In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to .
An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube.[1][2] The term measure polytope (originally from Elte, 1912)[3] is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.[4]
The hypercube is the special case of a hyperrectangle (also called an n-orthotope).
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2n points in Rn with each coordinate equal to 0 or 1 is called the unit hypercube.
- ^ Paul Dooren; Luc Ridder. "An adaptive algorithm for numerical integration over an n-dimensional cube".
- ^ Xiaofan Yang; Yuan Tang. "A (4n − 9)/3 diagnosis algorithm on n-dimensional cube network".
- ^ Elte, E. L. (1912). "IV, Five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Netherlands: University of Groningen. ISBN 141817968X.
- ^ Coxeter 1973, pp. 122–123, §7.2 see illustration Fig 7.2C.