Coxeter groups

Coxeter groups are mathematical structures that describe symmetrical patterns generated by reflections, like mirrors. Imagine reflecting a shape across lines or planes—these reflections, combined in certain ways, create groups that encode the symmetries of regular geometric objects such as polygons and polyhedra. They can be represented using diagrams called Coxeter diagrams, which specify the angles between reflection lines. These groups help mathematicians understand the symmetrical properties of diverse objects and spaces, including crystals, tilings, and even certain algebraic structures, providing a unified way to study symmetry in a rigorous mathematical framework.