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Doodling in math: Triangle party

Triangles! Created by Vi Hart.

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Video transcript

So you're me and you're in math class and-- triangles, triangles, triangles-- I don't know. The teacher keeps saying words, and you're supposed to be doing something with trigons, whatever those are. But you're bored and-- triangles, triangles, triangles. Sure you could draw your triangles separately, but then they get lonely. They're happiest when snuggled up together into a triangle party. Everybody knows triangles love parties. Sometimes they get together and do these triangle congo lines. If you keep adding new triangles on the same side, it gets all curvy and spirally. Or you can alternate and it goes pretty straight. In fact, since all the sides of the triangle are supposed to be straight lines, and since they're all lying on top of some previous straight line, then this whole line would have to be straight if these were actually triangles. Since it's not, it's proof that these aren't quite triangles. Maybe they've been partying a little too hard, but hey, at least you're not doing math. Speaking of which, the teacher is still going on about types of trigons, and you're supposed to be taking notes. But you're more interesting in types of triangles, which you already know all about. There are fat triangles, and pointy triangles, and perfect triangles, and cheese slice triangles which are a kind of pointy triangle but are symmetric like a slice of cheese or cake. Super pointy triangles are fun to stack into triangle stacks. You can put all the points facing one direction, but the stack starts to wobble too much towards that direction. So it's good to put some facing the other direction before you go too far. You'll notice pretty quickly, that the skinnier the triangle, the less wobble it adds to the stack. To compensate for a big wobble, you can put just one not so skinny triangle that's pointing the other way. Or maybe you want to wobble, because you have to navigate your triangle stack around your notes. In which case you can even alternate back and forth as long as you make the triangles point towards where you want to go, a little less skinny. The easy part about triangle stacks, is that there's really only one part of the triangle that's important, as far as the stack is concerned. And that's the pointy point. The other two angles can be fat and skinny, or skinny and fat, or both the same if it's a cheese slice, and it doesn't change the rest of the stack. Unless the top angle is really wide, because then you'll get two skinny points, and which side should you continue to stack on? Both? Also, instead of thinking fat and skinny, you should probably create code words that won't set off your teacher's mind reading alarms for non-math related thoughts. So you pick two words off the board, obtuse and acute, which by sheer coincidence I'm sure, just happen to mean fat angle and skinny angle. Of course, those are also kinds of triangles. Which doesn't make much sense, because the obtuser one angle of a triangle is, the more acute the other two get. Yet, if you make an acute triangle with the same perimeter, it has more area, which seems like an obtuse quality. And then, can you still call an obtuse equilateral triangle a cheese or cake-slice triangle, because these look more like [INAUDIBLE]. Point is, triangle terminology is tricky, but at least you're not paying attention to the stuff the teacher is saying about trigonometric functions. You'd rather think about the functions of triangles. And you already know some of those. There are sines, and cosines, but enough of this tangent. The thing to pay attention to is what affects your triangle how. If you start drawing the next triangle on your triangle stack this way, by this point, you already know what the full triangle would have to be. Because you just continue this edge until it meets this invisible line, and then fill the rest in. In fact, you can make an entire triangle stack just by piling on triangle parts and adding the points later to see what happens. There's some possible problems though. If you start a triangle like this, you can see that it's never going to close, no matter how far you extend the lines. Since this triangle isn't real, let's call it a Bermuda Triangle. This happens when two angles together are already more than 180 degrees. And since all the angles in a triangle add up to 180-- which, by the way, you can test by ripping one up and putting the three points into a line-- that means that if these are two 120 degree angles of a triangle, the third angle is off somewhere being negative 60. Of course, you have no problem being a Bermuda Triangle on a sphere, were angles always add up to more than 180, just the third point might be off in Australia. Which is fine, unless you're afraid your triangle might get eaten by sea monsters. Anyway, stacking triangles into a curve is nice, and you probably want to make a spiral. But if you're not careful, it'll crash into itself. So you'd better think about your angles. Though, if you do it just right, instead of a crash disaster, you'll get a wreath thing. Or you can get a different triangle circle by starting with a polygon, extending the sides in one direction, and then triangling around it, to get this sort of aperture shape. And then you should probably add more triangles, triangles, triangles-- One last game. Start with some sort of asterisk, then go around a triangle it up. Shade out from the obtusest angles, and it'll look pretty neat. You can then extend it with another layer of triangles, and another, and if you shade the inner parts of these triangles, it's guaranteed to be an awesome triangle party. And there's lots of other kinds of triangle parties just waiting to happen. Ah, the triangle. So simple, yet so beautiful. The essence of two-dimensionality, the fundamental object of Euclidean geometry, the three points that define a plane. You can have your fancy complex shapes, they're just made up of triangles. Triangles. Dissect a square into triangles, make symmetric arrangements, some reminiscent of spherical and hyperbolic geometry. Triangles branches into binary fractal trees. Numbers increasing exponentially with each iteration to infinity. Triangles, with the right proportion being a golden spiral of perfect right isosceles triangles. Put equilateral triangles on the middle third of the outside edges of equilateral triangles to infinity, and get a snow flake with the boundary of the Koch Curve. An infinitely long perimeter, continuous yet nowhere differentiable. With a fractional dimension of log four over log three, and-- uh oh, teacher's walking around, better pretend to be doing math.