Editing Face (geometry) (section)

==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.