Geodesics, tessellations, and triangle angle sums less than π in constant negative curvature
In the Poincaré disk model, "straight lines" (geodesics) are arcs of circles orthogonal to the boundary. Triangle angle sums are always less than π, with the deficit equal to the area (Gauss-Bonnet). The {p,q} tessellation tiles the hyperbolic plane with regular p-gons, q meeting at each vertex — valid when 1/p + 1/q < 1/2.