Facet (geometry)

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Short description: 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, 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.


  1. Bridge, N.J. (1974). "Facetting the dodecahedron". Acta Crystallographica A30 (4): 548–552. doi:10.1107/S0567739474001306. 
  2. Inchbald, G. (2006). "Facetting diagrams". The Mathematical Gazette 90 (518): 253–261. doi:10.1017/S0025557200179653. 
  3. Coxeter, H. S. M. (1973), "6 Star-Polyjedra", Regular Polytopes, Dover, p. 95, https://books.google.com/books?id=iWvXsVInpgMC&pg=PA95 
  4. 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 .
  5. 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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