Flag (geometry)

For a comparable but different concept of linear algebra, see flag manifold.

Face diagram of a square pyramid showing one of its flags

In geometry, a flag is a sequence of faces of a polytope, each contained in the next, with just 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 flag of a polyhedron is sometimes called a "dart".

A polytope may be regarded as regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.

References

• Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
• Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 9-521-55432-2