Face (geometry) - Formal Definition

Formal Definition

In convex geometry, a face of a polytope P is the intersection of any supporting hyperplane of P and P. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. For example, a polyhedron R3 is entirely on one hyperplane of R4. If R4 were spacetime, the hyperplane at t = 0 supports and contains the entire polyhedron. Thus, by the formal definition, the polyhedron is a face of itself.

All of the following are the n-faces of a 4-dimensional polytope:

• 4-face – the 4-dimensional 4-polytope itself
• 3-face – any 3-dimensional cell
• 2-face – any 2-dimensional polygonal face (using the common definition of face)
• 1-face – any 1-dimensional edge
• 0-face – any 0-dimensional vertex
• the empty set.