A unit cube, more formally a cube of side 1, is a
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
whose sides are 1 unit long.
[. See in particula]
p. 671
[.] The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.
[.]
Unit hypercube
The term ''unit cube'' or unit hypercube is also used for
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
s, or "cubes" in
''n''-dimensional spaces, for values of ''n'' other than 3 and edge length 1.
Sometimes the term "unit cube" refers in specific to the set
, 1sup>''n'' of all ''n''-tuples of numbers in the interval
, 1
The length of the longest diagonal of a unit hypercube of ''n'' dimensions is
, the square root of ''n'' and the (Euclidean) length of the vector (1,1,1,....1,1) in ''n''-dimensional space.
See also
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Doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...
*
''k''-cell
*
Robbins constant In geometry, the mean line segment length is the average length of a line segment connecting two points chosen uniformly at random in a given shape. In other words, it is the expected Euclidean distance between two random points, where each point ...
, the average distance between two random points in a unit cube
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Tychonoff cube In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tyc ...
, an infinite-dimensional analogue of the unit cube
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Unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinat ...
*
Unit sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
References
External links
* {{mathworld , urlname = UnitCube , title = Unit cube
Euclidean solid geometry
1 (number)
Cubes