in higher-dimensional geometry, the faces of a polytope are features of all dimensions. a face of dimension "k" is called a "k"-face. for example, the polygonal faces of an ordinary polyhedron are 2-faces. in set theory, the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. for any "n"-polytope ("n"-dimensional polytope), −1 ≤ "k" ≤ "n".