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. 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. Bibcode:1974AcCrA..30..548B. doi:10.1107/S0567739474001306.
^ Inchbald, G. (2006). "Facetting diagrams". The Mathematical Gazette. 90 (518): 253–261. doi:10.1017/S0025557200179653. S2CID 233358800.
^ Coxeter, H. S. M. (1973), "6 Star-Polyjedra", Regular Polytopes, Dover, p. 95
^ Matoušek, Jiří (2002), "5.3 Faces of a Convex Polytope", Lectures in Discrete Geometry, Graduate Texts in Mathematics, vol. 212, Springer, p. 86, ISBN 9780387953748.
^ De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco (2010), Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, vol. 25, Springer, p. 493, ISBN 9783642129711.

External links

Weisstein, Eric W. "Facet". MathWorld.

This set index article includes a list of related items that share the same name (or similar names).
If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.

This polyhedron-related article is a stub. You can help Wikipedia by expanding it.

[1] Bridge, N.J. (1974). "Facetting the dodecahedron". Acta Crystallographica. A30 (4): 548–552. Bibcode:1974AcCrA..30..548B. doi:10.1107/S0567739474001306.

[2] Inchbald, G. (2006). "Facetting diagrams". The Mathematical Gazette. 90 (518): 253–261. doi:10.1017/S0025557200179653. S2CID 233358800.

[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, vol. 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, vol. 25, Springer, p. 493, ISBN 9783642129711.

[1]

[2]

[3]

[4]

[5]