Face (geometry)
Template:Short description In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense.
In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense.Template:Sfn
Polygonal faceEdit
In elementary geometry, a face is a polygon<ref>Some other polygons, which are not faces, have also been considered for polyhedra and tilings. These include Petrie polygons, vertex figures and facets (flat polygons formed by coplanar vertices that do not lie in the same face of the polyhedron).</ref> on the boundary of a polyhedron.Template:Sfn<ref>Template:Citation.</ref> (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) 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.
| Polyhedron | Star polyhedron | Euclidean tiling | Hyperbolic tiling | 4-polytope |
|---|---|---|---|---|
| {4,3} | {5/2,5} | {4,4} | {4,5} | {4,3,3} |
| File:Hexahedron.png The cube has 3 square faces per vertex. |
File:Small stellated dodecahedron.png The small stellated dodecahedron has 5 pentagrammic faces per vertex. |
File:Tile 4,4.svg The square tiling in the Euclidean plane has 4 square faces per vertex. |
File:H2-5-4-primal.svg The order-5 square tiling has 5 square faces per vertex. |
File:Hypercube.svg The tesseract has 3 square faces per edge. |
Number of polygonal faces of a polyhedronEdit
Any convex polyhedron's surface has Euler characteristic
- <math>V - E + F = 2,</math>
where Template:Mvar is the number of vertices, Template:Mvar is the number of edges, and Template:Mvar 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-faceEdit
In higher-dimensional geometry, the faces of a polytope are features of all dimensions.Template:SfnTemplate:Sfn A face of dimension Template:Mvar is sometimes called a Template:Mvar-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 Template:Mvar-dimensional polytope, faces have dimension <math>k</math> with <math>-1 \leq k \leq n</math>.
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 Template:Mvar in Euclidean space <math>\mathbf{R}^n</math> is the intersection of Template:Mvar with any closed halfspace whose boundary is disjoint from the relative interior of Template:Mvar.<ref>Matoušek (2002) and Ziegler (1995) use a slightly different but equivalent definition, which amounts to intersecting Template:Mvar with either a hyperplane disjoint from the interior of Template:Mvar or the whole space.</ref> According to this definition, the set of faces of a polytope includes the polytope itself and the empty set.Template:SfnTemplate:Sfn 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 Template:Mvar-dimensional simplex (line segment (Template:Math), triangle (Template:Math), tetrahedron (Template:Math), etc.), defined by Template:Math 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 Template:Math faces in total. The number of Template:Mvar-faces, for Template:Math, is the binomial coefficient <math>\binom{n+1}{k+1}</math>.
There are specific names for Template:Mvar-faces depending on the value of Template:Mvar and, in some cases, how close Template:Mvar is to the dimension Template:Mvar of the polytope.
Vertex or 0-face Template:AnchorEdit
Vertex is the common name for a 0-face.
Edge or 1-face Template:AnchorEdit
Edge is the common name for a 1-face.
Face or 2-face Template:AnchorEdit
The use of face in a context where a specific Template:Mvar is meant for a Template:Mvar-face but is not explicitly specified is commonly a 2-face.
Cell or 3-face Template:AnchorEdit
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:
| 4-polytopes | 3-honeycombs | ||
|---|---|---|---|
| {4,3,3} | {5,3,3} | {4,3,4} | {5,3,4} |
| File:Hypercube.svg The tesseract has 3 cubic cells (3-faces) per edge. |
File:Schlegel wireframe 120-cell.png The 120-cell has 3 dodecahedral cells (3-faces) per edge. |
File:Partial cubic honeycomb.png The cubic honeycomb fills Euclidean 3-space with cubes, with 4 cells (3-faces) per edge. |
File:Hyperbolic orthogonal dodecahedral honeycomb.png The order-4 dodecahedral honeycomb fills 3-dimensional hyperbolic space with dodecahedra, 4 cells (3-faces) per edge. |
Facet or (n − 1)-face Template:AnchorEdit
In higher-dimensional geometry, the facets of a Template:Mvar-polytope are the (Template:Math)-faces (faces of dimension one less than the polytope itself).<ref>Template:Harvtxt, p. 87; Template:Harvtxt, p. 27; Template:Harvtxt, p. 17.</ref> 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.
- 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 Template:AnchorEdit
In related terminology, the (Template:Math)-faces of an Template:Mvar-polytope are called ridges (also subfacets).<ref>Template:Harvtxt, p. 87; Template:Harvtxt, p. 71.</ref> 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 or simply faces.
- The ridges of a 5D polytope or 4-honeycomb are its 3-faces or cells.
Peak or (n − 3)-face Template:AnchorEdit
The (Template:Math)-faces of an Template:Mvar-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 or simply faces.
Face of a convex setEdit
The notion of a face can be generalized from convex polytopes to all convex sets, as follows. Let <math>C</math> be a convex set in a real vector space <math>V</math>. A face of <math>C</math> is a convex subset <math>F\subseteq C</math> such that whenever a point <math>p\in F</math> lies strictly between two points <math>x</math> and <math>y</math> in <math>C</math>, both <math>x</math> and <math>y</math> must be in <math>F</math>. Equivalently, for any <math>x,y\in C</math> and any real number <math>0<\theta<1</math> such that <math>\theta x+(1-\theta)y</math> is in <math>F</math>, <math>x</math> and <math>y</math> must be in <math>F</math>.Template:Sfn
According to this definition, <math>C</math> itself and the empty set are faces of <math>C</math>; these are sometimes called the trivial faces of <math>C</math>.
An extreme point of <math>C</math> is a point <math>p\in C</math> such that <math>\{p\}</math> is a face of <math>C</math>.Template:Sfn That is, if <math>p</math> lies between two points <math>x,y\in C</math>, then <math>x=y=p</math>.
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 <math>\{ (x,y) \in \R^2: x^2+y^2 \leq 1 \}</math> are its extreme points, namely the points on the unit circle <math>S^1 = \{ (x,y) \in \R^2: x^2+y^2=1 \}</math>.
Let <math>C</math> be a convex set in <math>\R^n</math> that is compact (or equivalently, closed and bounded). Then <math>C</math> is the convex hull of its extreme points.Template:Sfn 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 <math>C</math> is the subset of points of <math>C</math> where a linear functional achieves its minimum on <math>C</math>. Thus, if <math>f</math> is a linear functional on <math>V</math> and <math>\alpha =\inf\{ f(c)\ \colon c\in C\}>-\infty</math>, then <math> \{c\in C\ \colon f(c)=\alpha\}</math> is an exposed face of <math>C</math>.
An exposed point of <math>C</math> is a point <math>p\in C</math> such that <math>\{p\}</math> is an exposed face of <math>C</math>. That is, <math>f(p) > f(c)</math> for all <math>c\in C\setminus\{p\}</math>. See the figure for examples of extreme points that are not exposed.
Competing definitionsEdit
Some authors do not include <math>C</math> and/or <math>\varnothing</math> as faces of <math>C</math>. Some authors require a face to be a closed subset; this is automatic for <math>C</math> a compact convex set in a vector space of finite dimension, but not in infinite dimensions.<ref>Template:Cite book</ref> In infinite dimensions, the functional <math>f</math> is usually assumed to be continuous in a given vector topology.
PropertiesEdit
An exposed face of a convex set is a face. In particular, it is a convex subset.
If <math>F</math> is a face of a convex set <math>C</math>, then a subset <math>E\subseteq F</math> is a face of <math>F</math> if and only if <math>E</math> is a face of <math>C</math>.