Affine symmetric group

Affine symmetric group.
The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane (pictured), as well as related higher-dimensional objects. They may also be defined as collections of permutations (rearrangements) of the integers that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in the fields of combinatorics and representation theory. Each of these groups is an infinite extension of a finite symmetric group, and many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups.