Editing Abstract polytope (section)

== Introductory concepts ==

=== Traditional versus abstract polytopes ===
[[Image:Isomorphic Tetragons.svg|thumb|275px|Isomorphic quadrilaterals.]]

In Euclidean geometry, two shapes that are not [[Similar (geometry)|similar]] can nonetheless share a common structure. For example, a [[square]] and a [[trapezoid]] both comprise an alternating chain of four [[vertex (geometry)|vertices]] and four sides, which makes them [[quadrilaterals]]. They are said to be [[isomorphic]] or “structure preserving”.

This common structure may be represented in an underlying abstract polytope, a purely algebraic partially ordered set which captures the pattern of connections (or ''incidences)'' between the various structural elements. The measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope.

What is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa. For example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope.<ref>{{Harvnb |McMullen |Schulte |2002 |loc=p. 31}}</ref>

====Realizations====
A traditional polytope is said to be a ''realization'' of the associated abstract polytope. A realization is a mapping or injection of the abstract object into a real space, typically [[Euclidean space|Euclidean]], to construct a traditional polytope as a real geometric figure.

The six quadrilaterals shown are all distinct realizations of the abstract quadrilateral, each with different geometric properties. Some of them do not conform to traditional definitions of a quadrilateral and are said to be ''unfaithful'' realizations. A conventional polytope is a faithful realization.

===Faces, ranks and ordering===
In an abstract polytope, each structural element (vertex, edge, cell, etc.) is associated with a corresponding member of the set. The term ''face'' is used to refer to any such element e.g. a vertex (0-face), edge (1-face) or a general ''k''-face, and not just a polygonal 2-face.

The faces are ''ranked'' according to their associated real dimension: vertices have rank 0, edges rank 1 and so on.

Incident faces of different ranks, for example, a vertex F of an edge G, are ordered by the relation  F < G. F is said to be a ''subface'' of G.

F, G are said to be ''incident'' if either F = G or F < G or G < F. This usage of "incidence" also occurs in [[finite geometry]], although it differs from traditional geometry and some other areas of mathematics. For example, in the square ''ABCD'', edges ''AB'' and ''BC'' are not abstractly incident (although they are both incident with vertex B).{{citation needed|date=December 2016|reason=equals sign undefined. F ranked equally to G is not sufficient condition for geometrical incidence, on the other hand F equating to G would mean they are the same face.}}

A polytope is then defined as a set of faces '''P''' with an order relation '''<'''. Formally, '''P''' (with '''<''') will be a (strict) [[partially ordered set]], or ''poset''.

===Least and greatest faces===
Just as the number zero is necessary in mathematics, so also every set has the [[empty set]] ∅ as a subset. In an abstract polytope ∅ is by convention identified as the ''least'' or ''null'' face and is a subface of all the others.{{why|date=July 2020|reason=Why is the empty set made a member? Many other subsets are not members.}} Since the least face is one level below the vertices or 0-faces, its rank is −1 and it may be denoted as ''F''<sub>−1</sub>. Thus F<sub>−1</sub> ≡ ∅ and the abstract polytope also contains the empty set as an element.<ref>{{Harvnb |McMullen |Schulte |2002 |loc=}}</ref> It is usually not realized, though the lack of its realization could be interpreted as it being realized as the set containing no points, the empty set.

There is also a single face of which all the others are subfaces. This is called the ''greatest'' face. In an ''n''-dimensional polytope, the greatest face has rank = ''n'' and may be denoted as ''F''<sub>''n''</sub>. It is sometimes realized as the interior of the geometric figure.

These least and greatest faces are sometimes called ''improper'' faces, with all others being ''proper'' faces.<ref name="ARP23"/>

===A simple example===
The faces of the abstract quadrilateral or square are shown in the table below:

{|class=wikitable
|- 
!|Face type 
!Rank (''k'')
!Count
!''k''-faces
|-
|Least ||−1 ||1||''F''<sub>−1</sub> 
|-
|Vertex ||0 ||4||'''a''', '''b''', '''c''', '''d'''
|-
|Edge ||1 ||4||W, X, Y, Z
|-
|Greatest ||2 ||1||G
|}

The relation < comprises a set of pairs, which here include

: ''F''<sub>−1</sub><'''a''', ... , ''F''<sub>−1</sub><X, ... , ''F''<sub>−1</sub><G, ... , '''b'''<Y, ... , '''c'''<G, ... , Z<G.

Order relations are [[Transitive relation|transitive]], i.e. F < G and  G < H implies that F < H. Therefore, to specify the hierarchy of faces, it is not necessary to give every case of F&nbsp;<&nbsp;H, only the pairs where one is the [[Covering relation|successor]] of the other, i.e. where F&nbsp;<&nbsp;H and no G satisfies F&nbsp;<&nbsp;G&nbsp;<&nbsp;H.

The edges W, X, Y and Z are sometimes written as '''ab''', '''ad''', '''bc''', and '''cd''' respectively, but such notation is not always appropriate.

All four edges are structurally similar and the same is true of the vertices. The figure therefore has the symmetries of a square and is usually referred to as the square.

===The Hasse diagram===
[[Image:A Square and its Hasse Diagram.PNG|thumb|300px|The [[Graph (discrete mathematics)|graph]] (left) and [[Hasse diagram]] of a quadrilateral, showing ranks (right)]]
Smaller posets, and polytopes in particular, are often best visualized in a [[Hasse diagram]], as shown. By convention, faces of equal rank are placed on the same vertical level. Each "line" between faces, say F, G, indicates an ordering relation < such that F < G where F is below G in the diagram.

The Hasse diagram defines the unique poset and therefore fully captures the structure of the polytope. Isomorphic polytopes give rise to isomorphic Hasse diagrams, and vice versa. The same is not generally true for the [[Graph (discrete mathematics)|graph]] representation of polytopes.

=== Rank ===
The ''rank'' of a face F is defined as (''m''&nbsp;−&nbsp;2), where ''m'' is the maximum number of faces in any [[Total order#Chains|chain]] (F',&nbsp;F",&nbsp;...&nbsp;,&nbsp;F) satisfying F'&nbsp;<&nbsp;F"&nbsp;<&nbsp;...&nbsp;<&nbsp;F. F' is always the least face, F<sub>−1</sub>.

The ''rank'' of an abstract polytope '''P''' is the maximum rank '''''n''''' of any face. It is always the rank of the greatest face F<sub>n</sub>.

The rank of a face or polytope usually corresponds to the ''dimension'' of its counterpart in traditional theory.

For some ranks, their face-types are named in the following table.
{|class=wikitable style="text-align: center;"
|- 
!! width="80" | Rank !! width="50" |  −1 !! width="50" |0 !! width="50" |1 !! width="50" |2 !! width="50" |3 !! width="30" | ... !! width="50" |''n'' − 2  !! width="50" |''n'' − 1  ||width="50" |''n'' 
|-
! Face Type 
| Least ||Vertex ||Edge ||† ||Cell ||   ||Subfacet or ridge<ref name=ARP23>{{Harvnb |McMullen |Schulte |2002 |loc=p. 23}}</ref> ||Facet<ref name=ARP23 /> ||Greatest
|}

† Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of ''any'' rank.

===Flags===

In geometry, a '''[[Flag (geometry)|flag]]''' is a maximal [[Total order#Chains|chain]] of faces, i.e. a (totally) ordered set Ψ of faces, each a subface of the next (if any), and such that Ψ is not a subset of any larger chain. Given any two distinct faces F, G in a flag, either F < G or  F > G.

For example, {'''ø''', '''a''', '''ab''', '''abc'''} is a flag in the triangle '''abc'''.

For a given polytope, all flags contain the same number of faces. Other posets do not, in general, satisfy this requirement.

===Sections===
[[Image:Triangular 3-Prism.PNG|thumb|540px|The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (<span style="color:red;">red</span>), and a 2-section (<span style="color:green;">green</span>).]]

Any subset P' of a poset P is a poset (with the same relation <, restricted to P').

In an abstract polytope, given any two faces ''F'', ''H'' of P with ''F'' ≤ ''H'', the set {''G'' | ''F'' ≤ ''G'' ≤ ''H''} is called a '''section''' of ''P'', and denoted ''H''/''F''. (In order theory, a section is called a [[Partially ordered set#Intervals|closed interval]] of the poset and denoted [''F'', ''H''].)

For example, in the prism  '''abcxyz''' (see diagram) the section '''xyz'''/'''ø''' (highlighted green) is the triangle

:{'''ø''', '''x''', '''y''', '''z''', '''xy''', '''xz''', '''yz''', '''xyz'''}.

A '''''k''-section''' is a section of rank ''k''.

P is thus a section of itself.

This concept of section ''does not'' have the same meaning as in traditional geometry.

====Facets====
The '''facet''' for a given ''j''-face ''F'' is the (''j''&minus;''1'')-section ''F''/∅, where ''F''<sub>''j''</sub> is the greatest face.

For example, in the triangle '''abc''', the facet at '''ab''' is '''ab'''/'''∅''' = {'''∅, a, b, ab'''}, which is a line segment.

The distinction between ''F'' and ''F''/∅ is not usually significant and the two are often treated as identical.

====Vertex figures====
The '''[[vertex figure]]''' at a given vertex ''V'' is the (''n''&minus;1)-section ''F''<sub>''n''</sub>/''V'', where ''F''<sub>''n''</sub> is the greatest face.

For example, in the triangle '''abc''', the vertex figure at '''b''' is '''abc'''/'''b''' = {'''b, ab, bc, abc'''}, which is a line segment.  The vertex figures of a cube are triangles.

====Connectedness====
A poset P is '''connected''' if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces

:H<sub>1</sub>, H<sub>2</sub>, ... ,H<sub>k</sub>

such that F = H<sub>1</sub>, G = H<sub>k</sub>, and each H<sub>i</sub>, i < k, is incident with its successor.

The above condition ensures that a pair of disjoint triangles '''abc''' and '''xyz''' is ''not'' a (single) polytope.

A poset P is '''strongly connected''' if every section of P (including P itself) is connected.

With this additional requirement, two pyramids that share just a vertex are also excluded. However, two square pyramids, for example, ''can'', be "glued" at their square faces - giving an octahedron. The "common face" is ''not'' then a face of the octahedron.