Doodling in math: Connecting dots

So you're me, and you're in math class. And your teacher's ranting on and on about this article about whether algebra should be taught in school – as if he doesn't realize that what he's teaching isn't even algebra – which could have been interesting – but how to manipulate symbols and some special cases of elementary algebra – which isn't. And so, instead of learning about self-consistent systems and logical thought, you spent all week memorizing how to graph parabolas. News flash: No one cares about parabolas. Which is why half the class is playing Angry Birds under their desks. But, since you don't have a smart phone yet, you have to resort to a more noble and outdated form of boredom relief – that is, doodling. And you've invented a game of your own. A doodle game that connects the dots in ways your math curriculum never will – except instead of connecting to the closest dots to discover the mysterious hidden picture you've got this precise method of skipping over some number of dots and connecting them that way. In the past you've characterized how this works if your dots are arranged in a circle – say 11 dots – and connect one to the dot four dots over, you get these awesome stars. And you can either draw the lines in the order of the dots, or you can just keep going around and maybe it will hit all the dots, or maybe it won't, depending on how many dots are in your circle, and how many dots you skip. But then there are other shapes. Circles are good friends with sine waves. And sine waves are good friends with square waves. And let's admit it, that's pretty cool looking. In fact, just two simple straight lines of dots connecting the dots from one line to the other in order somehow gives you this awesome woven curve shape. Another student is asking the teacher when he's ever going to need to know how to graph a parabola – even as he hides his multi-million dollar enterprise of a parabola graphing game under his desk. If your teacher thought about it, he would probably think shooting birds at things is a great reason to learn about parabolas because he's come to understand that education is about money and prestige and not about becoming a better human able to do great things. You yourself haven't done anything really great yet but you figure the path to your future greatness lies more in inventing awesome new connect-the-dot arrangements than in graphing parabolas or shooting birds at things. And that's when you begin to worry. What if this cool liney curvey thing you drew approximates a parabola? As if your teacher doesn't realize everyone has their phones under their desks, but he's underpaid and overworked and his whole word runs on plausible deniability, so he shouts state-mandated, pass the test, teach-to-the-middle nonsense at students who are not at all fooled by his false enthusiasm or false mathematics and he pretends he's teaching algebra and the students pretend to be taught algebra and everyone else involved in the system is too invested to do anything but pretend to believe them both. You think maybe it's a hyperbola, which is similar to the parabola in that they are both conic sections. A hyperbola is a nice vertical slice of cone, the cone itself being just like a line swirled around in a circle, which is why the cone is like two cones radiating both ways; the lovely hyperbola insecting both parts. Two perfect curves, looking disconnected when seen alone but sharing their common conic heritage. While the boring old parabola is a slice taken at an angle completely meant to miss the top part of the cone and to miss wrapping around the bottom like an ellipse would. And it's such a special, specific case of conic section that all parabolas are exactly the same, just bigger or smaller or moved around. Your teacher could just as well hand you parabolas already drawn and have you draw coordinate grids on parabolas rather than parabolas on coordinate grids. And it's stupid, and you hate it, and you don't wanna learn to graph them, even if it means not making a billion dollars from a game about shooting birds at things. Meanwhile anyone who actually learns how to think mathematically can then learn to graph a parabola or anything else they need in like five minutes. But teaching how to think is an individualized process that gives power and responsibility to individuals while teaching what to think can be done with one-size-fits-all bullet points and check-boxes and our culture of excuses demands that we do the latter, keeping ourselves placated in the comforting structure of tautology and clear expectations. Algebra has become a check-box subject and mathematics weeps alone in the top of the ivory tower prison to which she has been condemned. But you're not interested in check-boxes; you're interested in dots, and lines that connect them. Or maybe you could connect them with semicircles, to give visual structure to lines that would otherwise overlap. Or you could say one dot is the center of a circle and another defines a radius and draw the entire circle and do things that way. You could make rules about how every dot is the center of a circle with its neighbor being the radius, or say one dot stays the center of all circles, and all of the others define radii. But then you just get concentric circles, which I suppose should have been obvious. But what if you did it the other way around and said one dot always stays on the circle and all the other dots are centers, like this. Looks more promising. So you try putting all the dots in a circle and using them as circle centers and choose just one dot for the circles to go through and you get this awesome shape that looks kind of like a heart. So let's call it, oh I don't know, a cardioid. Which happens to be the same curve that you get when parallel lines like rays of light reflect off a circle the same heart of sunshine in a cup. Or maybe instead of circle centers you could have points all on the curve of a circle, which means you need three points to define a circle, maybe just a point and its two closest neighbors to start with. And of course, any collection of circles is two-colorable, which means you can contrast light and dark colors for a classy color scheme. Or maybe you could throw down some random points to make all possible circles. Only that would be a lot of circles, so you choose just ones you like. And then, against your will, you begin to wonder how many points it takes to define the boring old parabola. Because parabolas are actually a lot like circles in that both are like extreme ellipses, because a circle is like taking one focus of an ellipse and putting the other focus zero distance away. And a parabola is like an ellipse where one focus is infinity distance away. Which is why everyone lies to you and says throwing balls or shooting birds is all about parabolas when really it's about ellipses because the earth is a sphere and gravity doesn't actually go straight down. And the other focus of the elliptical orbit of your thrown object of choice may be very far away, but very far away is a great bit closer than infinity. So let's not fool ourselves. You can't look at everything that seems kind of parabolic and call it a parabola. Sure, if you connect two dots with a hanging string or chain, it looks parabolic, and so do structurally strong arches, but they're actually catenary arches and maybe you can't tell by sight, but if you're an architect you'd better know the difference. Though caternarys are quite related to parabolas, you get them by rolling around a parabola and tracing the focus which makes them a cousin of the ellipse and even a hyperbola is like an ellipse that got turned inside out or whose focus went through infinity and came out the other side or something. And of course parabolas and hyperbolas and ellipses are all conic sections which mean they all come from a line that got spun around. And a line is just what happens when a couple dots get connected. Or maybe what happens when your circle is so big that, like the extreme ellipse that becomes a parabola, the extreme circle is broken at infinity and becomes a line before getting larger than infinitely big which brings it back to the other side. This linear circle, the infinite in-between. Or maybe a line is what happens when you roll around a circle and trace the focus. Or rather the 2 foci, which are zero distance apart, in ellipse terms. Which makes you wonder what you get when you roll around an ellipse and trace the foci. In fact, there's lots of great shapes you can get by rolling around shapes on other shapes, like if you roll a circle around a circle and trace the focus, you get just another circle. But, if you trace a point on the edge, you get our awesome friend the cardioid again. So now it's related to circles in three ways , which means it's a close cousin of the ellipse and a second cousin to the infinite ellipse or, parabola. Except, not just that, if you take a parabola and invert it around the unit circle, reversing inside and outside, one half becomes two, one hundred becomes a hundredth, one stays one, infinity becomes zero, you get once again, the cardioid. The cardioid is the anti-parabola which is good because parabolas make you sad but you heart cardioids. And of course, any time you want to connect two dots on a piece of paper, instead of drawing the line you could fold the line. Here's the thing about connecting dots. You can have all the steps laid out for you, taking whatever next step is easiest and closest and be sure of what you're getting the whole time. This way is safe and comfortable. Or, you can try new ways of connecting dots and not know what you're going to get. Maybe it will be something great, maybe it will fail. And when it fails it will be your fault. You can't blame anyone else, not mathematics or the system or the check-boxes. But if I am to have faults I would rather they be my own.