Paper Models of Surfaces with Curvature

Howard Iseri

A model of a cone can be made from a piece of paper by removing a wedge and taping the edges together. I will discuss paper models that expand on this basic idea. Considering the "cone-points" formed in these paper models, it is relatively easy to see how the curvature of a surface affects the geometry of lines (or geodesics) on the surface. Elliptic and hyperbolic effects are quantifiable using basic geometry, and the Gauss-Bonnet theorem becomes intuitively clear after a few examples.

I will discuss the construction of the models, which consists of removing wedges to form "elliptic" cone-points and adding wedges to form "hyperbolic" cone-points. The models are flat everywhere, except at the cone-points, so the geodesics are locally straight lines in the natural sense. I define impulse curvature to be equal to the angle of the wedge "removed."

In the models, it easy to see, for example, that a triangle containing one or more cone-points will have an angle sum different from 180 degrees, and the difference is related precisely to the sum of the impulse curvatures within the triangle. Other examples include a point that lies on no parallels to a given line (elliptic geometry), and a point that lies on multiple parallels to a given line (hyperbolic geometry).