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Face (geometry)

From Wikipedia, the free encyclopedia

Planar surface that forms part of the boundary of a solid object

Insolid geometry, aface is a flatsurface (aplanar region) that forms part of the boundary of a solid object. For example, acube has six faces in this sense.

In more modern treatments of the geometry ofpolyhedra and higher-dimensionalpolytopes, 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.^[1]

Polygonal face

In elementary geometry, aface is apolygon^[2] on the boundary of apolyhedron.^[1]^[3] (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face includepolyhedron side and Euclidean planetile.

For example, any of the sixsquares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a4-polytope. With this meaning, the 4-dimensionaltesseract has 24 square faces, each sharing two of 8cubic cells.

Regular examples bySchläfli symbol
Polyhedron	Star polyhedron	Euclidean tiling	Hyperbolic tiling	4-polytope
{4,3}	{5/2,5}	{4,4}	{4,5}	{4,3,3}
The cube has 3 squarefaces per vertex.	Thesmall stellated dodecahedron has 5pentagrammic faces per vertex.	Thesquare tiling in the Euclidean plane has 4 squarefaces per vertex.	Theorder-5 square tiling has 5 squarefaces per vertex.	Thetesseract has 3 squarefaces per edge.

Number of polygonal faces of a polyhedron

Anyconvex polyhedron's surface hasEuler characteristic

V-E+F=2,

whereV is the number ofvertices,E is the number ofedges, andF is the number of faces. This equation is known asEuler'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 apolytope are features of all dimensions.^[4]^[5] A face of dimensionk is sometimes called ak-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 anyn-dimensional polytope, faces have dimension $k {\displaystyle k}$ 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 aspolyhedral combinatorics, a polytope is by definitionconvex. In this setting, there is a precise definition: a face of a polytopeP in Euclidean space $\mathbf {R} ^{n}$ is the intersection ofP with anyclosed halfspace whose boundary is disjoint from the relative interior ofP.^[6] According to this definition, the set of faces of a polytope includes the polytope itself and the empty set.^[4]^[5] For convex polytopes, this definition is equivalent to the general definition of a face of a convex set, givenbelow.

In other areas of mathematics, such as the theories ofabstract polytopes andstar polytopes, the requirement of convexity is relaxed. One precise combinatorial concept that generalizes some earlier types of polyhedra is the notion of asimplicial complex. More generally, there is the notion of apolytopal complex.

Ann-dimensionalsimplex (line segment (n = 1), triangle (n = 2), tetrahedron (n = 3), etc.), defined byn + 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 are2^{n + 1} faces in total. The number ofk-faces, fork ∈ {−1, 0, ...,n}, is thebinomial coefficient ${\binom {n+1}{k+1}}$ .

There are specific names fork-faces depending on the value ofk and, in some cases, how closek is to the dimensionn of the polytope.

Vertex or 0-face

Vertex is the common name for a 0-face.

Edge or 1-face

Edge is the common name for a 1-face.

Face or 2-face

The use offace in a context where a specifick is meant for ak-face but is not explicitly specified is commonly a 2-face.

Cell or 3-face

Acell is apolyhedral element (3-face) of a 4-dimensional polytope or 3-dimensional tessellation, or higher. Cells arefacets for 4-polytopes and 3-honeycombs.

Examples:

Regular examples bySchläfli symbol
4-polytopes		3-honeycombs
{4,3,3}	{5,3,3}	{4,3,4}	{5,3,4}
Thetesseract has 3 cubic cells (3-faces) per edge.	The120-cell has 3dodecahedral cells (3-faces) per edge.	Thecubic honeycomb fills Euclidean 3-space with cubes, with 4 cells (3-faces) per edge.	Theorder-4 dodecahedral honeycomb fills 3-dimensional hyperbolic space with dodecahedra, 4 cells (3-faces) per edge.

Facet or (n − 1)-face

Main article:Facet (geometry)

In higher-dimensional geometry, thefacets of an-polytope are the (n − 1)-faces (faces of dimension one less than the polytope itself).^[7] A polytope is bounded by its facets.

For example:

The facets of aline segment are its 0-faces orvertices.
The facets of apolygon are its 1-faces oredges.
The facets of apolyhedron or planetiling are its2-faces.
The facets of a4D polytope or3-honeycomb are its3-faces or cells.
The facets of a5D polytope or 4-honeycomb are its4-faces.

Ridge or (n − 2)-face

In related terminology, the (n − 2)-faces of ann-polytope are calledridges (alsosubfacets).^[8] A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb.

For example:

The ridges of a 2Dpolygon or 1D tiling are its 0-faces orvertices.
The ridges of a 3Dpolyhedron or planetiling are its 1-faces oredges.
The ridges of a4D polytope or3-honeycomb are its 2-faces or simplyfaces.
The ridges of a5D polytope or 4-honeycomb are its 3-faces orcells.

Peak or (n − 3)-face

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

For example:

The peaks of a 3Dpolyhedron or planetiling are its 0-faces orvertices.
The peaks of a4D polytope or3-honeycomb are its 1-faces oredges.
The peaks of a5D polytope or 4-honeycomb are its 2-faces or simplyfaces.

Face of a convex set

The two distinguished points are examples of extreme points of a convex set that are not exposed points. Therefore, not every face of a convex set is an exposed face.

The notion of a face can be generalized from convex polytopes to allconvex sets, as follows. Let $C {\displaystyle C}$ be a convex set in a realvector space $V {\displaystyle V}$ . Aface of $C {\displaystyle C}$ is a convex subset $F\subseteq C$ such that whenever a point $p\in F$ lies strictly between two points $x {\displaystyle x}$ and $y {\displaystyle y}$ in $C {\displaystyle C}$ , both $x {\displaystyle x}$ and $y {\displaystyle y}$ must be in $F {\displaystyle F}$ . Equivalently, for any $x,y\in C$ and any real number $0<\theta <1$ such that $\theta x+(1-\theta )y$ is in $F {\displaystyle F}$ , $x {\displaystyle x}$ and $y {\displaystyle y}$ must be in $F {\displaystyle F}$ .^[9]

According to this definition, $C {\displaystyle C}$ itself and the empty set are faces of $C {\displaystyle C}$ ; these are sometimes called thetrivial faces of $C {\displaystyle C}$ .

Anextreme point of $C {\displaystyle C}$ is a point $p\in C$ such that $\{p\}$ is a face of $C {\displaystyle C}$ .^[9] That is, if $p {\displaystyle p}$ lies between two points $x,y\in C$ , then $x=y=p$ .

For example:

Atriangle 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 theclosed unit disk $\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}\leq 1\}$ are its extreme points, namely the points on theunit circle $S^{1}=\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\}$ .

Let $C {\displaystyle C}$ be a convex set in $\mathbb {R} ^{n}$ that iscompact (or equivalently,closed andbounded). Then $C {\displaystyle C}$ is theconvex hull of its extreme points.^[10] More generally, each compact convex set in alocally convex topological vector space is the closed convex hull of its extreme points (theKrein–Milman theorem).

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

Anexposed point of $C {\displaystyle C}$ is a point $p\in C$ such that $\{p\}$ is an exposed face of $C {\displaystyle C}$ . 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 $C {\displaystyle C}$ and/or $\varnothing$ as faces of $C {\displaystyle C}$ . Some authors require a face to be a closed subset; this is automatic for $C {\displaystyle C}$ a compact convex set in a vector space of finite dimension, but not in infinite dimensions.^[11] In infinite dimensions, the functional $f {\displaystyle f}$ is usually assumed to be continuous in a givenvector topology.

Properties

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

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

See also

References

^^a ^bMatoušek 2002, p. 86.
^Some other polygons, which are not faces, have also been considered for polyhedra and tilings. These includePetrie polygons,vertex figures andfacets (flat polygons formed by coplanar vertices that do not lie in the same face of the polyhedron).
^Cromwell, Peter R. (1999),Polyhedra, Cambridge University Press, p. 13,ISBN 9780521664059.
^^a ^bGrünbaum 2003, p. 17.
^^a ^bZiegler 1995, p. 51.
^Matoušek (2002) and Ziegler (1995) use a slightly different but equivalent definition, which amounts to intersectingP with either a hyperplane disjoint from the interior ofP or the whole space.
^Matoušek (2002), p. 87;Grünbaum (2003), p. 27;Ziegler (1995), p. 17.
^Matoušek (2002), p. 87;Ziegler (1995), p. 71.
^^a ^bRockafellar 1997, p. 162.
^Rockafellar 1997, p. 166.
^Simon, Barry (2011).Convexity: an Analytic Viewpoint. Cambridge: Cambridge University Press. p. 123.ISBN 978-1-107-00731-4.MR 2814377.

Bibliography

Grünbaum, Branko (2003),Convex Polytopes, Graduate Texts in Mathematics, vol. 221 (2nd ed.), Springer,ISBN 0-387-00424-6,MR 1976856
Matoušek, Jiří (2002),Lectures in Discrete Geometry,Graduate Texts in Mathematics, vol. 212, Springer,ISBN 9780387953748,MR 1899299
Rockafellar, R. T. (1997) [1970].Convex Analysis. Princeton, NJ: Princeton University Press.ISBN 1-4008-7317-7.MR 0274683.
Ziegler, Günter M. (1995),Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer,ISBN 9780387943657,MR 1311028

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