Hyperbolic Tiling — Poincaré Disk

The Poincaré disk model: hyperbolic straight lines are circular arcs meeting the boundary at right angles. A {p,q} tiling has p-gons meeting q at each vertex; requires (p-2)(q-2)>4 for hyperbolic geometry. Generated by a Fuchsian group — a discrete subgroup of PSL(2,ℝ) acting by Möbius transformations z↦(az+b)/(c̄z+ā) with |a|²−|b|²=1. Colors encode generation depth.