Unit cube unit hypercube of "n" dimensions is formula_1, the square root of "n" and the (Euclidean) length of the vector (1,1,1...1,1) in "n"-dimensional space. Unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units. 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. Sometimes the term "unit cube" refers in specific
Principal root of unity Principal root of unity In mathematics, a principal "n"-th root of unity (where "n" is a positive integer) of a ring is an element formula_1 satisfying the equations In an integral domain, every primitive "n"-th root of unity is also a principal formula_3-th root of unity. In any ring, if formula_3 is a power of formula_5, then any formula_6-th root of formula_7 is a principal formula_3-th root of unity. A non-example is formula_9 in the ring of integers modulo formula_10; while formula_11 and thus formula_9 is a cube root of unity, formula_13 meaning that it is not a "principal" cube