Editing Hypercube (section)

== Construction ==
=== By the number of dimensions ===
[[File:From Point to Tesseract (Looped Version).gif|thumb|An animation showing how to create a tesseract from a point.]]

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 (topology)|skeleton]] of a hypercube is a [[hypercube graph]].

=== Vertex coordinates ===

[[File:8-cell.gif|thumb|Projection of a [[rotation|rotating]] [[tesseract]].]]

A unit hypercube of dimension <math>n</math> is the [[convex hull]] of all the <math>2^n</math> points whose <math>n</math> [[Cartesian coordinate system|Cartesian coordinates]] are each equal to either <math>0</math> or <math>1</math>. These points are its [[vertex (geometry)|vertices]]. The hypercube with these coordinates is also the [[cartesian product]] <math>[0,1]^n</math> of <math>n</math> copies of the unit [[interval (mathematics)|interval]] <math>[0,1]</math>. Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a [[translation (geometry)|translation]]. It is the convex hull of the <math>2^n</math> points whose vectors of Cartesian coordinates are

: <math>
\left(\pm \frac{1}{2}, \pm \frac{1}{2}, \cdots, \pm \frac{1}{2}\right)\!\!.
</math>

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

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