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[[File:Cube flag.png|200px|thumb|One of the 48 flags of the [[cube]] is highlighted.]]
A '''flag''' is a subset of a [[polytope]]'s [[element]]s that contains exactly one element of each [[rank]], such that any two elements in it are incident. That is to say, any flag must contain a [[vertex]] within an [[edge]]... within a [[facet]]. It must also include the [[improper element]]s of the polytope.

More formally, a flag is a maximal chain in the [[element lattice]]. Here, a '''chain''' is defined as a set of pairwise incident elements, and "maximal" means that there's no other chain that contains it as a strict subset.

Flags are central to the study of [[regular polytope]]s. They are also useful for computing certain properties such as [[volume]] or [[orientability]], or for carrying out certain constructions of highly symmetric polytopes such as [[truncate]]s. They even lead to further generalizations of polytopes, including [[complex]]es.

It is sometimes useful to think of flags as [[simplices]], whose vertices are given by points on each of the elements they contain. Doing this relates flags of regular polytopes to [[reflection group]]s, and allows one to treat a polytope as a [[simplicial complex]].

== Flag changes ==
[[File:Cube flag changes.png|thumb|300px|A map of all possible flag changes on a flag of a [[cube]].]]

An ''n''-polytope is [[dyadic]] if and only if, for every flag and every rank 0 ≤ ''j'' ≤ ''n'' &minus;1, there exists a unique flag that shares all elements but that of rank ''r''. This flag is said to be '''''j''-adjacent''' to the first.<ref>{{ARP|page=9}}</ref>

The operations on the set of flags that change a flag to its ''j''-indicent flag don't have a standard name. They're sometimes called '''flag changes''' within the amateur community.

Generally, flag changes don't [[w:Group action|act]] as a [[group]] on the set of flags. They do however act as a group on the set of permutations of flags – the group they generate is called the [[monodromy group]] of the polytope. A polytope is [[regular]] if and only if its monodromy group coincides with its [[automorphism group]].{{Citation needed}}

One may build, from any polytope, a colored graph where the vertices are the flags of the polytope, and an edge of color ''j'' connects two flags whenever they're ''j''-adjacent. This is known as the [[flag graph]] of the polytope. Flag graphs motivate a generalization of a polytope called a [[complex]].

== Flags as simplices ==
[[File:Cube flag simplex.png|thumb|200px|The [[simplex]] representing the [[cube]] flag shown previously.]]

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To every element of a [[planar]] polytope other than the least element, one may associate a point within the subspace of the element, often at the {{w|centroid}}. This process is known as [[barycentric division]].

== References ==
{{Reflist}}