Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.

In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:

• A facet of a geometric polyhedron is traditionally any polygon whose corners are vertices of the polyhedron. By extension to higher dimensions, it is any j-tope (j-dimensional polytope) whose vertices are shared by some n-tope (n-dimensional polytope where 0 < j < n). To facet a polytope is to find and join such facets to form a new polytope – this process is called facetting or faceting and is the reciprocal process to stellation.
• A facet of an n-polytope is, more recently, an (n−1)-dimensional face or (n−1)-face. The informal term side can mean the same thing, edges of a polygon and faces of a polyhedron.

  For example:

  1. The facets of a polygon are edges. (1-faces)
  2. The facets of a polyhedron or tiling are faces. (2-faces)
  3. The facets of a polychoron (4-polytope) or honeycomb are cells. (3-faces)
  4. The facets of a polyteron (5-polytope) or 4-honeycomb are hypercells. (4-faces)

  Exactly two facets meet at any ridge in a polytope. By extension, facet or j-facet is sometimes used to mean any j-dimensional element of a polytope.