WEBVTT 00:00:00.184 --> 00:00:01.685 So you're me and you're in math class 00:00:01.685 --> 00:00:02.898 and you're learning about graph theory, 00:00:02.898 --> 00:00:04.349 a subject too interesting to be included 00:00:04.349 --> 00:00:05.662 in most grade school curricula. 00:00:05.662 --> 00:00:06.877 So maybe you're in some special program 00:00:06.877 --> 00:00:07.846 or maybe you're in college 00:00:07.846 --> 00:00:09.705 and were somehow not scarred for life 00:00:09.705 --> 00:00:10.498 by your grade school math teachers. 00:00:10.498 --> 00:00:11.721 I'm not sure why you're not paying attention 00:00:11.721 --> 00:00:12.990 but maybe you have an incompetent teacher 00:00:12.990 --> 00:00:14.251 and it's too heart-breaking to watch him 00:00:14.251 --> 00:00:15.651 butcher what could be a fun subject, 00:00:15.651 --> 00:00:16.739 full of snakes and balloons. 00:00:16.739 --> 00:00:19.107 Snakes aren't really all that relevant to the mathematics here. 00:00:19.107 --> 00:00:20.655 But being able to draw them will be useful later, 00:00:20.655 --> 00:00:22.378 so you should probably start practicing now. 00:00:22.378 --> 00:00:24.936 I've got a family of 3 related doodle games to show you, 00:00:24.936 --> 00:00:27.098 all stemming from drawing squiggles all over the page. 00:00:27.098 --> 00:00:28.605 The first one goes like this: 00:00:28.605 --> 00:00:31.022 draw a squiggle- a closed curve that ends where it begins. 00:00:31.022 --> 00:00:32.528 The only real rule here is to make sure 00:00:32.528 --> 00:00:34.047 that all the crossings are distinct. 00:00:34.047 --> 00:00:35.956 Next, make it start weaving- 00:00:35.956 --> 00:00:36.968 follow the curve around and 00:00:36.968 --> 00:00:38.594 that each crossing alternate going under and over 00:00:38.594 --> 00:00:40.471 until you've assigned all the crossings. 00:00:40.471 --> 00:00:42.148 Then put on the finishing touches, and voila! 00:00:42.148 --> 00:00:44.822 You try it again, adding a little artistic flair to the lines. 00:00:44.822 --> 00:00:47.155 The cool part is that the weaving always works out perfectly, 00:00:47.155 --> 00:00:49.226 when you're going around alternating over and under 00:00:49.226 --> 00:00:50.788 and get to a crossing you've already assigned, 00:00:50.788 --> 00:00:52.462 it will always be the right one. 00:00:52.462 --> 00:00:55.037 This is very interesting, and we'll get back to it later. 00:00:55.037 --> 00:00:57.749 But first I'd like to point out 2 things: one is that 00:00:57.749 --> 00:01:00.654 this works for any number of closed curves on the plane. 00:01:00.654 --> 00:01:01.812 So go ahead and link stuff up 00:01:01.812 --> 00:01:03.943 or make a weaving out of 2 colors of yarn. 00:01:03.943 --> 00:01:04.881 The other is that 00:01:04.881 --> 00:01:07.279 this doodle also works out for snakes on a plane 00:01:07.279 --> 00:01:09.274 as long as you keep the head and tail on the outside 00:01:09.274 --> 00:01:10.435 or on the same inside face. 00:01:10.435 --> 00:01:12.716 because mathematically it's the same as if they linked up 00:01:12.716 --> 00:01:15.346 or just actually link up the head and tail into an Ouroboros. 00:01:15.346 --> 00:01:18.261 For example, here's 3 Ourobori in a configuration 00:01:18.261 --> 00:01:19.746 known as the Borromean Rings 00:01:19.746 --> 00:01:21.479 which has the neat property that 00:01:21.479 --> 00:01:23.865 no 2 snakes are actually linked with each other. 00:01:23.865 --> 00:01:25.448 Also because I like naming things, 00:01:25.448 --> 00:01:27.556 this design shall henceforth be known as 00:01:27.556 --> 00:01:29.237 the "OuroBorromean Rings". 00:01:29.237 --> 00:01:30.461 But you are me, after all, 00:01:30.461 --> 00:01:31.805 so you're finding a lot to think about 00:01:31.805 --> 00:01:34.082 even with just drawing one line that isn't a snake. 00:01:34.082 --> 00:01:35.812 Such as, "What kinds of knots are you drawing?" 00:01:35.812 --> 00:01:37.220 "And can you classify them?" 00:01:37.220 --> 00:01:39.388 For example, these 3 knots all have 5 crossings 00:01:39.388 --> 00:01:42.374 but 2 are essentially the same knot and one is different. 00:01:42.374 --> 00:01:45.258 Knot theory questions are actually really difficult and interesting 00:01:45.258 --> 00:01:47.055 but you're going to have to look that one up yourself. 00:01:47.055 --> 00:01:48.848 Oh, and you should also learn how to draw rope 00:01:48.848 --> 00:01:50.829 because it's an integral part of knot theory. 00:01:50.829 --> 00:01:51.687 So integral, in fact, that 00:01:51.687 --> 00:01:53.774 if you draw a bunch of integral signs in a row, 00:01:53.774 --> 00:01:56.234 a sight which is often quite daunting to a mathematician, 00:01:56.234 --> 00:01:58.526 you can just shade it in, and TA-DA. 00:01:58.526 --> 00:02:01.490 But, being able to draw snakes is also super useful 00:02:01.490 --> 00:02:03.346 especially as this doodle game is excellent for 00:02:03.346 --> 00:02:05.157 producing Dark Mark tattoo designs. 00:02:05.157 --> 00:02:06.860 Also, this doodle game can be combined 00:02:06.860 --> 00:02:08.020 with the stars doodle game. 00:02:08.020 --> 00:02:10.505 For example, if this pentagram gets knighted, 00:02:10.505 --> 00:02:13.331 it will henceforth be known as "Serpentagram" 00:02:13.331 --> 00:02:16.319 Also notice that this snake is a 5 twist Mobius strip 00:02:16.319 --> 00:02:17.946 so you could also call it a "Mobiaboros" 00:02:17.946 --> 00:02:19.784 but we'll get back to one-sidedness later. 00:02:19.784 --> 00:02:21.921 Or, if you want to draw something super complicated 00:02:21.921 --> 00:02:23.438 like the 8th square star, 00:02:23.438 --> 00:02:26.948 combining snakes and stars is a great technique for that too. 00:02:26.948 --> 00:02:29.328 Here's a boa that ate 8 8gons. 00:02:29.328 --> 00:02:31.808 The creativity that your mind is forced into 00:02:31.808 --> 00:02:34.736 during these boring classes, is both a gift and a burden. 00:02:34.736 --> 00:02:37.016 But here's a few authentic doodles using these techniques 00:02:37.016 --> 00:02:38.457 that I did when I was in college. 00:02:38.457 --> 00:02:40.079 Just to show you I'm not making all this up. 00:02:40.079 --> 00:02:41.895 These are from a freshmen music history class, 00:02:41.895 --> 00:02:43.963 because I happen to be able to find this notebook. 00:02:43.963 --> 00:02:45.807 But this is a doodle I actually did most often 00:02:45.807 --> 00:02:47.336 during my 9th grade Italian class. 00:02:47.336 --> 00:02:48.518 Language being another subject 00:02:48.518 --> 00:02:51.163 usually taught by unfathomably stupid methods. 00:02:51.163 --> 00:02:53.951 For example, these snakes are having trouble communicating 00:02:53.951 --> 00:02:55.369 because one speaks in Parseltongue 00:02:55.369 --> 00:02:56.798 and the other speaks in Python. 00:02:56.798 --> 00:02:59.022 And their language classes, much like math classes, 00:02:59.022 --> 00:03:02.193 focus too much on memorization and not enough on immersion 00:03:02.193 --> 00:03:03.380 But just pretend you're in math class, 00:03:03.380 --> 00:03:05.515 learning about graph theory so that I can draw the parallels 00:03:05.515 --> 00:03:07.227 because here's the 2nd doodle game 00:03:07.227 --> 00:03:08.987 which is very much mathematically related. 00:03:08.987 --> 00:03:10.983 Draw a squiggle all over the page 00:03:10.983 --> 00:03:12.679 and make sure it closes up. 00:03:12.679 --> 00:03:14.393 Pick an outside section and color it in. 00:03:14.393 --> 00:03:15.728 Now you want to alternate coloring 00:03:15.728 --> 00:03:18.426 so that no 2 faces of the same color touch. 00:03:18.426 --> 00:03:19.419 Curiously enough, 00:03:19.419 --> 00:03:20.923 much like the weaving game, 00:03:20.923 --> 00:03:23.325 this game always mathemagically works out. 00:03:23.325 --> 00:03:26.126 It also works really well if you make the lines spiky 00:03:26.126 --> 00:03:27.309 instead of a smooth curve 00:03:27.309 --> 00:03:29.188 and once again, it works with multiple lines too. 00:03:29.188 --> 00:03:30.364 It probably has something to do with 00:03:30.364 --> 00:03:32.332 the 2 colorability of graphs of even degree, 00:03:32.332 --> 00:03:33.611 which might even be what your teacher is trying to 00:03:33.611 --> 00:03:34.708 teach you at this very moment 00:03:34.708 --> 00:03:36.009 for all you're paying attention. 00:03:36.009 --> 00:03:37.524 But maybe you can chat with him after class about snakes 00:03:37.524 --> 00:03:38.692 and he'll explain it to you 00:03:38.692 --> 00:03:40.424 because I'd rather move on to the next doodle game. 00:03:40.424 --> 00:03:42.164 This is a combination of the last 2 00:03:42.164 --> 00:03:44.523 Step 1: draw a smooth closed curve 00:03:44.523 --> 00:03:46.374 Step 2: assign overs and unders 00:03:46.374 --> 00:03:48.423 Step 3: shade in every other face 00:03:48.423 --> 00:03:51.232 After that, it takes a little artistic finesse 00:03:51.232 --> 00:03:52.302 to get the shading right, 00:03:52.302 --> 00:03:53.372 but you end up with some sort of really neat surface. 00:03:53.372 --> 00:03:56.772 For example, this one only has one edge and one side 00:03:56.772 --> 00:03:58.117 but if you're interested in this, 00:03:58.117 --> 00:03:59.372 you should really be talking to 00:03:59.372 --> 00:04:00.924 your resident topology professor and not me. 00:04:00.924 --> 00:04:02.539 But here's the thing: 00:04:02.539 --> 00:04:03.840 if someone asked you 5 minutes ago 00:04:03.840 --> 00:04:05.702 what tangled up snakes, demented checkerboards, 00:04:05.702 --> 00:04:07.652 and crazy twisty surfaces have in common? 00:04:07.652 --> 00:04:08.955 what would you have answered? 00:04:08.955 --> 00:04:11.295 This is why I love mathematics: 00:04:11.295 --> 00:04:12.726 the moment when you realize that 00:04:12.726 --> 00:04:14.384 something seemingly arbitrary and confusing 00:04:14.384 --> 00:04:16.659 is actually part of something. 00:04:16.659 --> 00:04:18.739 It's better than the cleverest possible ending 00:04:18.739 --> 00:04:20.194 to any crime show or mystery novel, 00:04:20.194 --> 00:04:22.413 because that's only the beginning. 00:04:22.413 --> 00:04:24.691 Anyway, have fun with that.