• A facet of an n-dimensional simplex is one of its (n-1)-dimensional "faces."
• A facet of a simplicial complex is its maximal (under inclusion) face.

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 0facetting and is the reciprocal process to stellation.
• A facet of an abstract polytope is, more recently, an (n-1)-dimensional face. This accords with the above definition for an n-dimensional simplex, but not with tradition. Exactly two facets meet at any ridge in a polytope.

geometry | Algebraic topology | polyhedra