The basic idea here is that it's very easy to control holonomy, the excess turning as you circuit a region of a surface, a precise measure of the total curvature within.

In addition, the intrinsic curvature of a path, that is, the amount an insect would feel like is it turning as it walks along a path, gives the precise recipe for how to cut out strips of paper or steel, or any material for which the local curvature must remain flat. By using thin strips, we can build up surfaces with precisely controlled gaussian curvature.