Doodling in math: Snakes and graphs

So you're me and you're in math class, and you're learning about graph theory. A subject too interesting to be included in most grade school curricula, so maybe you're in some special program, or maybe you're in college and somehow not scarred for life by your grade school math teachers. I'm not sure why you're not paying attention, but maybe you have an incompetent teacher, and it's too heartbreaking to watch him butcher what could have been such a fun subject full of snakes and balloons. Snakes aren't really all that relevant to the mathematics here, but being able to draw them will be useful later. So you should probably start practicing now. I've got a family of three related doodle games to show you, all stemming from drawing squiggles all over the page. The first one goes like this. Draw a squiggle. A closed curve that ends where it begins. The only real rule here is to make sure all the crossings are distinct. Next, make it start weaving. Follow the curve around, and at each crossing, alternate going under and over until you've assigned all the crossing. Then put on the finishing touches, and voila. You try it again, adding a little artistic flair to the lines. The cool part is that the meeting always works out perfectly. When you're going around alternating over and under, and get to a crossing you've already assigned, it will always be the right one. This is very interesting, and we'll get back to it later. But first I'd like to point out two things. One is that this works for any number of closed curves on the plane, so go ahead and link stuff up or make a weaving out of two colors of yarn. The other is that this doodle also works for snakes on a plane, as long as you keep the head and tail on the outside or on the same inside face, because mathematically it's the same as if they linked up. Or just actually link up the head and tail, into an Ouroboros. For example, here's three Ouroborii in a configuration known as the Borromean Rings, which has the neat property that no two snakes are actually linked with each other. Also, because I like naming things, this design shall henceforth be known as the Ouroborromean Rings. But you are me, after all, so you're finding a lot to think about, even with just drawing one line that isn't a snake. Such as what kind of knot are you drawing? And can you classify them? For example, these three knots all have five crossings, but two are essentially the same knot, and one is different. Knot theory questions are actually really difficult and interesting, but you're going to have look that one up yourself. Oh, and you should also learn how to draw rope, because it's an integral part of knot theory. So integral in fact, that if you draw a bunch of integral signs in a row-- a sight which is often quite daunting to a mathematician-- you can just shade it in, and tada. But being able to draw snakes is also super useful, especially as this doodle game is excellent for producing Dark Mark tattoo designs. Also, this doodle game can be combined with the stars doodle game. For example, if this pentagram gets knighted, it will henceforth be known as Sir Pentagram. Also notice that this snakes is a five twist Mobius strip, so you could also call it a Mobiaborros, but we'll get back to one sidedness later. Or if you want to draw something super complicated, like the eighth square star, combining snakes and stars is a great technique for that to. Here is a boa that ate eight eight-gons. The creativity that your mind is forced into during these boring classes is both a gift and a burden, but here's a few authentic doodles using these techniques that I did when I was in college, just to show you that I'm not making all this up. These are from a freshman music history class, because I happen to be able to find this notebook-- but this is a doodle I actually did most often during my ninth grade Italian class-- language being another subject usually taught by unfathomably stupid methods. For example, these snakes are having trouble communicating, because one speaks in parseltongue, and the other speaks in Python. And their language classes, much like math classes, focus too much on memorization, and not enough on immersion. But just pretend you're in math class, learning about graph theory, so I can draw the parallels. Because here's the second doodle game, which is very much mathematically related. Draw a squiggle all over the page and make sure it closes up. Pick an outside section and color it in. Now you want to alternate coloring so that no two faces of the same color touch. Curiously enough, much like the weaving game, this game also always mathematically works out. It also works really well if you make the lines spiky instead of a smooth curve. And once again, it works with multiple lines too. It probably has something to do with the two-colorability of graphs of even degree, which might even be what your teacher is trying to teach you at this very moment, for all you're paying attention. But maybe you can chat with him after class about snakes and he'll explain it to you, because I'd rather move on to the next doodle game. This is a combination of the last two. Step one, draw a smooth, closed curve. Step two assign overs and unders. Step three, shade in every other face. After that, it takes a little artistic finesse to get the shading right, but you end up with some sort of really neat surface. For example, this one has only one edge and one side. But if you're interested in this, you should really be talking to your resident topology professor, and not me. But here's the thing, if someone asked you five minutes ago, what tangled up snakes, demented checkerboards, and crazy, twisty surfaces have in common, what would you have answered? This is why I love mathematics. The moment when you realize that something seemingly arbitrary and confusing is actually part of something. It's better than the cleverest possible ending to any crime show or mystery novel, because that's only the beginning. Anyway, have fun with that.