DFORMS AND KNIGHT'S VISOR 
Duc Nguyen '07 
Summer research 2005 
Advisor: Prof. Joe O'Rourke 
I worked with Stephanie Jakus and Professor O’Rourke on Dforms which are convex 3D objects constructed as follows. From two identical racetrack shaped pieces of paper, we picked a random point on the edge of each. Then we glued these 2 points together, and “zipped” close the remainder of the racetrack perimeters. The creased edge of this newly formed object is a closed curve in 3D that we have been studying. It is in general an unsolved problem to compute the shape of a Dform curve in space. The figure below (as shown to the right) shows the simplest Dform possible, where (as shown to the left) the midpoint of a circular arc of one racetrack is glued to the midpoint of one straight segment of the other racetrack. We partitioned this curve into three parts: straight touching straight, curve touching straight, and curve touching curve. Using the definition of Gaussian curvature as angle deficit, we proved that the Gaussian curvature for straightstraight parts is 0. Because Gaussian curvature is intrinsic, we used a Dform that had no curvecurve part to study the curvestraight parts. We knew that integral of Gaussian curvature over the whole curve is 4 p by the GaussBonnet theorem, and since there are four identical straightcurve parts, we concluded that the curvature for each point on those parts must be 1. Using the same logic combining with the notions of angle deficits, we also showed that the curvature at any point on the curvecurve parts must be 2. Thus we now know the distribution of Gaussian curvature over the whole surface, for it must be zero at every point not on the curve. The straightstraight parts on Dforms lead us to question: if we glue two rectangular pieces of paper together along a common edge, does that edge have to be a straight line in space? We proved that that was the case when the tangent planes of the two surfaces of any point on that edge do not coincide. (Supported by the National Science Foundation.)

dnguyen2@email.smith.edu 
Other member of Dforms group: Stephanie Jakus 