In (polyhedral)

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a flag is a sequence of faces of a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

, each contained in the next, with exactly one face from each

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

. More formally, a flag of an -polytope is a set such that and there is precisely one in for each , Since, however, the minimal face and the maximal face 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 In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

comprises one vertex, one

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

incident to that vertex, and one

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

al face incident to both, plus the two improper faces. A polytope is regular if, and only if, its

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

is transitive on its flags. This definition excludes

chiral polytope In the study of abstract polytopes, a chiral polytope is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms of the Group action (mathematics), action of the symmetry group of the polytope on its Flag (g ...

s. Two flags are -adjacent if they only differ by a face of rank . They are adjacent if they are -adjacent for some value of . Each flag is -adjacent to precisely one flag.

Incidence geometry

In the more abstract setting of

incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...

, which is a set having a symmetric and reflexive

relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...

called ''incidence'' defined on its elements, a flag is a set of elements that are mutually incident. This level of abstraction generalizes both the polyhedral concept given above as well as the related

flag A flag is a piece of textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...

concept from linear algebra. A flag is ''maximal'' if it is not contained in a larger flag. An incidence geometry (Ω, ) has rank if Ω can be partitioned into sets Ω₁, Ω₂, ..., Ω, 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 are called elements of type . Consequently, in a geometry of rank , each maximal flag has exactly 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). More formally, :An incidence structure is a triple D = (''V'', ''B'', ) where ''V'' and ''B'' are any two disjoint sets and is a binary relation between ''V'' and ''B'', that is, ⊆ ''V'' × ''B''. The elements of ''V'' will be called ''points'', those of ''B'' blocks and those of ''flags''. . 2nd ed. (1999)

Notes

References

* *{{citation , last1 = McMullen , first1 = Peter , author1-link = Peter McMullen , last2 = Schulte , first2 = Egon , isbn = 0-521-81496-0 , publisher = Cambridge University Press , title = Abstract Regular Polytopes , year = 2002 Incidence geometry Polygons Polyhedra 4-polytopes