On the irrationality of π, √2 and φ

A small interactive essay inspired by Numberphile

You are a flower and you want to grow your seeds as efficiently as possible around your center. How do you go about doing that?

But first, you don't even exist yet. Let's change that!

Or just skip the explanations and play around, if you so desire

There you are! How about you now grow some seeds?

That's not very efficient. Look at all that wasted space! Here's what we are going to do. You grow a seed, move foreward a bit, turn a third (of a full turn), and then grow your second seed.

What happens when you change how much you turn?


I'll use the terms "angle" and "turn" to refer to how much you turn compared to a full 360 degree turn. So turning 0.25 means turning by half a full turn.

That's nice, but we want to stop these spokes from appearing. What happens when we use a numerator on our fraction of a full turn?

Notice how turning \(\frac{1}{2} = 0.5\) is equivalent to turning \(\frac{3}{2} = 1.5\), because turning \(1\) gets you back to where you started... Basically only the decimals matter.

Well that didn't go as expected... Whenever we turn a fraction each time, we always get theses spokes appearing!

Introducing irrational numbers! Irrational numbers are numbers that can never be expressed as a fraction of two integers.

For instance, \(\frac{5}{100} = 0.05\) is a rational number. \(\frac{2}{3} = 0.\overline{6}\) is a rational number. But \(\sqrt{2}\) can't be expressed as a fraction of two integers, so it's irrational. What happens when we turn an irrational amount each step?

Looking at \(\pi\), we still see curved spokes. Why? Close to the center, you may notice that \(7\) spokes have formed. That comes from the fact that \(\pi\) is well approximated by \(\frac{22}{7} \approx 3.14\)!

You might want to try out your own numbers...

Please pause the animation before entering a new angle

What did we learn? The better an irrational number is approximated by a fraction, the more spokes form. So logically if we want less spokes, we want an irrational number that is badly approximated by a rational number. But Ben Sparks from Numberphile explains this way better than I ever could, so here you go!

TR;DW (Too Long, Didn't Watch)

Out of all the irrationals, the number below it the least well approximated by a rational. It equals \(\varphi\), and because it is so "irrational", it forms the least spokes and is the most efficient way of packing seeds together.

$$\varphi = 1+\frac{1}{1+\frac{1}{1+\frac{1}{\ddots}}}$$

$$\varphi = \frac{1+\sqrt{5}}{2}$$

Want to play around a bit more?

All the tings!
Why do you move away at \(\sqrt{n}\) each seed?

For more information, please check out the corresponding paper and of course, this excellent Numberphile video, just in case you didn't see it.

Thank you!