Poincaré disk model of hyperbolic geometry. Triangle groups (p,q,r) generate regular tilings. Euler characteristic χ = 2-2g determines genus g of the compact surface.
Triangle group (p,q,r): generated by reflections in sides of hyperbolic triangle with angles π/p, π/q, π/r.
Hyperbolic condition: 1/p + 1/q + 1/r < 1
Famous tilings:
(2,3,7): most efficient — 84(g-1) symmetries
(2,4,5): regular {4,5} tiling (squares)
(2,3,∞): modular group PSL(2,Z)
Gauss-Bonnet: Area = π(1 - 1/p - 1/q - 1/r)
χ = 2-2g = V-E+F for compact quotient.