Facet (geometry) explained

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.
• 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.^[3]
• 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.^[4] For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics.

Notes and References

1. Bridge . N.J. . Facetting the dodecahedron . Acta Crystallographica . A30 . 1974 . 4 . 548–552 . 10.1107/S0567739474001306 .
2. Inchbald . G. . Facetting diagrams . The Mathematical Gazette . 90 . 518 . 2006 . 253–261 . 10.1017/S0025557200179653. 233358800 .
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