Face (geometry)

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 (two-dimensional) polygons that bound higher-dimensional polytopes are also commonly called "faces". Formally, however, a face is "any" of the lower dimensional boundaries of the polytope, more specifically called an n-face.

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 R^{4. 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 polychoron:
* 4-face - the 4-dimensional polychoron 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.

Facets

If the polytope lies in "n"-dimensions, a face in the "(n-1)"-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 "(n-2)"-dimension is called a ridge.

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