How do we know the hyperbolic plane is 2 dimensional?

Question

Consider a disk in the hyperbolic plane with radius $R$, the area of this circle is given by $2 \pi \sinh(R) = 2\pi \frac{e^{R} - e^{-R}}{2}$

Usually an argument that the disk is 2-dimensional goes by observing that area of a circle quadruples when the radius is doubled.

In the hyperbolic case the area FAR MORE than doubles when the radius is doubled. In fact it cannot even be polynomially bounded. It would appear our notions of dimension depend in a fundamental way on the curvature of the ambient space we are in.

So the question: how can we still conclude the hyperbolic disk is 2-dimensional?

Answer 1

There are many notions of dimension, and you are confusing a couple of them.