A flag is a subset of a polytope's elements 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 elements 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 polytopes. 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 truncates. They even lead to further generalizations of polytopes, including complexes.
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 groups, and allows one to treat a polytope as a simplicial complex.
An n-polytope is dyadic if and only if, for every flag and every rank 0 ≤ j ≤ n −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.
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 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.
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
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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 centroid. This process is known as barycentric division.
- ↑ McMullen, Peter; Schulte, Egon (2002). Abstract Regular Polytopes. Cambridge University Press. p. 9.