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.

One of the 48 flags of the cube is highlighted.

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.

A symmetry of a polytope, other than the identity, can never map a flag to itself. Symmetries can be thought of as dividing an element into parts; an element's mirror symmetry divides it into two symmetric halves, rotational symmetry divides a part into n  distinct segments. Thus unlike edges, faces etc. flags cannot be meaningfully divided by the symmetry. From the perspective of symmetry, flags are the atomic building blocks of polytopes. This makes flags a useful notion for many things:

  • Flags are central to the study of regular polytopes.
  • They can be used to carry out certain constructions of highly symmetric polytopes from symmetry groups, in the Wythoffian construction.
  • They can be used for calculations of volume.
  • They are useful for the concept of orientability.
  • They can 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.

Flag changes Edit

 
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  , there exists a unique flag that shares all elements but that of rank j . This flag is said to be j -adjacent to the first.[1]

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.[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 Edit

 
The simplex representing the cube flag shown previously.

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.

References Edit

  1. McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes. Cambridge University Press. p. 9. ISBN 0-521-81496-0.