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 ${\displaystyle {\sqrt {n}}}$, 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. 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.
2. 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.

External links

• Weisstein, Eric W. "Unit cube". MathWorld.