# 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 , 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

- Ball, Keith (2010), "High-dimensional geometry and its probabilistic analogues", in Gowers, Timothy (ed.),
*The Princeton Companion to Mathematics*, Princeton University Press, pp. 670–680, ISBN 9781400830398. See in particular p. 671. - 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. -
*Geometry: Reteaching Masters*, Holt Rinehart & Winston, 2001, p. 74, ISBN 9780030543289.

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