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 The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

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 coord ...

. 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 o ...

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 ...

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 ...

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 ''inciden ...

, 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 empl ...

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