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Doodling in math: Spirals, Fibonacci, and being a plant [2 of 3]

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Say you're me and you're in math class, and you're doodling flowery petally things. If you want something with lots of overlapping petals, you're probably following a loose sort of rule that goes something like this. Add new petals where there's gaps between old petals. You can try doing this precisely. Start with some number of petals, say five, then add another layer in between. But the next layer, you have to add 10, then the next has 20. The inconvenient part of this is that you have to finish a layer before everything is even. Ideally, you'd have a rule that just lets you add petals until you get bored. Now imagine you're a plant, and you want to grow in a way that spreads out your leaves to catch the most possible sunlight. Unfortunately, and I hope I'm not presuming too much in thinking that, as a plant, you're not very smart. You don't know how to add number to create a series, you don't know geometry and proportions, and can't draw spirals, or rectangles, or slug cats. But maybe you could follow one simple rule. Botanists have noticed that plants seem to be fairly consistent when it comes to the angle between one leaf and the next. So let's see what you could do with that. So you grow your first leaf, and if you didn't change angle at all, then the next leaf you grow would be directly above it. So that's no good, because it blocks all the light or something. You can go 180 degrees, to have the next leaf directly opposite, which seems ideal. Only once you go 180 again, the third leaf is right over the first. In fact, any fraction of a circle with a whole number as a base is going to have complete overlap after that number of turns. And unlike when you're doodling, as a plant you're not smart enough to see you've gone all the way around and now should switch to adding things in between. If you try and postpone the overlap by making the fraction really small, you just get a ton of overlap in the beginning, and waste all this space, which is completely disastrous. Or maybe other fractions are good. The kind that position leaves in a star like pattern. It will be a while before it overlaps, and the leaves will be more evenly spaced in the meantime. But what if there were a fraction that never completely overlapped? For any rational fraction, eventually the star will close. But what if you used an irrational number? The kind of number that can't be expressed as a whole number ratio. What if you used the most irrational number? If you think it sounds weird to say one irrational number is more irrational than another, well, you might want to become a number theorist. If you are a number theorist, you might tell us that phi is the most irrational number. Or you might say, that's like saying, of all the integers, 1 is the integer-iest. Or you might disagree completely. But anyway, phi. It's more than 1, but less than 2, more than 3/2, less than 5/3. Greater than 8/5, but 13/8 is too big. 21/13 is just a little too small, and 34/21 is even closer, but too big, and so on. Each pair of adjacent Fibonacci numbers creates a ratio that gets closer and closer to Phi as the numbers increase. Those are the same numbers on the sides of these squares. Now stop being a number theorist, and start being a plant again. You put your first leaf somewhere, and the second leaf at an angle, which is one Phi-th of a circle. Which depending on whether you're going one way or the other, could be about 222.5 degrees, or about 137.5. Great, your second leaf is pretty far from the first, gets lots of space in the sun. And now let's add the next one a Phi-th of the circle away. And again, and again. You can see how new leaves tend to pop up in the spaces left between old leaves, but it never quite fills things evenly. So there's always room for one more leaf, without having to do a whole new layer. It's very practical and as a plant you probably like this. It would also be a good way to give lots of room to seed pods and petals and stuff. As a plant that follows this scheme, you'd be at an advantage. Where do spirals come in? Let's doodle a pinecone using this same method. By the way, you can make your own phi angle-a-tron by dog-earing a corner of your notebook. If you folded it so the edges of the line-- You have 45 degrees plus 90, which is 135. Pretty close to 137.5. If you're careful, you can slip in a couple more degrees. Detach your angle-a-tron and you're good to go. Add each new pine cone-y thing a phi angle around, and make them a little farther out each time. Which you can keep track of by marking the distance on your angle-a-tron. Check it out! The spirals form by themselves. And if we count the number of arms, look it's five and eight. If you're wondering why spirals would form, and why always with Fibonacci numbers, you could morph back into a number theorist, or a geometer or something. But here's just a little bit of intuition. One simple way to do a flower is to start with a certain number of petals, say five. And when you go back around, add the next layer close to the first, but bigger. Each layer adds five new petals and the five arms spiral out. Looks pretty spiral-ey to me. Now go back to phi. You put out three petals before you go back around. And if I make them really wide, the next lap adds three petals that overlap a bunch with the first, and so on. If I started with skinnier petals though, the second time you go around it doesn't quite overlap so much. And it takes eight petals before it goes around twice and they overlap enough for you to see the spirals. So this time I get 8 and 13. I mean none of the spirals are actually physically there on any of these plants, it's just the plant bits are close enough that you can see the pattern. So all the plant needs to do to get awesome Fibonacci numbers of spirals, is add new bits at 137.5 degree angles. The rest takes care of itself. That the Fibonacci series is in so many things really says less about those things, and more about mathematics. I mean that's what mathematics is all about. Simple rules, complex consequences. A process so easy that even a plant can do it, can turn into these amazing structures all around us. Just like a few simple postulates can give us an incredibly powerful geometry. I mean that's all assuming that a plant can do it. But measuring the angles between plant bits, you can see they obviously do do it, somehow. I mean, it's not like they have angle-trons, but plants have been around a long time, and have had a lot of practice, so that probably explains everything. And so we always get spirals of 5 and 8 on this flower, 5 and 8 on this artichoke, 5 and 8 on this pine cone, even this cauliflower has 1, 2, 3, 4, 5, 6, 7-- um, anyway, we always get 1, 2 3, 4, 5, 6, 7-- Huh. And 1, 2, 3, 4? It's easy to dismiss these as mutant anomalies, but just because they're different and unusual, doesn't mean we should ignore them. 4? 7? 11? What could these numbers mean? Maybe things aren't as simple as I thought.