geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a facet is a feature of a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

, or related geometric structure, generally of

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

one less than the structure itself. More specifically: * In three-dimensional geometry, a

facet of a polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

is any

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

whose corners are vertices of the polyhedron, and is not a ''

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

''. To ''

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

'' 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

s. * In polyhedral combinatorics and in the general theory of

s, a facet (or hyperface) of a polytope of dimension ''n'' is a

that has dimension ''n'' − 1. Facets may also be called (''n'' − 1)-faces. In three-dimensional geometry, they are often called "faces" without qualification. * A

facet of a simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.. For (boundary complexes of)

simplicial polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's ...

s this coincides with the meaning from polyhedral combinatorics.

References

External links

* Polyhedra Polyhedral combinatorics Polytopes Broad-concept articles {{Polyhedron-stub