# 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 (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics.

## References

- Bridge, N.J. Facetting the dodecahedron,
*Acta crystallographica***A30**(1974), pp. 548–552. - Inchbald, G. Facetting diagrams,
*The mathematical gazette*,**90**(2006), pp. 253–261. - Coxeter, H. S. M. (1973),
*Regular Polytopes*, Dover, p. 95. - Matoušek, Jiří (2002),
*Lectures in Discrete Geometry*, Graduate Texts in Mathematics,**212**, Springer, 5.3 Faces of a Convex Polytope, p. 86. - 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.

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