Face (geometry)

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Square tiling - four square faces per vertex

cube - three square faces per vertex

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.

[edit] 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 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

[edit] 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.