A polytope is a higher-dimensional analogue of a polygon or polyhedron. The number of possible regular polytopes depends on the number of dimensions. In two dimensions there are infinitely many possible regular polygons; in three dimensions there are five possible regular polyhedra; in four dimensions there are six possible regular polytopes; and in each number of dimensions higher than four, there are just three possible regular polytopes, analogous to the three-dimensional tetrahedron, cube, and octahedron. A four-dimensional polytope is also sometimes called a polyhedroid or a polychoron. Just as a polygon has vertices and edges, and a polyhedron has vertices, edges, and faces, a four-dimensional polytope has vertices, edges, faces, and cells, where a cell is a three-dimensional figure.



1. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover Publications, 1973.