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

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So say you're me and you're in math class and you're trying to ignore the teacher and doodle Fibonacci spirals, while simultaneously trying to fend off the local greenery. Only, you become interested in something that the teacher says by accident, and so you draw too many squares to start with. So you cross some out, but cross out too many, and then the teacher gets back on track, and the moment is over. So oh well, might as well try and just do the spiral from here. So you make a three by three square. And here's a four by four, and then seven, and then 11 This works because then you've got a spiral of squares. So you write down the numbers, 1, 3, 4, 7, 11, 18. It's kind of like the Fibonacci series, because 1 plus 3 is 4, 3 plus 4 is 7, and so on. Or maybe it starts at 2 plus 1, or negative 1 plus 2. Either way, it's a perfectly good series. And it's got another similarity with the Fibonacci series. The ratios of consecutive numbers also approach Phi. OK. So, a lot of plants have Fibonacci numbers of spirals, but to understand how they do it, we can learn from the exceptions. This pine cone that has seven spirals one way, and eleven the other, might be showing Lucas numbers. And since Fibonacci numbers and Lucas numbers are related, maybe that explains it. One theory was that plants get Fibonacci numbers by always growing new parts a Phi-th of a circle around. What angle would give Lucas numbers? In this pine cone, each new pine cone-y thing is about 100 degrees around from the last. We're going to need a Lucas angle-a-tron. It's easy to get a 90 degree angle-a-tron, and if I take a third of a third of that, that's a ninth of 90, which is another 10 degrees. There. Now you can use it to get spiral patterns like what you see on the Lucas number plants. It's an easy way to grow Lucas spirals, if plants have an internal angle-a-tron. Thing is, 100 is pretty far from 137.5. If plants were somehow measuring angle, you'd think the anomalous ones would show angles close to a Phi-th of a circle. Not jump all the way to 100. Maybe I'd believe different species use different angles, but two pine cones from the same tree? Two spirals on the same cauliflower? And that's not the only exception. A lot of plants don't grow spirally at all. Like this thing. With leaves growing opposite from each other. And some plants have alternating leaves, 180 degrees from each other, which is far from both Phi and Lucas angles. And you could say that these don't count because they have a fundamentally different growth pattern, and are in a different class of plant or something, but wouldn't it be even better if there were one simple reason for all of these things? These variations are a good clue that maybe these plants get this angle, and Fibonacci numbers, as a consequence of some other process and not just because it mathematically optimizes sunlight exposure if the sun is right overhead, which it pretty much never is, and if the plan are perfectly facing straight up, which they aren't. So how do they do it? Well, you could try observing them. That would be like science. If you zoom in on the tip of a plant, the growing part, there's this part called the meristem. That's where new plant bits form. The biggest plant bits where the first to form off the meristem, and the little ones around the center are newer. As the plants grow, they get pushed away from the meristem, but they all started there. The important part is that a science observer would see the plant bits pushing away, not just from the meristem, but from each other. A couple physicists once tried this thing where they dropped drops of a magnetized liquid in a dish of oil. The drops repelled each other, kind of like plant bits do, and were attracted to the edge of the dish, just like how plant bits move away from the center. The first couple jobs would head in opposite directions from each other, but then the third was repelled by both, but pushed farther by the more recent, and closer, drop. It, and each new drop, would set off at a Phi angle relative to the drop before, and the drops ended up forming Fibonacci numbers spirals. So all a plant would need to do to get Fibonacci numbers spirals is figure out how to make the plant bits repel each other. We don't know all the details, but here's what we do know. There's a hormone that tells plant bits to grow. A plant bit might use up the hormone around it, but there's more further away, so it'll grow in that direction. That makes plant bits move out from the meristem after they form. Meanwhile, the meristem keeps forming new plant bits, and they're going to grow in places that aren't too crowded, because that's where there's the most growth hormone. This leads them to move further out into the space left by the other outward moving plant bits. And once everything gets locked into a pattern, it's hard to get out of it. Because there's no way for this plant bit to wander off unless there were an empty space with a trail of plant hormone to lead it out of its spot. But if there were, all the nearer plant bits would use up the hormone grow to fill in the space. Mathematicians and programmers made their own simulations, and found the same thing. The best way to fit new things in with the most space, has them pop up at that angle, not because the plant knows about the angle, but because that's where the most hormone has built up. Once it gets started, it's a self perpetuating cycle. All these flower bits are doing is growing where there's the most room for them. The rest happens auto-mathically. It's not weird that all these plants show Fibonacci numbers. It would be weird if they didn't. It had to be this way. The best thing about that theory is that it explains why Lucas pine cones would happen. If something goes a bit differently in the very beginning, the meristem will settle into a different, but stable, pattern of where there's the most room to add new plant bits. That is 100 degrees away. It even explains alternating leaf patterns. If the leaves are far enough apart, relative to how much growth hormone they like, that these leaves don't have any repelling force with each other, then all this leaf cares about is being farthest away from the two above and below it, which makes 180 degrees optimal. And when leaves grow in pairs that are opposite each other, the answer of where there's the most room for both of those leaves, is at 90 degrees from the one below it. And if you look hard, you can discover even more unusual patterns. The dots on the neck of this whatever-it-is, come in spirals of 14 and 22, which maybe is like double the Lucas numbers. And this pine cone has 6 and 10, double Fibonacci numbers. So how is a pineapple like a pine cone? What do daisies and Brussels sprouts have in common? It's not the numbers they show, it's how they grow. This pattern is not just useful, not just beautiful, it's inevitable. This is why science and mathematics are so much fun. You discover things that seem impossible to be true, and then get to figure out why it's impossible for them not to be. To get this far in our understanding of these things, it took the combined effort of mathematicians, physicists, botanists, and biochemists. And we've certainly learned a lot. But there's much more to be discovered. Maybe if you keep doodling in math class, you can help figure it out.