The heart and soul of elliptic, flat (Euclidean), and hyperbolic spaces

... is constant, polynomial, and exponential.

Think about building cellphone towers in a Euclidean plane. First, we choose the central station arbitrarily at point $O$, and call it the "origin". To cover all space that's within a distance $R$ to the origin, we need $O(R^2)$ stations.

This is polynomial growth.

Then think about building cellphone towers on a spherical surface, on a little asteroid. Then it's clear that there's only a constant number of towers needed no matter how big $R$ is chosen, since the whole surface has finite area. So we need $O(1)$ stations.

This is constant "growth".

Then, if you know how to play in hyperbolic space (if not, go play Hyperrogue!), you'd know that the number of stations grows as $O(\exp{(CR)})$, where $C=(-K)^{-1/2}$, and $K$ is the Gaussian curvature of the plane. This can be derived by using the formula of the area of a hyperbolic circle of radius $R$:
$$\frac{4\pi}{-K} \sinh^2 \frac{r}{2\sqrt{-K}} $$

We normalize $K = -1$ to get the growth rate $O(\exp{R})$.