# Facet (geometry)

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In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

• In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face.   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 facet of a polytope of dimension n is a face 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 polytopes this coincides with the meaning from polyhedral combinatorics.

## Related Research Articles In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.

In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way. A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets. In geometry, a d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges. The vertex figure of a simple d-polytope is a (d − 1)-simplex. In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.

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.

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.” In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher-dimensional polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional polyhedron or polytope, then the resulting graph is at least d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair. Polymake is software for the algorithmic treatment of convex polyhedra.

In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a shallow pyramid. Kleetopes are named after Victor Klee.

In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric polytope has at least 3d nonempty faces.

In polyhedral combinatorics, a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.

In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.

1. Bridge, N.J. Facetting the dodecahedron, Acta crystallographicaA30 (1974), pp. 548–552.
2. Inchbald, G. Facetting diagrams, The mathematical gazette, 90 (2006), pp. 253–261.
3. Coxeter, H. S. M. (1973), Regular Polytopes, Dover, p. 95.
4. Matoušek, Jiří (2002), Lectures in Discrete Geometry, Graduate Texts in Mathematics, 212, Springer, 5.3 Faces of a Convex Polytope, 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, 25, Springer, p. 493, ISBN   9783642129711 .