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.
- For example:
Famous quotes containing the word facet:
“It is not however, adulthood itself, but parenthood that forms the glass shroud of memory. For there is an interesting quirk in the memory of women. At 30, women see their adolescence quite clearly. At 30 a woman’s adolescence remains a facet fitting into her current self.... At 40, however, memories of adolescence are blurred. Women of this age look much more to their earlier childhood for memories of themselves and of their mothers. This links up to her typical parenting phase.”
—Terri Apter (20th century)