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{{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;<ref>{{cite book | title = [[Merriam-Webster's Collegiate Dictionary]] | edition = Eleventh | publisher = [[Merriam-Webster]] | location = Springfield, MA | year = 2004}}</ref> a three-dimensional solid bounded exclusively by faces is a ''[[polyhedron]]''.
 
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:<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 (geometry)|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.<ref name="m"/><ref name="g">{{citation|title=Convex Polytopes|title-link=Convex Polytopes|first=Branko|last=Grünbaum|author-link=Branko Grünbaum|page=[https://books.google.com/books?id=ISHO86XJ1CsC&pg=PA17 17]|volume=221|edition=2nd|series=Graduate Texts in Mathematics|publisher=Springer|year=2003}}.</ref><ref name="z">{{citation|first=Günter M.|last=Ziegler|author-link=Günter M. Ziegler|title=Lectures on Polytopes|at=Definition 2.1, p. 51|url=https://books.google.com/books?id=xd25TXSSUcgC&pg=PA51|volume=152|series=Graduate Texts in Mathematics|publisher=Springer|year=1995|isbn=9780387943657}}.</ref> A face of dimension ''{{mvar|k''}} is called a ''{{mvar|k''}}-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In [[set theory]], the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. For any ''{{mvar|n''}}-polytope (''{{mvar|n''}}-dimensional polytope), {{math|−1 ≤ ''k'' ≤ ''n''}}.
 
For example, with this meaning, the faces of a [[cube]] comprise the cube itself (3-face), its (square) [[Face (geometry)#Facet or (n-1)-face|facets]] (2-faces), its (linearline segment) edges (1-faces), its (point) vertices (0-faces), and the empty set. The following are the '''faces''' of a [[4-polytope|4-dimensional polytope]]:
 
In some areas of mathematics, such as [[polyhedral combinatorics]], a polytope is by definition convex. Formally, a face of a polytope ''{{mvar|P''}} is the intersection of ''{{mvar|P''}} with any [[closed set|closed]] [[Half-space (geometry)|halfspace]] whose boundary is disjoint from the interior of ''{{mvar|P''}}.<ref>{{harvtxt|Matoušek|2002}} and {{harvtxt|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> From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.<ref name="g"/><ref name="z"/>▼
*4-face – the 4-dimensional [[4-polytope]] itself
*3-faces – 3-dimensional [[cell (geometry)|cell]]s ([[polyhedron|polyhedral]] faces)
*2-faces – 2-dimensional [[Face (geometry)#Ridge or (n-2)-face|ridges]] ([[polygon]]al faces)
*1-faces – 1-dimensional [[edge (geometry)|edge]]s
*0-faces – 0-dimensional [[vertex (geometry)|vertices]]
*the empty set, which has dimension −1
 
▲In some areas of mathematics, such as [[polyhedral combinatorics]], a polytope is by definition convex. Formally, a face of a polytope ''P'' is the intersection of ''P'' with any [[closed set|closed]] [[Half-space (geometry)|halfspace]] whose boundary is disjoint from the interior of ''P''.<ref>{{harvtxt|Matoušek|2002}} and {{harvtxt|Ziegler|1995}} use a slightly different but equivalent definition, which amounts to intersecting ''P'' with either a hyperplane disjoint from the interior of ''P'' or the whole space.</ref> From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.<ref name="g"/><ref name="z"/>
 
In other areas of mathematics, such as the theories of [[abstract polytope]]s and [[star polytope]]s, the requirement for convexity is relaxed. Abstract theory still requires that the set of faces include the polytope itself and the empty set.
 
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 them that are {{mvar|k}}-faces, for {{math|''k'' ∈ {{mset|−1, 0, ..., ''n''}}}}, is the [[binomial coefficient]] <math>\binom{n+1}{k+1}</math>.
===Cell or 3-face===▼
 
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 dimensionality {{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.
 
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|}
 
===Facet or (''n'' − 1)-face {{anchor|Facet}}===
{{main article|Facet (geometry)}}
In higher-dimensional geometry, the '''facets''' (also called ''hyperfaces'')<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite symmetry groups'', 11.1 Polytopes and Honeycombs, p.225</ref> of a ''n''-polytope are the (''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.▼
 
▲In higher-dimensional geometry, the '''facets''' (also called '''hyperfaces''')<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite symmetry groups'', 11.1 Polytopes and Honeycombs, p.225</ref> 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:
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*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''&nbsp; −&nbsp; 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:
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*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''&nbsp; −&nbsp; 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:
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