⭕ Ford circles

What you’re seeing

Every fraction $\tfrac{p}{q}$ between 0 and 1, written in lowest terms, gets a circle. The circle sits on the number line at $\tfrac{p}{q}$, and its radius is $\tfrac{1}{2q^2}$. So $\tfrac12$ gets a big circle, $\tfrac13$ and $\tfrac23$ smaller ones, and fractions with big bottoms get tiny ones. Lester R. Ford described these circles in 1938.

Look closely: no two circles overlap. Many of them touch. And in the curved gap between two touching circles there is always a smaller circle that touches both. Zoom in on any gap, and the same pattern of circles and gaps appears again, as far as you care to look.

Why it works

When do two circles touch? Take the circles for $\tfrac{p}{q}$ and $\tfrac{r}{s}$. Their centers are at $\left(\tfrac{p}{q}, \tfrac{1}{2q^2}\right)$ and $\left(\tfrac{r}{s}, \tfrac{1}{2s^2}\right)$. Work out the square of the distance between the centers and subtract the square of the sum of the radii. Everything cancels except

$$\frac{(ps - qr)^2 - 1}{q^2 s^2}.$$

So the circles touch when $|ps - qr| = 1$, and they are apart when $|ps - qr| \ge 2$. Because the fractions are different, $ps - qr$ is never 0, so the circles can never overlap.

The circle in the gap. If $\tfrac{p}{q}$ and $\tfrac{r}{s}$ touch, try the fraction $\tfrac{p + r}{q + s}$ (add the tops and add the bottoms, the mediant). Check: $p(q + s) - q(p + r) = ps - qr = \pm 1$, and the same for the other side. So the new circle touches both. This is why the gaps never run out, and why the picture repeats at every zoom.

Falling toward a number. Draw a straight line down to a number, such as $\pi - 3 = 0.14159\ldots$ The circles the line passes through are some of the best fractions for that number: its continued-fraction convergents, $\tfrac17$ and $\tfrac{16}{113}$ (that is $\tfrac{22}{7}$ and $\tfrac{355}{113}$ for pi). A circle is pierced exactly when the fraction is within $\tfrac{1}{2q^2}$ of the number, and Legendre proved that any fraction that close must be a convergent.

Try this

• Find the circle that touches both $\tfrac12$ and $\tfrac13$. Check your answer with the mediant rule.
• Zoom in toward pi. Which circles does the dashed line go through? Which convergent does it skip?
• Choose the golden ratio. Its convergents are ratios of Fibonacci numbers, and after $\tfrac01$ the line goes through the circle of every one of them.
• The circles of $\tfrac01$ and $\tfrac11$ have radius $\tfrac12$. How big is the circle of $\tfrac{1}{10}$ compared with them?