Regular tilings of the hyperbolic plane in the Poincaré disk model — infinitely many tiles fit in a bounded disk.
The Poincaré disk model maps the entire hyperbolic plane into a unit disk. Geodesics are circular arcs perpendicular to the boundary. Triangle group (p,q,r) tessellates when 1/p+1/q+1/r < 1. The {7,3} tiling (Klein quartic) tiles the hyperbolic plane with 168-fold symmetry — the maximum for a surface of genus 3.