Polytope - Elements

Elements

The elements of a polytope are its vertices, edges, faces, cells and so on. The terminology for these is not entirely consistent across different authors. To give just a few examples: Some authors use face to refer to an (n−1)-dimensional element while others use face to denote a 2-face specifically, and others use j-face or k-face to indicate an element of j or k dimensions. Some sources use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an (n−1)-dimensional element.

An n-dimensional polytope is bounded by a number of (n−1)-dimensional facets. These facets are themselves polytopes, whose facets are (n−2)-dimensional ridges of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (n−3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.