hyperpolytope

A hyperpolytope is a generalization of the notion of a polytope to higher dimensions. While a polytope in three dimensions is a three-dimensional shape with flat sides (a polyhedron), and in two dimensions is a polygon, a hyperpolytope can exist in any number of dimensions. The term "hyperpolytope" was introduced by George Olshevsky in 1975.

In two dimensions, a hyperpolytope is simply a polygon. In three dimensions, it is a polyhedron. In four dimensions, it is a 4-polytope, such as a tesseract or a 24-cell. In higher dimensions, the concept becomes increasingly abstract, with examples including the 8-cell, 16-cell, and 24-cell in four dimensions, and the 600-cell in five dimensions.

Hyperpolytopes can be characterized by their number of vertices, edges, faces, and higher-dimensional analogs. For example, a tetrahedron has 4 vertices, 6 edges, and 4 triangular faces, while a 5-cell has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells.

The study of hyperpolytopes is a branch of convex geometry and combinatorial geometry. It has applications in various fields, including mathematics, physics, and computer science, particularly in the study of higher-dimensional spaces and in the development of algorithms for processing and analyzing high-dimensional data.