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:
*:#The facets of a polygon are edges. (1-faces)
*:#The facets of a polyhedron or tiling are faces. (2-faces)
*:#The facets of a polychoron (4-polytope) or honeycomb are cells. (3-faces)
*:#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.

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