# Unit cube

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{{short description|Cube with edge length one}}
thumb|300px|right|Unit cube
A '''unit cube''', more formally a '''cube of side 1''', is a [cube](/source/cube) whose sides are 1 unit long.<ref name="pcm">{{citation|contribution=High-dimensional geometry and its probabilistic analogues|first=Keith|last=Ball|title=[The Princeton Companion to Mathematics](/source/The_Princeton_Companion_to_Mathematics)|editor-first=Timothy|editor-last=Gowers|editor-link=Timothy Gowers|publisher=Princeton University Press|year=2010|isbn=9781400830398|pages=670–680}}. See in particular [https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA671 p.&nbsp;671].</ref><ref name="cbm">{{citation|contribution=Chapter 13: Hypercubes|first=Martin|last=Gardner|title=The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics|publisher=W. W. Norton & Company|year=2001|isbn=9780393020236|pages=162–174|url=https://books.google.com/books?id=orz0SDEakpYC&pg=PA162}}.</ref> The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.<ref>{{citation|title=Geometry: Reteaching Masters|publisher=Holt Rinehart & Winston|isbn=9780030543289|year=2001|page=74}}.</ref>

== Unit hypercube ==
The term ''unit cube'' or '''unit hypercube''' is also used for [hypercube](/source/hypercube)s, or "cubes" in [''n''-dimensional spaces](/source/N-dimensional_space), for values of ''n'' other than 3 and edge length 1.<ref name="pcm"/><ref name="cbm"/>

Sometimes the term "unit cube" refers in specific to the set [0,&nbsp;1]<sup>''n''</sup> of all ''n''-tuples of numbers in the interval [0,&nbsp;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 (1,1,1,....1,1) in ''n''-dimensional space.<ref name="cbm"/>

== See also ==
*[Doubling the cube](/source/Doubling_the_cube)
*[''k''-cell](/source/k-cell_(mathematics))
*[Robbins constant](/source/Robbins_constant), the average distance between two random points in a unit cube
*[Tychonoff cube](/source/Tychonoff_cube), an infinite-dimensional analogue of the unit cube
*[Unit square](/source/Unit_square)
*[Unit sphere](/source/Unit_sphere)

==References==
{{reflist}}

== External links ==
* {{mathworld | urlname = UnitCube | title = Unit cube}}

Category:Euclidean solid geometry
Category:1 (number)
Category:Cubes

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