The connection of Fibonacci’s sequence to the Golden Ratio.
Expand φ as a continued fraction, and find the successive partial sums. (Previous page.) Both the numerators and the denominators follow the same, famous sequence named after Fibonacci, or Leonardo of Pisa: 1,1,2,3,5,8,13,21,34... And the numerators are one step ahead of the denominators: 2/1, 3/2, 5/3, 8/5... What’s up with this?
Fibonacci’s sequence is defined this way. Start with 0 and 1. The next term is the sum of the previous two: that is, 1. So now we have 0,1,1. Keep doing that. You next have 0,1,1,2. Then 0,1,1,2,3. Et cetera. So why do these numbers crop up in the Golden Ratio? It all comes from the simple way that fractions add.
Here you can watch the evolution of the Fibonacci sequence. Each time you click the button, you add the last two numbers find the next value. To start over, click the button.
This shows how the Fibonacci sequence is constructed. In the next page, we will concentrate on the operation that takes you from the current trailing pair of numbers to the next trailing pair. Why? It turns out to be a lot like the operation you perform in adding up a continued fraction.