Fuchsian groups

Fuchsian groups are mathematical structures consisting of transformations that preserve the geometry of the hyperbolic plane, a shape with constant negative curvature. They are made up of special functions called Möbius transformations that map the hyperbolic plane onto itself, often in a repeating pattern. These groups help mathematicians understand the symmetries and tessellations of hyperbolic surfaces, which are fundamentally different from flat or spherical shapes. Fuchsian groups play a key role in fields like geometry, complex analysis, and number theory, providing a framework to study intricate patterns and the properties of hyperbolic spaces.