# 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}, *F*_{0}, ..., *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.

## Incidence geometry

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.[1] This level of abstraction generalizes both the polyhedral concept given above as well as the related flag concept from linear algebra.

A flag is *maximal* if it is not contained in a larger flag. An incidence geometry (Ω, I) has **rank** r if Ω can be partitioned into sets Ω_{1}, Ω_{2}, ..., Ω_{r}, such that each maximal flag of the geometry intersects each of these sets in exactly one element. In this case, the elements of set Ω_{j} are called elements of **type j**.

Consequently, in a geometry of rank r, each maximal flag has exactly r elements.

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).[2] More formally,

- An incidence structure is a triple
**D**= (*V*,*B*, I) where*V*and*B*are any two disjoint sets and I is a binary relation between*V*and*B*, that is, I ⊆*V*×*B*. The elements of*V*will be called*points*, those of*B*blocks and those of I*flags*.[3]

## Notes

- Beutelspacher & Rosenbaum 1998, pg. 3
- Beutelspacher & Rosenbaum 1998, pg. 5
- Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986).
*Design Theory*. Cambridge University Press. p. 15.. 2nd ed. (1999) ISBN 978-0-521-44432-3

## References

- Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
*Projective Geometry: from foundations to applications*, Cambridge: Cambridge University Press, ISBN 0-521-48277-1 - Peter R. Cromwell,
*Polyhedra*, Cambridge University Press 1997, ISBN 0-521-55432-2 - Peter McMullen, Egon Schulte,
*Abstract Regular Polytopes*, Cambridge University Press, 2002. ISBN 0-521-81496-0