# Difference between revisions of "Unit cube"

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en>Tamfang (add a necessary condition; remove something not distinctive of *unit* cubes; split a paragraph of two unrelated sentences) |
en>Cydebot m (Robot - Moving category One to Category:1 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.) |
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Sometimes the term "unit cube" refers in specific to the set [0, 1]<sup>''n''</sup> of all ''n''-tuples of numbers in the interval [0, 1].<ref name="pcm"/> | Sometimes the term "unit cube" refers in specific to the set [0, 1]<sup>''n''</sup> of all ''n''-tuples of numbers in the interval [0, 1].<ref name="pcm"/> | ||

The length of the longest diagonal of a unit hypercube of ''n'' dimensions is <math>\sqrt n</math>, the square root of ''n'' and the (Euclidean) length of the vector | The length of the longest diagonal of a unit hypercube of ''n'' dimensions is <math>\sqrt n</math>, the square root of ''n'' and the (Euclidean) length of the vector (1,1,1,....1,1) in ''n''-dimensional space.<ref name="cbm"/> | ||

== See also == | == See also == | ||

*[[K-cell (mathematics)|K-cell]] | |||

*[[Robbins constant]], the average distance between two random points in a unit cube | *[[Robbins constant]], the average distance between two random points in a unit cube | ||

*[[Tychonoff cube]], an infinite-dimensional analogue of the unit cube | *[[Tychonoff cube]], an infinite-dimensional analogue of the unit cube | ||

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[[Category:Euclidean solid geometry]] | [[Category:Euclidean solid geometry]] | ||

[[Category: | [[Category:1 (number)]] | ||

[[Category:Cubes]] | [[Category:Cubes]] | ||

{{Elementary-geometry-stub}} | {{Elementary-geometry-stub}} |

## Latest revision as of 22:01, 28 November 2014

A **unit cube**, more formally a **cube of side 1**, is a cube whose sides are 1 unit long.^{[1]}^{[2]} The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.^{[3]}

## Unit hypercube

The term *unit cube* or **unit hypercube** is also used for hypercubes, or "cubes" in *n*-dimensional spaces, for values of *n* other than 3 and edge length 1.^{[1]}^{[2]}

Sometimes the term "unit cube" refers in specific to the set [0, 1]^{n} of all *n*-tuples of numbers in the interval [0, 1].^{[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.^{[2]}

## See also

- K-cell
- Robbins constant, the average distance between two random points in a unit cube
- Tychonoff cube, an infinite-dimensional analogue of the unit cube
- Unit square
- Unit sphere