Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

• In three-dimensional geometry a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face.^[1][2]To facet a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higher-dimensional polytopes.^[3]
• In polyhedral combinatorics and in the general theory of polytopes, a facet of a polytope of dimension n is a face that has dimension n − 1. Facets may also be called (n − 1)-faces. In three-dimensional geometry, they are often called "faces" without qualification.^[4]
• A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.^[5] For simplicial polytopes this coincides with the meaning from polyhedral combinatorics.

References

1. Bridge, N.J. Facetting the dodecahedron, Acta crystallographica A30 (1974), pp. 548–552.
2. Inchbald, G. Facetting diagrams, The mathematical gazette, 90 (2006), pp. 253–261.
3. Coxeter, H. S. M. (1973), Regular Polytopes, Dover, p. 95.
4. Matoušek, Jiří (2002), Lectures in Discrete Geometry, Graduate Texts in Mathematics 212, Springer, 5.3 Faces of a Convex Polytope, p. 86.
5. De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco (2010), Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics 25, Springer, p. 493, ISBN 9783642129711.

External links

• Weisstein, Eric W., "Facet", MathWorld.
• Olshevsky, George, Facet at Glossary for Hyperspace.