Flag (geometry)

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 may be regarded as regular if, and only if, its symmetry group is transitive on its flags.

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.

An incidence geometry (Ω, I) has rank r if Ω can be partitioned into sets Ω1, Ω2, ..., Ωr, such that each maximal flag of the geometry intersects each of these sets in exactly one element.

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).