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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).