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

External links

Weisstein, Eric W. "Facet". MathWorld.

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

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