In (polyhedral)

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 coor ...

. 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 (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

comprises one

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

, 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 b ...

incident to that vertex, and one

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

al face incident to both, plus the two improper faces. A polytope may be regarded as 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 Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

polytopes.

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

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

* * Peter R. Cromwell, ''Polyhedra'', Cambridge University Press 1997, *

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

, Egon Schulte, ''Abstract Regular Polytopes'', Cambridge University Press, 2002. {{ISBN, 0-521-81496-0 Incidence geometry Polygons Polyhedra 4-polytopes