Face (geometry)

Polygonal face

In elementary geometry, polyhedra are defined in various ways as shapes defined by systems of vertices (points), edges (line segments), and faces (polygons), that in many but not all of these definitions are required to form a surface that encloses a solid volume; the faces are the two-dimensional polygons of these definitions. Other names for a polygonal face include polyhedron side and Euclidean plane tile.

For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells.

Number of polygonal faces of a polyhedron

Any convex polyhedron's surface has Euler characteristic

:V - E + F = 2,

where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.

''k''-face

In higher-dimensional geometry, the faces of a polytope are features of all dimensions. A face of dimension k is sometimes called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. The word "face" is defined differently in different areas of mathematics. For example, many but not all authors allow the polytope itself and the empty set as faces of a polytope, where the empty set is for consistency given a "dimension" of −1. For any n-dimensional polytope, faces have dimension k with -1 \leq k \leq n.

For example, with this meaning, the faces of a cube comprise the cube itself (a 3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.

In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. In this setting, there is a precise definition: a face of a polytope P in Euclidean space \mathbf{R}^n is the intersection of P with any closed halfspace whose boundary is disjoint from the relative interior of P. According to this definition, the set of faces of a polytope includes the polytope itself and the empty set. For convex polytopes, this definition is equivalent to the general definition of a face of a convex set, given below.

In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement of convexity is relaxed. One precise combinatorial concept that generalizes some earlier types of polyhedra is the notion of a simplicial complex. More generally, there is the notion of a polytopal complex.

An n-dimensional simplex (line segment (), triangle (), tetrahedron (), etc.), defined by n + 1 vertices, has a face for each subset of the vertices, from the empty set up through the set of all vertices. In particular, there are 2 faces in total. The number of k-faces, for k ∈ , is the binomial coefficient \binom{n+1}{k+1}.

There are specific names for k-faces depending on the value of k and, in some cases, how close k is to the dimension n of the polytope.

Vertex or 0-face {{anchor|Vertex}}

Vertex is the common name for a 0-face.

Edge or 1-face {{anchor|Edge}}

Edge is the common name for a 1-face.

Face or 2-face {{anchor|Edge}}

The use of face in a context where a specific k is meant for a k-face but is not explicitly specified is commonly a 2-face.

Cell or 3-face {{anchor|Cell}}

A cell is a polyhedral element (3-face) of a 4-dimensional polytope or 3-dimensional tessellation, or higher. Cells are facets for 4-polytopes and 3-honeycombs.

Examples:

Facet or (''n'' − 1)-face {{anchor|Facet}}

Main article: Facet (geometry)

In higher-dimensional geometry, the facets of a n-polytope are the (n − 1)-faces (faces of dimension one less than the polytope itself). A polytope is bounded by its facets.

For example:

• The facets of a line segment are its 0-faces or vertices.
• The facets of a polygon are its 1-faces or edges.
• The facets of a polyhedron or plane tiling are its 2-faces. However, in some contexts, a facet of a polyhedron means any polygon formed from a subset of three or more vertices of a 2-face.
• The facets of a 4D polytope or 3-honeycomb are its 3-faces or cells.
• The facets of a 5D polytope or 4-honeycomb are its 4-faces.

Ridge or (''n'' − 2)-face {{anchor|Ridge}}

In related terminology, the (n − 2)-faces of an n-polytope are called ridges (also subfacets). A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb.

For example:

• The ridges of a 2D polygon or 1D tiling are its 0-faces or vertices.
• The ridges of a 3D polyhedron or plane tiling are its 1-faces or edges.
• The ridges of a 4D polytope or 3-honeycomb are its 2-faces.
• The ridges of a 5D polytope or 4-honeycomb are its 3-faces or cells.

Peak or (''n'' − 3)-face {{anchor|Peak}}

The (n − 3)-faces of an n-polytope are called peaks. A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb.

For example:

• The peaks of a 3D polyhedron or plane tiling are its 0-faces or vertices.
• The peaks of a 4D polytope or 3-honeycomb are its 1-faces or edges.
• The peaks of a 5D polytope or 4-honeycomb are its 2-faces.

Face of a convex set

The notion of a face can be generalized from convex polytopes to all convex sets, as follows. Let C be a convex set in a real vector space V. A face of C is a convex subset F\subseteq C such that whenever a point p\in F lies strictly between two points x and y in C, both x and y must be in F. Equivalently, for any x,y\in C and any real number 0 such that \theta x+(1-\theta)y is in F, x and y must be in F.

According to this definition, C itself and the empty set are faces of C; these are sometimes called the trivial faces of C.

An extreme point of C is a point p\in C such that {p} is a face of C. That is, if p lies between two points x,y\in C, then x=y=p.

For example:

• A triangle in the plane (including the region inside) is a convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extreme points are the three vertices.)
• The only nontrivial faces of the closed unit disk { (x,y) \in \R^2: x^2+y^2 \leq 1 } are its extreme points, namely the points on the unit circle S^1 = { (x,y) \in \R^2: x^2+y^2=1 }.

Let C be a convex set in \R^n that is compact (or equivalently, closed and bounded). Then C is the convex hull of its extreme points. More generally, each compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points (the Krein–Milman theorem).

An exposed face of C is the subset of points of C where a linear functional achieves its minimum on C. Thus, if f is a linear functional on V and \alpha =\inf{ f(c)\ \colon c\in C}-\infty, then {c\in C\ \colon f(c)=\alpha} is an exposed face of C.

An exposed point of C is a point p\in C such that {p} is an exposed face of C. That is, f(p) f(c) for all c\in C\setminus{p}. See the figure for examples of extreme points that are not exposed.

Competing definitions

Some authors do not include C and/or \varnothing as faces of C. Some authors require a face to be a closed subset; this is automatic for C a compact convex set in a vector space of finite dimension, but not in infinite dimensions. In infinite dimensions, the functional f is usually assumed to be continuous in a given vector topology.

Properties

An exposed face of a convex set is a face. In particular, it is a convex subset.

If F is a face of a convex set C, then a subset E\subseteq F is a face of F if and only if E is a face of C.

Notes

References

Bibliography

References

1. Cromwell, Peter R.. (1999). "Polyhedra". Cambridge University Press.
2. Matoušek (2002) and Ziegler (1995) use a slightly different but equivalent definition, which amounts to intersecting {{mvar. P with either a hyperplane disjoint from the interior of {{mvar. P or the whole space.
3. {{harvtxt. Matoušek. 2002, p. 87; {{harvtxt. Grünbaum. 2003, p. 27; {{harvtxt. Ziegler. 1995, p. 17.
4. {{harvtxt. Matoušek. 2002, p. 87; {{harvtxt. Ziegler. 1995, p. 71.
5. Simon, Barry. (2011). "Convexity: an Analytic Viewpoint". Cambridge University Press.