Talk:Fibonacci number/Phyllotaxis

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People make much of the way φ pops up everywhere. Probably the Greeks put it into the Parthenon on purpose. Probably a lot of the appearances people cite are coincidental. For example, your belly button divides your height into two parts, of which the larger is about φ times the smaller. That might be mysterious, or it might just be that a lot of things in the world are about one and three-eighths as big as a lot of other things.

But one place place φ really does appear is in botany. Suppose you're a plant. You have a central stalk, and you're going to put out leaves (or branches) on the stalk. Each leaf will be a little higher than the previous one because your stalk is always growing upwards and you put out new leaves at the tip of the stalk. But you don't want to put the leaf out pointing in the same direction as an earlier leaf, because then the upper leaf will block all the sun from the lower leaf. You'll put out the new leaf at an angle to the previous leaf.

If you picked an angle like 90 degrees, you would be making a mistake, because then when the time came to put out leaf #5, you would have gone all the way around your stalk and leaf #5 would point in the same direction as leaf #1 and keep the sun from reaching it. So 90 degree angles is not the way to go.

You could avoid this by picking a small angle, like say 5 degrees, because then you won't repeat your leaf position until leaf #37. But that is risky for another reason. If you put out 36 leaves, 5 degree angles is great. You get 36 leaves evenly spaced around the whole stalk. But you might not get to put out 36 leaves. You might put out only 18 leaves, and then you'll look like a dork with all your leaves on one side of the stalk. More to the point, you wasted half your stalk space, and you are only getting half as much sun as you could have gotten if you had used a 10-degree angle. The problem is that you do not know in advance how many leaves you will be able to put out. Even if you do get to put out 36 leaves, you have no good place to put leaf 37.

What you want is a simple strategy for putting out leaves that leads to a good leaf distribution regardless of how many leaves you actually put on the stalk. It turns out that if you pick an angle of 360/φ degrees, which is about 222 degrees, you will do very well. The first leaf comes out in direction 0, and then the second leaf comes out at an angle of 222 degrees. If you were going to have only two leaves, 180 degrees would be optimal, but you might want to put out a third leaf, and having two leaves in opposite directions would not leave a good space for the third leaf. 222 degrees is not too far off of 180, and it leaves a nice big space for leaf 3, which you put at an angle of 85 degrees (720/φ), in between leaves 1 and 2. It's not as good as if you spaced them evenly, with 120 degrees between each pair of leaves, but it is pretty good, and it leaves you in a better position for leaf 4, which comes out at about 307.5 degrees. This leaves a wide gap between leaves 2 and 3, but leaf 5 goes right in the gap. (I put illustrations of this at http://www.plover.com/~mjd/misc/leaves/ .)

It turns out that if you're a plant that doesn't know in advance how many leaves you will put out, and you want to put each one out at the same angle from the previous one, 360/φ is the best angle you can choose. There are some gaps between your leaves where the sun can get through, especially if you have only a few leaves. But every time you put out a new leaf, it happens to go almost in the middle of the widest gap you have, which means it's getting as much sun as it can get. It doesn't go exactly in the middle of the gap, but it's close. (It turns out that of the two parts of the gap on either side of the new leaf, the larger is φ times as big as the smaller.)

This might seem like more planning than plants can accomplish, but it's actually very easy for plants to execute this plan. There's some chemical in the stalk which promotes the growth of a new leaf or branch. The plant manufactures this chemical uniformly, and it builds up in the stem over time. The leaf comes out where the concentration is greatest. But when the leaf comes out, it uses up the supply of the chemical in its vicinity, so that the next leaf or branch comes out somewhere else. most likely far away from the previous one, because the chemical is still at a high concentration on the other side of the stalk.

This simple model results in leaf growth in exactly the pattern we just saw.

Because this growth pattern is simply achieved and leads to optimal leaf placement, it is very common. The centers of daisies and sunflowers have spiral structures in which the spirals have 8 components in one direction and 13 in the other (Fibonacci numbers again), or, in larger specimens, 13 in one direction and 21 in the other, or 21 and 34, or 34 and 55, or even 55 and 89. This is the result of the φ-related growing process. Many other plants, such as pineapples, show similar structures.