Editing Face (geometry) (section)

==''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'''.