From David Henderson: The act of constructing the surface will give you a feel for hyperbolic planes that is difficult to get any other way.... A paper model of the hyperbolic plane may be constructed as follows: Cut out many identical annular ("annulus" is the region between two concentric circles) strips as in Figure 5.2 [see below]. Attach the strips together by taping the inner circle of one to the outer circle of the other. It is crucial that all the annular strips have the same inner radius and the same outer radius, but the lengths of the annular strips do not matter. You can also cut an annular strip shorter or extend an annular strip by taping two strips together along their straight ends. The resulting surface is of course only an approximation of the desired surface. The actual hyperbolic plane is obtained by letting δ -> 0 while holding the radius ρ fixed.
Figure 5.2
When you tape enough annuli together, it will be clear to you that the geometry is not that of flat annuli [just like the strake, the curvature of the outer helix, especially when we went to the auger, showed us that we had a different geometry than a flat annulus]. The more annuli you put on, the more you will ruffle at the edges, until the ruffles become unmanageable. Tape together enough of the annuli so that you get a ruffled surface.

Annular Hyperboli Planes by Daina Taimina
Annular Strips