- Unit hypercube
- See also
- References
- External links
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
- Doubling the cube
- 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
References
1. ^1 2 {{citation|contribution=High-dimensional geometry and its probabilistic analogues|first=Keith|last=Ball|title=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. 671].
2. ^1 2 {{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}}.
3. ^{{citation|title=Geometry: Reteaching Masters|publisher=Holt Rinehart & Winston|isbn=9780030543289|year=2001|page=74}}.
2. ^1 2 {{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}}.
3. ^{{citation|title=Geometry: Reteaching Masters|publisher=Holt Rinehart & Winston|isbn=9780030543289|year=2001|page=74}}.
External links
- {{mathworld | urlname = UnitCube | title = Unit cube}}
3 : Euclidean solid geometry|1 (number)|Cubes
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