The Fibonacci sequence and the golden quadratic

Jose C. Iniguez, B. Argentina Iniguez

The fact that the golden mean ($\Phi=1.61803\dots$) appears both as the limit of the ratio of consecutive Fibonacci numbers, as well as one of the solutions of the golden quadratic, prompted us to conduct a graphical analysis of this equation in order to ascertain what kind of connection its geometry has with the Fibonacci sequence. Our results indicate that the following are all subsumed by the geometry of this equation: the Fibonacci sequence, a sequence of powers of $\Phi$, Division in Extreme and Mean Ratio, $\Phi$, as well as the golden rectangle.