Poincaré disk model — exponential growth of hyperbolic space
The Poincaré disk is a conformal model of the hyperbolic plane H² with constant negative curvature K < 0. The number of nodes at radius r grows like e^{√|K|·r}, in stark contrast to Euclidean space where growth is polynomial. The Cayley tree of a group with k generators embeds isometrically in H² — a key insight for network models and AdS/CFT duality.