n×Hn

n×Hn, also known as the n-dimensional hypercube or n-cube, is a generalization of the cube in n-dimensional Euclidean space. It is a regular convex polytope with vertices and edges defined by the Cartesian product of n copies of the interval [0,1]. The n×Hn can be thought of as a cube with each of its vertices replaced by another cube, and this process repeated recursively for each dimension.

The 0-dimensional hypercube is a single point. The 1-dimensional hypercube is a line segment with two vertices. The 2-dimensional hypercube is a square with four vertices and four edges. The 3-dimensional hypercube is a cube with eight vertices and twelve edges. In general, an n-dimensional hypercube has 2^n vertices and n×2^(n-1) edges.

Hypercubes are important in various fields of mathematics and computer science. In topology, they are used to study the properties of spaces. In combinatorics, they are used to model problems involving binary choices. In computer science, they are used in the design of algorithms and data structures, particularly in the context of binary search trees and hash functions.

The n×Hn is a special case of a more general class of polytopes known as zonotopes. It is also related to the concept of a Boolean lattice, which is a partially ordered set that can be used to model the power set of a finite set. The hypercube is a fundamental object in the study of discrete geometry and has been the subject of extensive research in these fields.