Doodling in math: Infinity elephants

So you mean you're in math class, yet again, because they make you go every single day. And you're learning about, I don't know, the sums of infinite series. That's a high school topic, right? Which is odd, because it's a cool topic. But they somehow manage to ruin it anyway. So I guess that's why they allow infinite serieses in the curriculum. So, in a quite understandable need for distraction, you're doodling and thinking more about what the plural of series should be than about the topic at hand. Serieses, serises, seriesen, seri? Or is it that the singular should be changed? One serie, or seris, or serum? Just like the singular of sheep should be shoop. But the whole concept of things like 1/2 plus 1/4 plus 1/8 plus 1/16, and so on approaches 1 is useful if, say, you want to draw a line of elephants, each holding the tail of the next one. Normal elephant, young elephant, baby elephant, dog-sized elephant, puppy-sized elephant, all the way down to Mr. Tusks, and beyond. Which is at least a tiny bit awesome because you can get an infinite number of elephants in a line and still have it fit across a single notebook page. But there's questions, like what if you started with a camel, which, being smaller than an elephant, only goes across a third of the page. How big should the next camel be in order to properly approach the end of the page? Certainly you could calculate an answer to this question, and it's cool that that's possible. But I'm not really interested in doing calculations. So we'll come back to camels. Here's a fractal. You start with these circles in a circle, and then keep drawing the biggest circle that fits in the spaces between. This is called an Apollonian Gasket. And you can choose a different starting set of circles, and it still works nicely. It's well known in some circles because it has some very interesting properties involving the relative curvature of the circles, which is neat, and all. But it also looks cool and suggests an awesome doodle game. Step 1, draw any shape. Step 2, draw the biggest circle you can within this shape. Step 3, draw the biggest circle you can in the space left. Step 4, see step 3. As long as there is space left over after the first circle, meaning don't start with a circle, this method turns any shape into a fractal. You can do this with triangles. You can do this with stars. And don't forget to embellish. You can do this with elephants, or snakes, or a profile of one of your friends. I choose Abraham Lincoln. Awesome. OK, but what about other shapes besides circles? For example, equilateral triangles, say, filling this other triangle, which works because the filler triangles are the opposite orientation to the outside triangles, and orientation matters. This yields our friend, Sierpinski's triangle, which, by the way, you can also make out of Abraham Lincoln. But triangles seem to work beautifully in this case. But that's a special case. And the problem with triangles is that they don't always fit snugly. For example, with this blobby shape, the biggest equilateral triangle has this lonely hanging corner. And sure, you don't have to let that stop you, and it's a fun doodle game. But I think it lacks some of the beauty of the circle game. Or what if you could change the orientation of the triangle to get the biggest possible one? What if you didn't have to keep it equilateral? Well, for polygonal shapes, the game runs out pretty quickly, so that's no good. But in curvy, complicated shapes, the process itself becomes difficult. How do you find the biggest triangle? It's not always obvious which triangle has more area, especially when you're starting shape is not very well defined. This is an interesting sort of question because there is a correct answer, but if you were going to write a computer program that filled a given shape with another shape, following even the simpler version of the rules, you might need to learn some computational geometry. And certainly, we can move beyond triangles to squares, or even elephants. But the circle is great because it's just so fantastically round. Oh, just a quick little side doodle challenge. A circle can be defined by three points. So draw three, arbitrary points, and then try and find the circle they belong to. So one of the things that intrigues me about the circle game is that, whenever you have one of these sorts of corners, you know there's going to be an infinite number of circles heading down into it. Thing is, for every one of those infinite circles, you create a few more little corners that are going to need an infinite number of circles. And for every one of those, and so on. You just get an incredible number of circles breeding more circles. And you can see just how dense infinity can be. Though the astounding thing is that this kind of infinity is still the smallest, countable kind of infinity. And there are kinds of infinity that are just mind bogglingly infiniter. But wait, here's an interesting thing. If you call this distance 1 arbitrary length unit, then this distance plus this, dot, dot, dot, is an infinite series that approaches 1. And this is another, different, series that still approaches 1. And here's another, and another. And as long as the outside shape is well defined, so will the series be. But if you want the simple kind of series, where each circle's diameter is a certain percentage of the one before it, you get straight lines. Which makes sense if you know how the slope of a straight line is defined. This is good because it suggests a wonderful, mathematical, and doodle-able way to solve our camel problem, with no calculations necessary. If instead of camels, we had circles, we could make the right infinite series just by drawing an angle that ends where the page does and filling it up. Replace circles with camels and, voila, infinite Saharan caravan fading into the distance. No numbers necessary. Well, I have an infinite amount of information I'd like to share with you in this last sentence. [VOICE SPEEDS UP]