# Facet (geometry)

Short description: Feature of a polyhedron, polytope, etc.

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

• 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 face that has dimension n − 1 (an (n − 1)-face or hyperface) is also called a facet.[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 (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics.

## References

1. Bridge, N.J. (1974). "Facetting the dodecahedron". Acta Crystallographica A30 (4): 548–552. doi:10.1107/S0567739474001306.
2. Inchbald, G. (2006). "Facetting diagrams". The Mathematical Gazette 90 (518): 253–261. doi:10.1017/S0025557200179653.
3. Coxeter, H. S. M. (1973), "6 Star-Polyjedra", Regular Polytopes, Dover, p. 95
4. Matoušek, Jiří (2002), "5.3 Faces of a Convex Polytope", Lectures in Discrete Geometry, Graduate Texts in Mathematics, 212, Springer, p. 86, ISBN 9780387953748 .
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 .