003251-0039 1 Proving the parametric equations, volume, and surface area of the torus in comparison to the given formulasIntroduction:My attendance at a conference concerning the rise and fall of the stock markets piqued my interest in the area of topology. The speaker of the conference posed the question of the three-dot puzzle (“Can you connect all nine dots using 4 lines?”). Of course, my brother and I having solved and seen the puzzle easily produced the solution shown in Figure 1. However, this started an interesting conversation pertaining to the ambiguity of the question and whether the puzzle could be solved in fewer moves. Over the days, my brother and I started proposing a three- or two-line solution, but suddenly, I had a eureka moment. I proposed using a Mobius Strip and drawing a single continuous line which would intersect all three points after two loops (depicted in Appendix. I; same concept as the sphere shown, however around the Mobius strip). To solidify my theory, I conversed with one of my distant relatives who was a professor of Mathematics and had a PhD in Topology. He replied that it was a genuine answer to the problem and started an interesting discussion pertaining to the Mobius Strip and other topological figures. What interested me the most, however, was not the model of the figure itself but the properties, and how they are formed from an initial plane. The differences originate from the way the vertices on the plane are manipulated: either the create an internal closed path, or mirrored path; or, the opposite in which a path where the points are incongruent over the axes. The construction of these sorts of planes are called nonorientable and orientablesurfaces Figure 1

003251-0039 2 respectively (Hedegaard -Orientable). The figures are also given an equation with a general form, which are generally parametric due given their given three-dimensional property; however, the third dimension is produced by integrating over the y-, z- or r- axes in polar coordinates. Thus, I decided to deconstruct a topological structure, the torus, into its 2-dimension geometric forms, and try to proof the given parametric, volume, and surface area equations. Modeling the TorusThe topological structure which I will try to model is the torus. The torus is a topological figure with an orientable surface. However, the most distinguishing element to this topographical structure is that there three different classifications for a torus: a ring torus, a horn torus, and a spindle torus which I have depicted below using Wolfram Mathematica’s 3D Parametric Equation Plotter. The tori have the same general parametric equation form: ? = (? + ? cos ?) cos ?? = (? + ? cos ?) sin ?? = ? sin ?If ? > ?, then it is a ring; ? = ?, then it is a horn; Ring torusHorn torus Spindle Torus

003251-0039 3 ? < ?, then it is a spindle.