Editing Polytope (section)

==Elements==
A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors use ''face'' to refer to an (''n''&nbsp;−&nbsp;1)-dimensional element while others use ''face'' to denote a 2-face specifically. Authors may use ''j''-face or ''j''-facet to indicate an element of ''j'' dimensions. Some use ''edge'' to refer to a ridge, while [[H. S. M. Coxeter]] uses ''cell'' to denote an (''n''&nbsp;−&nbsp;1)-dimensional element.<ref>Regular polytopes, p. 127 ''The part of the polytope that lies in one of the hyperplanes is called a cell''</ref>{{citation needed|date=February 2015|reason=need to cite each definition claimed}} <!-- Note that "each definition claimed" means "each definition claimed" and this tag should remain until each definition claimed has been cited -->

The terms adopted in this article are given in the table below:

{|class="wikitable"
!Dimension<br>of element
!Term<br>(in an ''n''-polytope)
|-
|align=center|−1
|Nullity (necessary in [[Abstract polytope|abstract]] theory)<ref name="johnson224">Johnson, Norman W.; ''Geometries and Transformations'', Cambridge University Press, 2018, p.224.</ref>
|-
|align=center|0
|[[Vertex (geometry)|Vertex]]
|-
|align=center|1
|[[Edge (geometry)|Edge]]
|-
|align=center|2
|[[Face (geometry)|Face]]
|-
|align=center|3
|[[Cell (geometry)|Cell]]
|-
|align=center|<math>\vdots</math>
|&nbsp;<math>\vdots</math>
|-
|align=center|''j''
|''j''-face – element of rank ''j'' = −1, 0, 1, 2, 3, ..., ''n''
|-
|align=center|<math>\vdots</math>
|&nbsp;<math>\vdots</math>
|-
|align=center|''n'' − 3
|[[Peak (geometry)|Peak]] – (''n'' − 3)-face
|-
|align=center|''n'' − 2
|[[Ridge (geometry)|Ridge]] or subfacet – (''n'' − 2)-face
|-
|align=center|''n'' − 1
|[[Facet (mathematics)|Facet]] – (''n'' − 1)-face
|-
|align=center|''n''
|The polytope itself
|}

An ''n''-dimensional polytope is bounded by a number of (''n''&nbsp;−&nbsp;1)-dimensional ''[[facet (mathematics)|facets]]''. These facets are themselves polytopes, whose facets are (''n''&nbsp;−&nbsp;2)-dimensional ''[[Ridge (geometry)|ridges]]'' of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (''n''&nbsp;−&nbsp;3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as [[Face (geometry)|faces]], or specifically ''j''-dimensional faces or ''j''-faces. A 0-dimensional face is called a ''vertex'', and consists of a single point. A 1-dimensional face is called an ''edge'', and consists of a line segment. A 2-dimensional face consists of a [[polygon]], and a 3-dimensional face, sometimes called a ''[[Cell (mathematics)|cell]]'', consists of a [[polyhedron]].