Facet (geometry)

Last updated February 28, 2025

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, some authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not faces .^[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 called a facet. In this terminology, every facet is a face.^[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 .

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[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 .

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