In geometry: commonly, a face of a polytope is any of its 2-dimensional polygonal boundaries. For example, one of the squares that bound a cube is a face of the cube.

Formally, however, a face is any of the lower dimensional boundaries of the polytope. This includes the polytope itself and the empty set, see notes below. For example, all of the following are faces of a 4-dimensional polychoron:

• the 4-dimensional polychoron itself (see below))
• any 3-dimensional cell
• any 2-dimensional polygonal "face" (using the common definition of face)
• any 1-dimensional edge
• any 0-dimensional vertex
• the empty set (see below)

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.

Formal Definition

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In convex geometry, a face of a polytope P is the intersection of any supporting hyperplane of P and P.

Notes

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• Notice that a polyhedron in R³ is entirely on one hyperplane of R^{4. For example, 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.}

Elementary geometry | Convex geometry