A polytope is a geometric object with flat sides, existing in any number of dimensions. In two dimensions, a polytope is a polygon, while in three dimensions, it is a polyhedron. Polytopes can be defined mathematically as the convex hull of a finite set of points in a Euclidean space. Polytopes have various properties, such as vertices, edges, and faces, which can be studied in different dimensions. The study of polytopes is important in fields like mathematics, computer science, and optimization, where they help in understanding complex shapes and solving multidimensional problems.