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Facet (geometry) - Wikipedia

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Feature of a polyhedron, polytope, etc.

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.

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.

• Weisstein, Eric W. "Facet". MathWorld.

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