A Surprise Visitor

Create two columns, one starting with the numbers 12 and 18 and the other with 5 and 5. Continue each column, deriving each new number by adding the two that precede it:

JRM pi ratios

In the Journal of Recreational Mathematics, James Davis writes, “Forming successive pairs with adjoining numbers from each column one finds the ratio of the two numbers in each pair converges to π!” How can this be?

“The alert reader will suspect there is a trick in this method, as I did when π first presented it to me. The labor of several hours of computation coupled with trial and error produced half of the secret of the method. It is obviously based somehow on the fact that φ (the golden mean, which equals \frac{1+\sqrt{5}}{2}, can be closely approximated by the nifty pseudo equation below:”

1.2 \times \varphi ^{2}=\pi

“Can the reader decipher π’s technique for making herself with φ?”