Expanded Patterns of Phi

Welcome to my video on an expanded pattern of Phi, or the Golden Ratio. I'm going to be showing how this pattern is manifested in a series of parabolas.

Part I shows the basics of this expanded Phi pattern in the parabolas and their respective quadratic equations;

In Part II, I add right triangles to the parabolas, and show how the ratios of their sides are connected in an unexpected way;

Part III shows how the Golden ratio and the Silver ratio are both essential parts of the parabola described by x^2-x-1;

And in Part IV we look at how the Fibonacci series enters into all this. The means of using the Fibonacci numbers to get this expanded Phi pattern is similar to, but not the same as, the way it's used to get the Silver and other Metal ratios.