Few months ago I gave a lecture on Non-euclidean geometry and it was a bit difficult for me to give audience an example of hyperbolic surface from their day-to-day life. While reading Donal O’ Shea’s book on Poincaré Conjecture I came across following interesting example on pp. 97 :
Estrogen causes fat to be stored in the buttocks, thighs, and hips in women. Thus females generally have relatively narrow waists and large buttocks, and this along with wide hips make for a wider hip section and a lower waist-hip ratio compared to men. The saddle-shaped area on a woman’s side above her hip has negative curvature.
One can imagine cloth (it is flexible but does not stretch, hence an isometry) that would drape it perfectly. Here the region inside a circle of given radius contains more material than the same circle on the plane, and to make the cloth the tailor might start with a flat piece of fabric, make a cut as if he/she were going to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric or a gusset. Negatively curved cloth would have lots of folds if one tried to lay it flat in dresser.
If one tries to extend a cloth with constant positive curvature (like a cap), in all directions, it would close up, making a sphere. On the other hand, if one imagines extending a cloth with constant negative curvature in all directions, one gets a surface called hyperbolic plane.