Unit cube explained

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]

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

Notes and References

1. . See in particular p. 671.
2. .
3. .