Knowra H. S. M. Coxeter H. S. M. Coxeter Harold Scott MacDonald Coxeter (1907–2003) was a British-Canadian geometer whose work shaped the study of regular polytopes, reflection groups, and geometric symmetry.
Regular polytope : A polytope whose symmetries act transitively on its flags, generalizing regular polygons and polyhedra. Coxeter made regular polytopes a central subject, extending their study beyond ordinary three-dimensional figures.
Uniform polyhedron : A polyhedron whose faces are regular polygons and whose vertices are equivalent under its symmetries. Coxeter’s symmetry methods helped systematize the classification of these highly symmetric solids.
Donald Coxeter : The mathematician Harold Scott MacDonald Coxeter, commonly known by his initials H. S. M. Coxeter. His publications and mathematical legacy established the name Coxeter in modern geometry.
Regular polytopes in four dimensions : The six convex regular polytopes that exist in four-dimensional Euclidean space. Their classification illustrates how Coxeter extended familiar three-dimensional geometry into higher dimensions.
Reflection group : A group generated by reflections across hyperplanes or, in two dimensions, lines. Coxeter groups describe the symmetries behind many of the regular figures he investigated.
Regular tessellation : A tiling of a space by congruent regular polytopes, with the same arrangement around every vertex. Coxeter analyzed regular tessellations in spherical, Euclidean, and hyperbolic geometry.
Harold Scott MacDonald Coxeter : The full name of the geometer known professionally as H. S. M. Coxeter. This is Coxeter’s expanded name, distinct from the abbreviated form used in most mathematical writing.
Icosahedral symmetry : The symmetry group of the regular icosahedron, a finite group of order 120 including reflections. Coxeter’s treatment of the icosahedron connects classical geometry with finite reflection groups.
Coxeter group : A group defined by generators that are involutions and relations specifying the orders of their pairwise products. Coxeter developed the systematic notation and theory for these reflection-generated groups.
Crystallographic group : A discrete group of Euclidean isometries compatible with a lattice of translations. Reflection groups and symmetry classifications connect Coxeter’s work to the mathematics of crystal patterns.
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