A New Approach

These last couple of days I have looked at the Fibonacci spiral more in depth by taking it apart and separating the quarter circles that make it up.  

I wrote their circumferences in terms of one another to then be able to write their trigonometric functions in terms of one another, all the while using Binet's formula from past posts. When trying to put it back together, though, this is what I got.

In this image, the angle I am trying to work with appears all over the place, and will continue to do so as the spiral continues. So instead, what I did was stack the circles on top of each other while switching the coordinate axis from sinx to cosx and vice versa, after every circle. The result came as follows:

Having this new view of the Fibonacci spiral and using the properties of similar triangles, as well as Thales’ theorem (also known as the intercept theorem), will hopefully open a doorway to more discoveries (or inventions, as you wish).