Editing Face (geometry)

{{Short description|Planar surface that forms part of the boundary of a solid object}}
In [[solid geometry]], a '''face''' is a flat [[surface]] (a [[Plane (geometry)|planar]] [[region (mathematics)|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 [[polytope]]s, 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.{{sfn|Matoušek|2002|p=86}}

==Polygonal face==
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 polygon]]s, [[vertex figures]] and [[Facet (geometry)|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]].{{sfn|Matoušek|2002|p=86}}<ref>{{citation|title=Polyhedra|first=Peter R.|last=Cromwell|publisher=Cambridge University Press|year=1999|page=13|isbn=9780521664059|url=https://books.google.com/books?id=OJowej1QWpoC&pg=PA13}}.</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 ''[[tessellation|tile]]''.

For example, any of the six [[square (geometry)|square]]s 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 [[cube|cubic]] cells.

{| class=wikitable width=640
|+ Regular examples by [[Schläfli symbol]]
|-
![[Platonic solid|Polyhedron]]
![[Kepler-Poinsot polyhedron|Star polyhedron]]
![[Regular tiling#Regular tilings|Euclidean tiling]]
![[List of regular polytopes#Hyperbolic tilings|Hyperbolic tiling]]
![[Convex regular polychoron|4-polytope]]
|-
![[cube|{4,3}]]
![[Small stellated dodecahedron|{5/2,5}]]
![[square tiling|{4,4}]]
![[Order-5 square tiling|{4,5}]]
![[tesseract|{4,3,3}]]
|- align=center valign=top
|[[Image:hexahedron.png|100px]]<BR>The cube has 3 square ''faces'' per vertex.
|[[File:Small stellated dodecahedron.png|100px]]<BR>The [[small stellated dodecahedron]] has 5 [[pentagram]]mic faces per vertex.
|[[Image:Tile 4,4.svg|100px]]<BR>The [[square tiling]] in the Euclidean plane has 4 square ''faces'' per vertex.
|[[File:H2-5-4-primal.svg|100px]]<BR>The [[order-5 square tiling]] has 5 square ''faces'' per vertex.
|[[File:Hypercube.svg|100px]]<BR>The [[tesseract]] has 3 square ''faces'' per edge.
|}

===Number of polygonal faces of a polyhedron===
Any [[convex polyhedron]]'s surface has [[Euler characteristic]]

:<math>V - E + F = 2,</math>

where {{mvar|V}} is the number of [[vertex (geometry)|vertices]], {{mvar|E}} is the number of [[edge (geometry)|edges]], and {{mvar|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.{{sfn|Grünbaum|2003|p=17}}{{sfn|Ziegler|1995|p=51}} A face of dimension {{mvar|k}} is sometimes called a {{mvar|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 {{mvar|n}}-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) [[Face (geometry)#Facet|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 set|convex]]. In this setting, there is a precise definition: a face of a polytope {{mvar|P}} in Euclidean space <math>\mathbf{R}^n</math> is the intersection of {{mvar|P}} with any [[closed set|closed]] [[Half-space (geometry)|halfspace]] whose boundary is disjoint from the relative interior of {{mvar|P}}.<ref>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.</ref> According to this definition, the set of faces of a polytope includes the polytope itself and the empty set.{{sfn|Grünbaum|2003|p=17}}{{sfn|Ziegler|1995|p=51}} For convex polytopes, this definition is equivalent to the general definition of a face of a convex set, given [[Face (geometry)#Face of a convex set|below]].

In other areas of mathematics, such as the theories of [[abstract polytope]]s and [[star polytope]]s, 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 {{mvar|n}}-dimensional [[simplex]] (line segment ({{math|1=''n'' = 1}}), triangle ({{math|1=''n'' = 2}}), tetrahedron ({{math|1=''n'' = 3}}), etc.), defined by {{math|''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 {{math|2{{sup|''n'' + 1}}}} faces in total.  The number of {{mvar|k}}-faces, for {{math|''k'' ∈ {{mset|−1, 0, ..., ''n''}}}}, is the [[binomial coefficient]] <math>\binom{n+1}{k+1}</math>.

There are specific names for {{mvar|k}}-faces depending on the value of {{mvar|k}} and, in some cases, how close {{mvar|k}} is to the dimension {{mvar|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 {{mvar|k}} is meant for a {{mvar|k}}-face but is not explicitly specified is commonly a 2-face.

===Cell or 3-face {{anchor|Cell}}===
A '''cell''' is a [[polyhedron|polyhedral]] element ('''3-face''') of a 4-dimensional polytope or 3-dimensional tessellation, or higher. Cells are [[Facet (geometry)|facets]] for 4-polytopes and 3-honeycombs.

Examples:
{| class=wikitable width=640
|+ Regular examples by [[Schläfli symbol]]
|-
!colspan=2|4-polytopes
!colspan=2|3-honeycombs
|- align=center
![[tesseract|{4,3,3}]]
![[120-cell|{5,3,3}]]
![[cubic honeycomb|{4,3,4}]]
![[Order-4 dodecahedral honeycomb|{5,3,4}]]
|- valign=top align=center
|[[Image:Hypercube.svg|120px]]<BR>The [[tesseract]] has 3 cubic cells (3-faces) per edge.
|[[File:Schlegel wireframe 120-cell.png|120px]]<BR>The [[120-cell]] has 3 [[dodecahedron|dodecahedral]] cells (3-faces) per edge.
|[[Image:Partial cubic honeycomb.png|120px]]<BR>The [[cubic honeycomb]] fills Euclidean 3-space with cubes, with 4 cells (3-faces) per edge.
|[[File:Hyperbolic orthogonal dodecahedral honeycomb.png|120px]]<BR>The [[order-4 dodecahedral honeycomb]] fills 3-dimensional hyperbolic space with dodecahedra, 4 cells (3-faces) per edge.
|}

===Facet or (''n'' − 1)-face {{anchor|Facet}}===
{{main article|Facet (geometry)}}

In higher-dimensional geometry, the '''facets''' of a {{mvar|n}}-polytope are the ({{math|''n'' − 1}})-faces (faces of dimension one less than the polytope itself).<ref>{{harvtxt|Matoušek|2002}}, p. 87; {{harvtxt|Grünbaum|2003}}, p. 27; {{harvtxt|Ziegler|1995}}, p. 17.</ref> A polytope is bounded by its facets.

For example:
*The facets of a [[line segment]] are its 0-faces or [[Vertex (geometry)|vertices]].
*The facets of a [[polygon]] are its 1-faces or [[Edge (geometry)|edges]].
*The facets of a [[polyhedron]] or plane [[uniform tiling|tiling]] are its [[2-face]]s.
*The facets of a [[4-polytope|4D polytope]] or [[convex uniform honeycomb|3-honeycomb]] are its [[3-face]]s or cells.
*The facets of a [[5-polytope|5D polytope]] or 4-honeycomb are its [[4-face]]s.

===Ridge or (''n'' − 2)-face {{anchor|Ridge}}===
In related terminology, the ({{math|''n'' − 2}})-''face''s of an {{mvar|n}}-polytope are called '''ridges''' (also '''subfacets''').<ref>{{harvtxt|Matoušek|2002}}, p. 87; {{harvtxt|Ziegler|1995}}, 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 [[Vertex (geometry)|vertices]].
*The ridges of a 3D [[polyhedron]] or plane [[uniform tiling|tiling]] are its 1-faces or [[Edge (geometry)|edges]].
*The ridges of a [[4-polytope|4D polytope]] or [[convex uniform honeycomb|3-honeycomb]] are its 2-faces or simply '''faces'''.
*The ridges of a [[5-polytope|5D polytope]] or 4-honeycomb are its 3-faces or [[Cell (geometry)|cells]].

===Peak or (''n'' − 3)-face {{anchor|Peak}}===
The ({{math|''n'' − 3}})-''face''s of an {{mvar|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 [[uniform tiling|tiling]] are its 0-faces or [[Vertex (geometry)|vertices]].
*The peaks of a [[4-polytope|4D polytope]] or [[convex uniform honeycomb|3-honeycomb]] are its 1-faces or [[Edge (geometry)|edges]].
*The peaks of a [[5-polytope|5D polytope]] or 4-honeycomb are its 2-faces or simply '''faces'''.

==Face of a convex set==
[[File:Extremenotexposed.png|thumb|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 all [[convex set]]s, 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>.{{sfn|Rockafellar|1997|p=162}}

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>.{{sfn|Rockafellar|1997|p=162}} 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 space|compact]] (or equivalently, [[closed set|closed]] and [[bounded set|bounded]]). Then <math>C</math> is the [[convex hull]] of its extreme points.{{sfn|Rockafellar|1997|p=166}} 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 definitions ===
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>{{cite book | last=Simon | first=Barry | author-link=Barry Simon | title=Convexity: an Analytic Viewpoint |mr=2814377|publisher=Cambridge University Press | location=Cambridge | year=2011 | page=123|isbn=978-1-107-00731-4 |url=https://books.google.com/books?id=xWCs0lWGxjkC }}</ref> In infinite dimensions, the functional <math>f</math> 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 <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>.

==See also==
* [[Face lattice]]
* [[Polyhedral combinatorics]]
* [[Discrete geometry]]

== References ==
{{reflist|30em}}

==Bibliography==
* {{citation|title=Convex Polytopes|first=Branko|last=Grünbaum|author-link=Branko Grünbaum|volume=221|edition=2nd|series=Graduate Texts in Mathematics|publisher=Springer|year=2003|mr=1976856|isbn=0-387-00424-6|url=https://books.google.com/books?id=5iV75P9gIUgC}}
* {{citation|title=Lectures in Discrete Geometry|url=https://books.google.com/books?id=0N5RVe5lKQUC&pg=PA86|volume=212|series=[[Graduate Texts in Mathematics]]|publisher=Springer|year=2002|author-link=Jiří Matoušek (mathematician)|first=Jiří|last=Matoušek|mr=1899299|isbn = 9780387953748}}
* {{cite book | last=Rockafellar | first=R. T. | author-link=R. Tyrrell Rockafellar | title=Convex Analysis |mr=0274683|publisher=Princeton University Press | location=Princeton, NJ | orig-year=1970 | year=1997 | isbn=1-4008-7317-7 |url=https://books.google.com/books?id=1TiOka9bx3sC }}
* {{citation|first=Günter M.|last=Ziegler|author-link=Günter M. Ziegler|title=Lectures on Polytopes|url=https://books.google.com/books?id=xd25TXSSUcgC&pg=PA51|volume=152|series=Graduate Texts in Mathematics|publisher=Springer|year=1995|mr=1311028|isbn=9780387943657}}

==External links==
* {{mathworld |urlname=Face |title=Face}}
* {{mathworld |urlname=Facet |title=Facet}}
* {{mathworld |urlname=Side |title=Side}}

[[Category:Elementary geometry]]
[[Category:Convex geometry]]
[[Category:Polyhedra]]
[[Category:Planar surfaces]]

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