In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix hedron is derived from the Greek word hedra which means face.
The (twodimensional) polygons that bound higherdimensional polytopes are also commonly called faces. Formally, however, a face is any of the lower dimensional boundaries of the polytope, more specifically called an nface.
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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 R^{3} is entirely on one hyperplane of R^{4}. If R^{4} 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 nfaces of a 4dimensional polytope:
 4face  the 4dimensional 4polytope itself
 3face  any 3dimensional cell
 2face  any 2dimensional polygonal face (using the common definition of face)
 1face  any 1dimensional edge
 0face  any 0dimensional vertex
 the empty set.
Facets
If the polytope lies in mdimensions, a face in the (m1)dimension is called a facet. For example, a cell of a polychoron is a facet, a "face" of a polyhedron is a facet, an edge of a polygon is a facet, etc. A face in the (n2)dimension is called a ridge.
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