In
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 ...
, 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 a ...
() and a
cube (). It is a
closed,
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 polytop ...
figure whose 1-
skeleton 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 ...
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 coor ...
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 ca ...
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 (originally from Elte, 1912) is also used, notably in the work of
H. S. M. Coxeter who also labels the hypercubes the γ
n 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 ...
(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'' (a ...
. 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
![From Point to Tesseract (Looped Version)](https://upload.wikimedia.org/wikipedia/commons/d/d9/From_Point_to_Tesseract_%28Looped_Version%29.gif)
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 ca ...
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 e ...
).
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 i ...
.
The 1-
skeleton 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, e ...
.
Vertex coordinates
![8-cell](https://upload.wikimedia.org/wikipedia/commons/d/d7/8-cell.gif)
A unit hypercube of dimension
is the
convex hull of all the points whose
Cartesian coordinates are each equal to either
or
. This hypercube is also the
cartesian product of
copies of the unit
interval