Hypercubes

Hypercube, also called an n-cube, is the generalization of a cube to n dimensions. It is a regular convex polytope with 2^n vertices, formed as the Cartesian product of n segments. In standard coordinates it can be taken as the set of all points with coordinates xi in {0,1} (yielding the cube [0,1]^n) or, in a symmetric form, as the 2^n points (±1, ..., ±1), yielding the cube [-1,1]^n. Its edges connect pairs of vertices that differ in exactly one coordinate, and all edges have the same length.

The n-cube has a well-defined hierarchy of faces: the number of k-dimensional faces is C(n, k) 2^{n-k}. In particular, it has 2^n vertices, n 2^{n-1} edges, and C(n,2) 2^{n-2} square faces. It is a regular polytope with Schläfli symbol {4, 3, ..., 3}, where there are n−1 threes. The dual polytope is the cross-polytope (orthoplex).

The skeleton of the hypercube forms the n-dimensional hypercube graph Q_n, whose vertices correspond to the 2^n corners and whose edges join vertices that differ in one coordinate. The full symmetry group of an n-cube is the hyperoctahedral group of order 2^n n!.

Special cases include the 1-cube (a line segment), the 2-cube (a square), the 3-cube (a regular cube), and the 4-cube (a tesseract).

Hypercubes appear in various contexts, including high-dimensional geometry, computing models, coding theory, and network design, where their regular structure and symmetries provide useful properties.