# Facet (geometry)

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__: Feature of a polyhedron, polytope, etc.__

**Short description**

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

- ↑ Bridge, N.J. (1974). "Facetting the dodecahedron".
*Acta Crystallographica***A30**(4): 548–552. doi:10.1107/S0567739474001306. - ↑ Inchbald, G. (2006). "Facetting diagrams".
*The Mathematical Gazette***90**(518): 253–261. doi:10.1017/S0025557200179653. - ↑ Coxeter, H. S. M. (1973), "6 Star-Polyjedra",
*Regular Polytopes*, Dover, p. 95, https://books.google.com/books?id=iWvXsVInpgMC&pg=PA95 - ↑ Matoušek, Jiří (2002), "5.3 Faces of a Convex Polytope",
*Lectures in Discrete Geometry*, Graduate Texts in Mathematics,**212**, Springer, p. 86, ISBN 9780387953748, https://books.google.com/books?id=0N5RVe5lKQUC&pg=PA86. - ↑ 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, https://books.google.com/books?id=SxY1Xrr12DwC&pg=PA493.

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