BrightKidz Library
Subjects
The one cut that gives a sunflower its turn On the left a ring with a single cut passing right through it, leaving a gap at each of the two places the cut crosses, so the ring lies in two pieces of clearly different size. There is only one place that cut can go so that the larger piece compares to the smaller piece in exactly the same way the whole ring compares to the larger piece, and this is that place. An arrow leads across to the right, to a sunflower head packed with seeds. Each new seed is set the smaller piece of that ring further round than the one before it, and because a turn like that never comes back round to where it started, every seed drops into a gap instead of lining up behind an earlier one.

Where the Sunflower Gets Its Turn

About 11 minutes

You have the turn already. A growing tip puts each new seed about 137.5 degrees around from the last one, and you know why a turn like that works: it never comes back around to where it started, so seeds keep dropping into gaps instead of lining up.

But 137.5 is a very strange number to arrive at. It is not a half or a quarter or anything you would pick on purpose. No plant measured it and no gardener chose it.

So where does a number like that come from?

One cut, and only one place to put it

Take a circle and cut it into two pieces, a bigger one and a smaller one. Slide the cut around and the two pieces trade size.

Now hunt for one particular place. You want the cut to sit where this is true:

the big piece compares to the small piece in exactly the same way the whole circle compares to the big piece.

That is a fussy demand. It asks for one relationship to hold twice over, at two different scales at once. Almost everywhere you put the cut, the two comparisons disagree — the whole circle is a lot more than twice the big piece while the big piece is barely more than the small one, or the other way about.

There is exactly one place where they match. And when the cut is there, the smaller piece measures 137.5077… degrees.

That is where the plant's turn comes from. Nobody picked it. It is simply what is left over when a circle is divided that way.

The parts of the golden cut of a circle A disc divided by a single straight cut running from the rim through the centre and out to the rim again. The smaller shaded piece is the turn a growing plant makes between one seed and the next. The larger piece is the rest of the circle. The whole disc, the larger piece and the smaller piece are three quantities, and the cut is placed so that one single number relates the whole to the larger piece and the larger piece to the smaller one. the turn the rest One cut. Only one place it can go.

Pick a part of the divided circle to see what it has to do with the others.

  • The whole circle — all 360 degrees of it. Compare this with the larger piece and note the answer.
  • The larger piece — about 222.5 degrees. Now compare this with the smaller piece. You get the same answer as before, which is the whole point of putting the cut here.
  • The smaller piece — 137.5 degrees, and this is the turn a plant makes between one seed and the next.
  • The cut itself — the only position where those two comparisons agree. Move it anywhere else and they come apart.

The number doing the comparing

Both of those comparisons come out as the same number, and it is this:

1.618033988…

It carries on, it never settles into a repeating pattern, and it is called the golden ratio.

It has a habit most numbers do not have. Take one away from it and you get 0.618033988… Now instead divide 1 by it, and you get 0.618033988… again — the same digits, from a completely different sum. Try it on a calculator, because it looks like a mistake until you have done it yourself.

It does the same thing upward. Multiply it by itself and you get 2.618033988…, which is just the number with 1 added on.

Numbers do not usually behave like this. This one keeps handing you back a copy of itself.

Where your sequence has been all along

Here is the part that was never said.

Take the sequence you already know — the one where you add the last two numbers to get the next. Pick any number in it and divide it by the number just before it. Write down what you get.

Now move further along the list and do it again. And again.

The answers do not wander. They close in, overshooting a little then undershooting a little, and every one of them is nearer the mark than the last. What they are closing in on is 1.618033988… They come as near as you like and never land on it exactly, however far along the list you go.

So the sequence and the plant's turn are not two separate curiosities. They are one number, met twice: once as a list you build by adding, and once as an angle you get by dividing a circle.

Why never quite landing is the whole point

You already saw what happens to a turn that divides neatly into a circle. The seeds march back into the same directions over and over and the head opens into empty spokes.

The golden ratio is the number that resists exactly that. Every simple fraction you try against it comes close and misses — and each better attempt still misses. There is no fraction sitting at the end of that chase waiting to be found.

That is what "it never comes back around to where it started" really means. A seed placed at this turn misses the earlier seeds not by luck and not for the first hundred seeds only, but always, because the turn is built out of a number no fraction can pin down.

Where does the plant's turn of 137.5 degrees come from?

Now go and count something

The counting instructions have not changed. Find a pinecone, mark one scale, and count the spirals going one way and then the other.

What you already know is that those counts come out as numbers from your sequence. What you know now is that the angle the plant turns by and the sequence you count with are the same number wearing two different coats.

Which makes the counts less of a coincidence and much more of a clue — and the best bit is that you can go and get the numbers yourself, off a real cone, before anybody tells you what they are going to be.