Coxeters

Coxeters refers to a family of mathematical concepts and structures named after the British mathematician Harold Scott MacDonald Coxeter (1907–2003), who made significant contributions to geometry, group theory, and combinatorics. The term is most commonly associated with Coxeter groups, which are algebraic structures that generalize the symmetries of regular polygons and polyhedra.

A Coxeter group is defined by a set of generators and a set of relations, often represented as a Coxeter graph. These groups are fundamental in the study of reflection groups and have applications in various areas of mathematics, including Lie theory, root systems, and the classification of finite simple groups. Coxeter groups can be finite, affine, or hyperbolic, depending on their properties.

Coxeter also introduced the concept of Coxeter complexes, which are geometric realizations of Coxeter groups in hyperbolic space. These structures help visualize the symmetries and relationships within the group. His work on regular polytopes, including the discovery of new higher-dimensional shapes, expanded the understanding of geometric configurations beyond three dimensions.

Beyond pure mathematics, Coxeter’s ideas have influenced fields such as crystallography, theoretical physics, and computer science, particularly in the study of tilings and symmetry in patterns. His collaborations with other mathematicians, including T. G. Room and J. A. Todd, further expanded the reach of these concepts.

Harold Coxeter’s contributions remain influential in modern mathematics, and his name is often associated with foundational work in symmetry, group theory, and geometric structures. His book *Regular Polytopes* (co-authored with W. O. J. Moser) remains a seminal reference in the field.