Incompatibilities of Algebra and Geometry

Jennifer Batel, RHS ‘08

My topic discusses the incompatibilities of algebra and geometry. Often these mathematical concepts are used to solve problems. However, my paper discusses several circumstances in which there are incompatibilities. In a situation where there is an incompatibility, the solutions obtained algebraically and geometrically in a problem, contradict each other. Specifically, I analyzed problems involving regular polygons, regular and semi-regular tessellations, a problem called the trapezoid fallacy, and another problem involving circles. For example, an incompatibility occurred while attempting to create a tessellation. In this problem, three polygons were able to form a single vertex algebraically, however, geometrically the pattern was unable to be repeated. Further expansion on this topic led to the derivation of a formula that had been used to discover one of the incompatibilities. I also created my own problem involving regular polygons which revealed yet another incompatibility. This topic allowed me to analyze and compare both algebra and geometry as well as explore other areas of mathematics.