Joseph Martin, Susan Phillips, Alejandra Marquez, Bennedy Ferrer, Fatima Cabanas, & Dr. Jorge Garcia
Whereas there have been analytical proofs for Apery’s constant (the sum of the reciprocals of cubes), there have been no geometrical proofs. In our search for such a proof, we constructed a new type of triangular spiral and associated it with the Apery’s constant equation. Through an infinite process, we stacked right-triangles with specific values on top of each other, obtaining a growing spiral. We extended this construction to many series of similar form, namely, real values of the Riemann-Zeta function. There is a well known closed form for the Riemann-Zeta function values at even integers, but not at odd integers. The spirals we constructed apply to both even and odd integers. We investigated the properties of these spirals through drawings and computer programming. In addition to spirals, we built interesting angles associated with these Riemann-Zeta values. Our work may contribute to the pursuit of a closed form expression of Apery’s constant, which has been a dream of many mathematicians for nearly three centuries.