Reflection groups are mathematical structures that consist of a set of reflections across hyperplanes in a vector space. These reflections can be combined to generate symmetries, forming a group that captures the geometric properties of the space. Reflection groups are important in various fields, including geometry, algebra, and physics, as they help describe symmetrical patterns. One of the most well-known types of reflection groups is the Coxeter group, which is defined by a set of generators and relations. Reflection groups can also be classified into finite and infinite types, with finite groups often associated with regular polytopes and root systems in Lie algebra theory.