Flag (geometry)

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 ψ of an n-polytope is a set {F_–1, F₀, ..., F_n} such that F_i ≤ F_i+1 (–1 ≤ i ≤ n – 1) and there is precisely one F_i in ψ for each i, (–1 ≤ i ≤ n). Since, however, the minimal face F_–1 and the maximal face F_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 may be regarded as regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.