{{Short description|Aspect of geometry}} {{Dark mode invert|[[File:Pyramid flag.svg|thumb|upright=1.8|Face diagram of a square pyramid showing one of its flags]]}}
In (polyhedral) geometry, a '''flag''' is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.
More formally, a '''flag''' {{mvar|ψ}} of an {{mvar|n}}-polytope is a set {{math|{''F''{{sub|-1}}, ''F''{{sub|0}}, ..., ''F''{{sub|''n''}}} }} such that {{math|''F''{{sub|''i''}} ≤ ''F''{{sub|''i''+1}}}} {{math|(-1 ≤ ''i'' ≤ ''n'' – 1)}} and there is precisely one {{math|''F''{{sub|''i''}}}} in {{mvar|ψ}} for each {{mvar|i}}, {{math|(-1 ≤ ''i'' ≤ ''n'').}} Since, however, the minimal face {{math|''F''{{sub|–1}}}} and the maximal face {{math|''F''{{sub|''n''}}}} must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called '''improper''' faces.
For example, a flag of a polyhedron comprises one vertex, one edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces.
A polytope is regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.
Two flags are '''{{mvar|j}}-adjacent''' if they only differ by a face of rank {{mvar|j}}. They are '''adjacent''' if they are {{mvar|j}}-adjacent for some value of {{mvar|j}}. Each flag is {{mvar|j}}-adjacent to precisely one flag.<ref>{{harvnb|McMullen|Schulte|2002|loc=pg. 9}}</ref>
==Incidence geometry== In the more abstract setting of incidence geometry, which is a set having a symmetric and reflexive relation called ''incidence'' defined on its elements, a '''flag''' is a set of elements that are mutually incident.<ref>{{harvnb|Beutelspacher|Rosenbaum|1998|loc=pg. 3}}</ref> This level of abstraction generalizes both the polyhedral concept given above as well as the related flag concept from linear algebra.
A flag is ''maximal'' if it is not contained in a larger flag. An incidence geometry (Ω, {{mvar|I}}) has '''rank''' {{mvar|r}} if Ω can be partitioned into sets Ω<sub>1</sub>, Ω<sub>2</sub>, ..., Ω<sub>{{mvar|r}}</sub>, such that each maximal flag of the geometry intersects each of these sets in exactly one element. In this case, the elements of set Ω<sub>{{mvar|j}}</sub> are called elements of '''type {{mvar|j}}'''.
Consequently, in a geometry of rank {{mvar|r}}, each maximal flag has exactly {{mvar|r}} elements.
An incidence geometry of rank 2 is commonly called an ''incidence structure'' with elements of type 1 called points and elements of type 2 called blocks (or lines in some situations).<ref>{{harvnb|Beutelspacher|Rosenbaum|1998|loc=pg. 5}}</ref> More formally, :An incidence structure is a triple '''D''' = (''V'', ''B'', {{mvar|I}}) where ''V'' and ''B'' are any two disjoint sets and {{mvar|I}} is a binary relation between ''V'' and ''B'', that is, {{mvar|I}} ⊆ ''V'' × ''B''. The elements of ''V'' will be called ''points'', those of ''B'' blocks and those of {{mvar|I}} ''flags''.<ref> {{cite book|first1=Thomas|last1=Beth|first2=Dieter|last2=Jungnickel|authorlink2=Dieter Jungnickel|first3=Hanfried|last3=Lenz|authorlink3=Hanfried Lenz|title=Design Theory|publisher=Cambridge University Press|page=15|year=1986}}. 2nd ed. (1999) {{ISBN|978-0-521-44432-3}}</ref>
==See also== * Flag (linear algebra)
==Notes== {{reflist}}
== References == *{{citation | last1 = Beutelspacher | first1 = Albrecht | last2 = Rosenbaum | first2 = Ute | isbn = 0-521-48277-1 | publisher = Cambridge University Press | title = Projective Geometry: From Foundations to Applications | year = 1998}} *{{citation | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | last2 = Schulte | first2 = Egon | isbn = 0-521-81496-0 | publisher = Cambridge University Press | title = Abstract Regular Polytopes | year = 2002}}
Category:Incidence geometry Category:Polygons Category:Polyhedra Category:4-polytopes