7-faces.... Hexadeca-8-tope	256	1024	1792	1792	1120	448	112	16	1
9 9-polytope In geometry, a nine-dimensional polytope, or 9-polytope, is a polytope in 9-dimensional space. Each 7-polytope Ridge being shared by exactly two 8-polytope Facet ....	?₉	9-cube		Enneract Enneract An enneract is a nine-dimensional hypercube with 512 Vertex , 2304 Edge s, 4608 square Face , 5376 cubic Cell , 4032 tesseract hypercell, 2016 penteract 5-faces, 672 hexeract 6-faces, 144 hepteract 7-faces, and 18 octeract 8-faces.... Octadeca-9-tope	512	2304	4608	5376	4032	2016	672	144	18	1
10 10-polytope In geometry, a ten-dimensional polytope, or 10-polytope, is a polytope in 10-dimensional space, each 8-polytope Ridge being shared by exactly two 9-polytope Facet ....	?₁₀	10-cube		10-cube 10-cube A 10-cube or dekeract is a ten-dimensional hypercube with 1024 Vertex , 5120 Edge s, 11520 square Face , 15360 cubic Cell , 13440 tesseract hypercell, 8064 penteract 5-faces, 3360 hexeract 6-faces, 960 hepteract 7-faces, 180 octeract 8-faces, and 20 enneract 9-faces.... icosa-10-tope	1024	5120	11520	15360	13440	8064	3360	960	180	20

n-cube 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 our natural 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

Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
to the Hasse diagram

Hasse diagram

In the mathematics discipline known as order theory, a Hasse diagram is a simple picture of a finite partially ordered set, forming a Graph drawing of the transitive reduction of the partial order....
of the (n-1)-simplex

Simplex

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn....
. 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.

External links

(Rotation of 4D – 7D-Cube)
by Enrique Zeleny, Wolfram Demonstrations Project
Wolfram Demonstrations Project

The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
.

Hypercube

Coordinates

Related families of polytopes

Elements

n-cube rotation

Relation to n-simplices

See also

External links