Golden Ratios, Golden Rectangles

And here we are at the brow chakra, the tippy-top! Roger sends us to the book again, to follow the adventures of Fibonacci, the early Renaissance Italian mathematician. While a lot of his fame comes from his investigations into the recovered Greek/Roman/Arabian literature emerging from Grenada and other cultural centers, his insight into the Fibonacci series seems to be his own advance (there was some arcane knowledge of the series in India from way back, but Fibonacci's 'breeding bunnies' example is much clearer).

While Fibonacci explicitly set the series out to 233, he did not note how the successive series pairs would form a ratio that converged on a 'golden ratio' - but we can easily watch the series convergence here:

Notice how quick the ratio converges! - each step brings the percentage closer by more than half, and each succeeding step nicely 'rings' around the convergent value, first over, then under the projected goal.

Fibonacci did not relate the ratio to sides of a golden rectangle, either - but that idea came soon enough.

Roger poses this next question. I have not found his answer in any media about Fibonacci or phi - I had to guess. Can anyone find this documented anywhere?

We will return for the answer next time.