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 o ...

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 mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

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

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 In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...

'' and may also be applied to higher-dimensional

s. * In

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral co ...

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 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 this coincides with the meaning from polyhedral combinatorics.

References

External links

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