1 00:00:00,697 --> 00:00:02,763 Okay, let's say you're me and you're in math class 2 00:00:02,763 --> 00:00:04,371 and you're supposed to be learning about 3 00:00:04,371 --> 00:00:05,690 exponential functions, but you're having 4 00:00:05,690 --> 00:00:07,674 trouble caring about exponential functions 5 00:00:07,674 --> 00:00:10,077 because unfortunately your math class 6 00:00:10,077 --> 00:00:11,745 is probably not terribly engaging. 7 00:00:11,745 --> 00:00:14,721 You're supposed to be drawing and labeling some axes 8 00:00:14,721 --> 00:00:16,904 so that you can graph this y equals 2 to the x thing 9 00:00:16,904 --> 00:00:18,274 and your teacher seems to think 10 00:00:18,274 --> 00:00:19,876 that drawing and labeling axes 11 00:00:19,876 --> 00:00:21,949 is the very essence of mathematics 12 00:00:21,949 --> 00:00:24,543 but you're bored and you can't help but wonder... 13 00:00:24,543 --> 00:00:25,797 Why? 14 00:00:25,797 --> 00:00:28,398 So you do what any conscientious student would do 15 00:00:28,398 --> 00:00:29,697 in this situation and start doodling 16 00:00:29,697 --> 00:00:31,207 and because you're me 17 00:00:31,207 --> 00:00:33,321 you like to play games with yourself when you doodle. 18 00:00:33,321 --> 00:00:34,505 Here's one game. 19 00:00:34,505 --> 00:00:35,991 You're drawing a line, but when it crosses 20 00:00:35,991 --> 00:00:37,941 one of the blue lines on your little piece of paper 21 00:00:37,941 --> 00:00:39,427 it splits into two lines. 22 00:00:39,427 --> 00:00:42,191 Maybe this line is like the neck of the mythical hydra 23 00:00:42,191 --> 00:00:44,257 where every time one of its heads gets chopped off 24 00:00:44,257 --> 00:00:47,601 by a blue line, it grows two more in its place. 25 00:00:47,601 --> 00:00:49,163 You want to see if you can get all the way 26 00:00:49,163 --> 00:00:50,829 to the bottom of the page following this rule 27 00:00:50,829 --> 00:00:52,864 because if you do, you can draw 28 00:00:52,864 --> 00:00:54,991 all of the little hydra heads at the end. 29 00:00:54,991 --> 00:00:56,977 But you don't get very far on your first try. 30 00:00:56,977 --> 00:00:58,793 You decide to try again, this time 31 00:00:58,793 --> 00:01:00,883 spacing things out a little more at the beginning. 32 00:01:00,883 --> 00:01:02,368 Unfortunately, things are filling up fast, 33 00:01:02,368 --> 00:01:03,832 though you got farther than last time. 34 00:01:03,832 --> 00:01:04,903 Maybe if you had more room. 35 00:01:04,903 --> 00:01:06,754 Or maybe if you sharpened your pencil more, 36 00:01:06,754 --> 00:01:08,239 you could get to the bottom of the page. 37 00:01:08,239 --> 00:01:10,333 Oh, and don't forget to draw and label your axes. 38 00:01:10,333 --> 00:01:12,831 If each broad swing of Hercules' sword chops off 39 00:01:12,831 --> 00:01:14,613 all the heads thus doubling your number, well, 40 00:01:14,613 --> 00:01:16,281 you can see where I'm going. 41 00:01:16,281 --> 00:01:17,767 I'm not going to try and teach you math, 42 00:01:17,767 --> 00:01:19,616 just how to wield it for doodling purposes. 43 00:01:19,616 --> 00:01:21,704 In this case, that's going to be a lot of heads. 44 00:01:21,704 --> 00:01:23,615 Good luck Hercules. 45 00:01:23,615 --> 00:01:26,448 But maybe drawing binary trees all straight like that 46 00:01:26,448 --> 00:01:28,845 is not an interesting game to hold your attention for long 47 00:01:28,845 --> 00:01:30,847 so you start drawing them in arbitrary shapes. 48 00:01:30,847 --> 00:01:32,907 Or less arbitrary shapes. 49 00:01:32,907 --> 00:01:35,002 Maybe you start drawing a binary tree 50 00:01:35,002 --> 00:01:36,237 that looks like a tree. 51 00:01:36,237 --> 00:01:38,172 And maybe you can't see this tree in very high quality 52 00:01:38,172 --> 00:01:40,207 because your camera, very much like your math class 53 00:01:40,207 --> 00:01:43,544 is fuzzy, unfocused, and all together not very good. 54 00:01:43,544 --> 00:01:45,379 Maybe you changed the rules slightly and 55 00:01:45,379 --> 00:01:47,601 make a ternary bush where each branch sprouts 56 00:01:47,601 --> 00:01:48,963 three more branches. 57 00:01:48,963 --> 00:01:51,837 Unfortunately, your math class is 45 minutes long 58 00:01:51,837 --> 00:01:54,206 and soon you need a more interesting doodle game. 59 00:01:54,206 --> 00:01:56,241 Say you go back to the game 60 00:01:56,241 --> 00:01:57,642 where your line splits at every level. 61 00:01:57,642 --> 00:02:00,120 Only this time, instead of trying to squish all the lines in, 62 00:02:00,120 --> 00:02:02,481 you let them hit each other. 63 00:02:02,481 --> 00:02:04,601 And when they crash, there's a firey explosion 64 00:02:04,601 --> 00:02:07,810 and the crashing lines end there. 65 00:02:07,810 --> 00:02:09,256 Maybe you turn your notebook sideways 66 00:02:09,256 --> 00:02:10,370 so that you can make sure 67 00:02:10,370 --> 00:02:12,479 you're getting the horizontal spacing right. 68 00:02:12,479 --> 00:02:14,225 Maybe, to go back to mythology, 69 00:02:14,225 --> 00:02:16,212 Hercules has a method where, instead of cauterizing 70 00:02:16,212 --> 00:02:19,194 the necks of the Hydra to keep them from growing back 71 00:02:19,209 --> 00:02:21,978 he's found that the necks stick together if they get too close 72 00:02:21,978 --> 00:02:23,917 And instead of growing new heads, 73 00:02:23,917 --> 00:02:26,216 they just fill up with blood. 74 00:02:26,216 --> 00:02:28,035 It might seem a little morbid for math class, 75 00:02:28,035 --> 00:02:31,255 but maybe, if the curriculum wasn't so appalling 76 00:02:31,255 --> 00:02:33,447 and the teaching methods wern't so atrocious, 77 00:02:33,447 --> 00:02:35,231 you wouldn't have to entertain yourself 78 00:02:35,231 --> 00:02:37,151 with these stories and games. 79 00:02:37,151 --> 00:02:38,634 Speaking of this doodle game, 80 00:02:38,634 --> 00:02:40,491 something very interesting is happening. 81 00:02:40,491 --> 00:02:42,574 Looks like your simple rules about splitting 82 00:02:42,574 --> 00:02:45,628 and crashing are creating Sierpinski's triangle. 83 00:02:45,628 --> 00:02:47,112 Which is a pretty awesome fractal. 84 00:02:47,112 --> 00:02:48,514 But the point is not to learn about fractals 85 00:02:48,514 --> 00:02:50,767 or cellular automata or Sierpinski, 86 00:02:50,767 --> 00:02:52,369 but to show that simple doodle games 87 00:02:52,369 --> 00:02:53,668 can lead to mathematical results 88 00:02:53,668 --> 00:02:56,806 so cool and beautiful that they're famous. 89 00:02:56,806 --> 00:02:59,304 At least, famous to people like me. 90 00:02:59,304 --> 00:03:00,651 And if you're good at inventing doodle games, 91 00:03:00,651 --> 00:03:01,793 you might even end up 92 00:03:01,793 --> 00:03:05,760 doing some real mathematics during your math class. 93 00:03:05,760 --> 00:03:08,709 Anyway, maybe you don't care about accuracy. 94 00:03:08,709 --> 00:03:09,581 Maybe you try the game again, 95 00:03:09,581 --> 00:03:11,506 only you don't keep track of spacing, 96 00:03:11,506 --> 00:03:12,116 and when you make a mistake and 97 00:03:12,116 --> 00:03:14,020 accidentally grow heads where you shouldn't 98 00:03:14,020 --> 00:03:15,785 you just roll with it. 99 00:03:15,785 --> 00:03:18,043 Now you've introduced an element of random error 100 00:03:18,043 --> 00:03:19,313 and you want to know 101 00:03:19,313 --> 00:03:20,982 how this will affect the final picture. 102 00:03:20,982 --> 00:03:22,966 It still looks like a pretty awesome doodle, 103 00:03:22,966 --> 00:03:24,452 and has many of the same elements, 104 00:03:24,452 --> 00:03:25,810 though it lacks the structure. 105 00:03:25,810 --> 00:03:27,001 Speaking of the structure, 106 00:03:27,001 --> 00:03:28,916 maybe because you're really super bored 107 00:03:28,916 --> 00:03:31,696 and your class is seemingly never going to end, 108 00:03:31,696 --> 00:03:34,613 you start looking at the number of necks at each level 109 00:03:34,613 --> 00:03:37,199 and trying to figure out the pattern. 110 00:03:37,199 --> 00:03:40,056 Maybe you haven't forgotten about powers of two. 111 00:03:40,056 --> 00:03:41,453 Anyway, I hope I've provided you with 112 00:03:41,453 --> 00:03:44,583 something entertaining to do next time you're bored. 113 00:03:44,583 --> 00:03:48,000 Good luck with your math class.