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.
Read more about this topic: Face (geometry)
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