In my last post I talked about using Farey numbers as coordinates for linked list elements. Here’s what that looks like. Click a link to insert a new element into the list.

### Rationalizing the Irrational

We can use the Farey Linked List to find rational approximations of irrational numbers between zero and one. If we want to approximate a number greater than one, first we approximate the reciprocal of that number, then we take the reciprocal of that approximation. Let’s do $$\displaystyle\pi.$$ $$\displaystyle\pi$$ is greater than one, so first we take its reciprocal which is ~$$0.318$$. The empty Farey Linked List has one edge that spans $$\displaystyle0-1.$$ $$0.318$$ is in that range, so we’ll insert a node there. Now we have two edges. The first edge spans $$0-\frac{1}{2}$$, and the second edge spans $$\frac{1}{2}-1.$$ $$0.318$$ is in the first range, so we insert there. We can continue inserting until our approximation is good enough. Not every insertion we do will give us a better approximation immediately, but it will take us in the correct direction. The process looks like this (click the approximate button below).

$$\displaystyle\pi =$$0

That might take awhile, let’s do something else in the meantime. Here’s $$\displaystyle\phi.$$

$$\displaystyle\phi =$$0

Watch both closely. $$\displaystyle\pi$$ takes long runs in the same direction, while $$\displaystyle\phi$$ alternates directions every step. Alternating the direction the way $$\displaystyle\phi$$ does turns out to be the fastest way to make the numerator and denominator of the Farey Numbers grow. This is one manifestation of $$\displaystyle\phi's$$ reputation as “The Most Irrational Number”.