Flag (geometry)
Encyclopedia
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, ..., Fn} such that Fi ≤ Fi+1 (−1 ≤ i ≤ n − 1) and there is precisely one Fi in ψ for each i, (−1 ≤ i ≤ n). Since, however, the minimal face F−1 and the maximal face Fn 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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a flag is a sequence of faces of a polytope
Abstract polytope
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
, 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, ..., Fn} such that Fi ≤ Fi+1 (−1 ≤ i ≤ n − 1) and there is precisely one Fi in ψ for each i, (−1 ≤ i ≤ n). Since, however, the minimal face F−1 and the maximal face Fn 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
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
is transitive on its flags. This definition excludes chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
polytopes.