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, ..., Fn} such that Fi ≤ Fi+1 (-1 ≤ i ≤ n – 1) and there is precisely one Fi in ψ for each i, (-1 ≤ i ≤ n). Since, however, the minimal face F–1 and the maximal face Fn 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 is regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.
Two flags are j-adjacent if they only differ by a face of rank j. They are adjacent if they are j-adjacent for some value of j. Each flag is j-adjacent to precisely one flag.