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Say you're me and you're in math class and you're doodling flowery petaly 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 are gaps between old petals. You can try doing this precisely start with some number of petals, say 5, then add another layer in between. But the next layer you have to add 10, and 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 numbers to create a series, you don't know geometry and proportions, and can't draw spirals or rectangles, or slug cats. But maybe you can 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 directly above it. So that's no good because it blocks out 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 should now 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 overlapping in the beginning and waste all this space, which is completely disastrous. Or maybe other fractions are good, the kind that positions leaves in a starlike pattern. It'll be a while before it overlaps, and the leafs will be more evenly spaced in the mean time. But what if there a fraction that never completely overlapped? For any rational fraction, eventually the star will close, but what if you use an irrational number? The kind of number that can't be expressed as whole numbered ratio. What if you used the most irrational number? If you think it weird to say that one irrational number is more irrational than another...

Well you might wanna become a number theorist. If you're 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 integeriest, or you might disagree completely. But anyway, phi - is more than one, but less than two; more than 3/2, less than 5/3; greater than 8/5, but 13/8 is too big. 21/13 just a little to small, and 34/21 is even close than 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 that is 1 phi-th of a circle, which, depending on whether you going one way than the other, could be about 225.5 degrees or about 137.5. Great! Your second leaf is pretty far from the first, gets lots of space and sun.

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 numbers to create a series, you don't know geometry and proportions, and can't draw spirals or rectangles, or slug cats. But maybe you can 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 directly above it. So that's no good because it blocks out 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 should now 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 overlapping in the beginning and waste all this space, which is completely disastrous. Or maybe other fractions are good, the kind that positions leaves in a starlike pattern. It'll be a while before it overlaps, and the leafs will be more evenly spaced in the mean time. But what if there a fraction that never completely overlapped? For any rational fraction, eventually the star will close, but what if you use an irrational number? The kind of number that can't be expressed as whole numbered ratio. What if you used the most irrational number? If you think it weird to say that one irrational number is more irrational than another...

Well you might wanna become a number theorist. If you're 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 integeriest, or you might disagree completely. But anyway, phi - is more than one, but less than two; more than 3/2, less than 5/3; greater than 8/5, but 13/8 is too big. 21/13 just a little to small, and 34/21 is even close than 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 that is 1 phi-th of a circle, which, depending on whether you going one way than the other, could be about 225.5 degrees or about 137.5. Great! Your second leaf is pretty far from the first, gets lots of space and sun.

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