Hypercubic

Hypercubic refers to objects related to the hypercube, the generalization of the cube to higher dimensions. An n-dimensional hypercube, or n-cube, is a regular convex polytope with Schläfli symbol {4,3,...,3} (with n−1 copies of 3). It can be realized as the Cartesian product of n line segments, for example as the set of points in R^n with coordinates in [0,1] (or [-1,1]).

Key properties include that an n-cube has 2^n vertices, n·2^(n−1) edges, and higher-dimensional faces counted by combinatorial formulas: the number of k-faces is binomial(n,k)·2^(n−k). The dual polytope of the hypercube is the cross-polytope (orthoplex). The hypercube also forms a standard example of a regular polytope in every dimension.

Symmetry and group theory play a notable role. The full symmetry group of the n-cube is the hyperoctahedral group, of order 2^n·n!, consisting of all coordinate permutations combined with sign changes of coordinates. This reflects the invariance under changing axes and reflecting along coordinate hyperplanes.

Examples across dimensions illustrate the concept: the 0-cube is a point, the 1-cube a line segment, the 2-cube a square, the 3-cube a regular cube, and the 4-cube is the tesseract. Hypercubic structures appear in mathematics, computer graphics, and physical models, including lattice formulations of space and high-dimensional data representations.