hypercube

A hypercube, also called an n-cube, is the n-dimensional analogue of a cube. It can be defined as the Cartesian product [0,1]^n in R^n or as the convex hull of the 2^n points with coordinates 0 or 1. Its center lies at (1/2, ..., 1/2). The unit hypercube has volume 1.

The hypercube has 2^n vertices, each labeled by a binary n-tuple. Edges connect pairs of vertices that differ in exactly one coordinate, so every vertex has degree n. The boundary consists of 2n facets, each an (n-1)-cube. The numbers of k-dimensional faces are given by f_k = binomial(n, k) 2^{n-k}.

Symmetry and duality are notable features. The symmetry group of the n-cube is the hyperoctahedral group B_n, of order 2^n n!. The dual polytope of the n-cube is the n-orthoplex (cross-polytope). The n-cube is a zonotope, expressible as the Minkowski sum of n line segments along coordinate axes.

In low dimensions, standard names arise: the 1-cube is a line segment, the 2-cube a square, the 3-cube a regular cube, and the 4-cube a tesseract. The Schläfli symbol for an n-cube is {4,3,...,3}, with n−1 threes. Projections into lower dimensions can yield recognizable patterns, such as the cube-within-a-cube appearance associated with the tesseract.

Applications span mathematics and computer science. Hypercubes model high-dimensional geometry, serve as network topologies in parallel computing, and provide a convenient structure for multi-dimensional data analysis and coding theory.