Hyperbolic Tessellation — Poincaré Disk

Poincaré Disk Model (Poincaré, 1882): The unit disk D = {|z|<1} models the hyperbolic plane with metric ds² = 4|dz|²/(1−|z|²)². Geodesics are circular arcs orthogonal to the boundary circle. A {p,q} tessellation tiles the disk with regular p-gons, q meeting at each vertex. It exists in hyperbolic geometry when (p−2)(q−2) > 4. The symmetry group is generated by reflections in tile edges — a Fuchsian group Γ ⊂ PSL(2,ℝ). Möbius transformations z↦(az+b)/(b̄z+ā) with |a|²−|b|²=1 are the orientation-preserving isometries. M.C. Escher's "Circle Limit" series used {6,4} and {4,6} tilings.