# Unit cube

Unit cube

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 $\sqrt n$, the square root of n and the (Euclidean) length of the vector (1,1,1,....1,1) in n-dimensional space.[2]

## References

1. ^ a b c Ball, Keith (2010), "High-dimensional geometry and its probabilistic analogues", in Gowers, Timothy, The Princeton Companion to Mathematics, Princeton University Press, pp. 670–680, ISBN 9781400830398. See in particular p. 671.
2. ^ a b c Gardner, Martin (2001), "Chapter 13: Hypercubes", 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, W. W. Norton & Company, pp. 162–174, ISBN 9780393020236.
3. ^ Geometry: Reteaching Masters, Holt Rinehart & Winston, 2001, p. 74, ISBN 9780030543289.