We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…

Given is a proof on the classification of surfaces that involves some simple graph theory. It could serve as an introduction to some methods of modern differential topology. (MNS)

Outlines a guided discovery activity for ascertaining the conditions which allow for the correct tracing of networks. (JP)