WEBVTT 00:00:00.000 --> 00:00:03.276 Let's say you're me and you're in math class and you're supposed to be learning about factoring. 00:00:03.276 --> 00:00:07.144 Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill 00:00:07.144 --> 00:00:10.580 for the average person to know with real-world applications ranging from passing your state exams 00:00:10.580 --> 00:00:12.026 all the way to getting a higher SAT score 00:00:12.026 --> 00:00:14.376 and unfortunately does not have the time to show you 00:00:14.376 --> 00:00:16.387 why factoring is actually interesting. 00:00:16.387 --> 00:00:19.362 It's perfectly reasonable for you to get bored in this situation. 00:00:19.362 --> 00:00:21.932 So like any reasonable person, you start doodling. 00:00:21.932 --> 00:00:24.507 Maybe it's because your teacher's sophorific voice reminds you of a lullaby 00:00:24.522 --> 00:00:27.007 but you're drawing stars. 00:00:27.007 --> 00:00:29.373 And because you're me, you quickly get bored of the usual 5-pointed star 00:00:29.373 --> 00:00:31.696 and get to wondering: why five? 00:00:31.696 --> 00:00:33.478 So you start exploring. 00:00:33.478 --> 00:00:35.846 It seems obvious that a 5-pointed star is the simplest one- 00:00:35.846 --> 00:00:38.249 the one that takes the least number of strokes to draw. 00:00:38.249 --> 00:00:41.000 Sure you can make a star with 4 points but that's not really a star 00:00:41.000 --> 00:00:42.578 the way you're defining stars. 00:00:42.578 --> 00:00:44.966 Then there's the 6-pointed star which is also familiar 00:00:44.981 --> 00:00:46.898 but totally different from the 5-pointed star because 00:00:46.898 --> 00:00:48.669 it takes 2 seperate lines to make. 00:00:48.669 --> 00:00:50.944 And then you're thinking about how 00:00:50.944 --> 00:00:52.973 much like you can put 2 triangles together to make a 6-pointed star, 00:00:52.973 --> 00:00:55.380 you can put two squares together to make an 8-pointed star. 00:00:55.380 --> 00:01:00.012 And any even numbered star with "p" points can be made of 2 "p over 2" gons. 00:01:00.014 --> 00:01:03.878 It is at this point that you realize if you wanted to avoid thinking about factoring 00:01:03.878 --> 00:01:06.505 maybe drawing stars was not the brightest idea. 00:01:06.505 --> 00:01:09.341 But wait! 4 would be an even number of points 00:01:09.341 --> 00:01:11.276 but that would mean you could make it out of 2 "2-gons" 00:01:11.276 --> 00:01:13.778 Maybe you were taught polygons with only two sides can't exist 00:01:13.778 --> 00:01:16.881 but for the purposes of drawing stars it works out rather well. 00:01:16.881 --> 00:01:19.203 Sure, the 4-pointed star doesn't look too star-like 00:01:19.203 --> 00:01:22.547 But then you realize that you can make a 6-pointed star out of 3 of these things 00:01:22.547 --> 00:01:25.380 and you've got an asterisk, which is definitely a legitimate star. 00:01:25.380 --> 00:01:28.478 In fact, for any star with a number of points that is divisible by 2 00:01:28.478 --> 00:01:30.713 you can draw it asterisk style. 00:01:30.713 --> 00:01:32.181 But that's not quite what you're looking for 00:01:32.181 --> 00:01:34.335 what you want is a doodle game, and here it is: 00:01:34.335 --> 00:01:36.337 draw "p" points in a circle, evenly spaced. 00:01:36.337 --> 00:01:38.197 Pick a number "q". 00:01:38.197 --> 00:01:41.075 Starting at one point, go around the circle and connect to the point q places over. 00:01:41.075 --> 00:01:42.676 Repeat. 00:01:42.676 --> 00:01:44.578 If you get to the starting place before you've covered all the points 00:01:44.578 --> 00:01:46.046 jump to a lonely point and keep going. 00:01:46.046 --> 00:01:47.381 That's how you draw stars. 00:01:47.381 --> 00:01:49.835 And it's a successfull game in that previously you were considering 00:01:49.835 --> 00:01:51.336 running, screaming, from the room 00:01:51.336 --> 00:01:53.383 or the window is open so that's an option too. 00:01:53.383 --> 00:01:55.049 But now you're not only entertained, 00:01:55.049 --> 00:01:57.743 but beginning to become curious about the nature of this game. 00:01:57.743 --> 00:01:59.444 The interesting thing is that the more points you have, 00:01:59.444 --> 00:02:01.680 the more different ways there is to draw the star. 00:02:01.680 --> 00:02:05.039 I happen to like 7-pointed stars because there's two really good ways to draw them. 00:02:05.039 --> 00:02:06.536 but they're still simple. 00:02:06.536 --> 00:02:09.814 I would like to note here that I have never actually left a math class via the window, 00:02:09.814 --> 00:02:12.008 not that I can say the same for other subjects. 00:02:12.008 --> 00:02:14.925 8 is interesting too, because not only are there a couple nice ways to draw it, 00:02:14.925 --> 00:02:16.412 but one's a composite of two polygons 00:02:16.412 --> 00:02:19.148 while another can be drawn without picking up the pencil. 00:02:19.148 --> 00:02:20.573 Then there's 9, 00:02:20.573 --> 00:02:22.872 which, in addition to a couple of other nice versions, you can make out of 3 triangles. 00:02:22.872 --> 00:02:25.434 And, because you're me, and you're a nerd, and you like to amuse yourself, 00:02:25.434 --> 00:02:27.842 you decide to call this kind of star a Square Star. 00:02:27.842 --> 00:02:29.844 because that's kind of a funny name. 00:02:29.844 --> 00:02:31.445 So you start drawing other square stars. 00:02:31.445 --> 00:02:32.333 4 4-gons, 00:02:32.333 --> 00:02:33.512 2 2-gons, 00:02:33.512 --> 00:02:35.847 even the completely degenerate case of 1 1-gon. 00:02:35.847 --> 00:02:38.754 Unfortunately 5 pentagons is already difficult to discern, 00:02:38.754 --> 00:02:41.808 and beyond that it's very hard to see and appreciate the structure of square stars. 00:02:41.808 --> 00:02:44.176 So you get bored and move on to 10 dots and a circle, 00:02:44.176 --> 00:02:47.379 which is interesting because this is the first number where you can make a star 00:02:47.379 --> 00:02:48.991 as a composite of smaller stars, 00:02:48.991 --> 00:02:50.924 that is, 2 boring old 5-pointed stars. 00:02:50.924 --> 00:02:54.261 Unless you count asterisk stars, in which case 8 was 2 4s, or 4 2's. or 2 2's and a 4. 00:02:54.261 --> 00:02:57.749 But 10 is interesting, because you can make it as a composite in more than one way. 00:02:57.749 --> 00:03:01.511 because it's divisible by five which itself can be made in 2 ways. 00:03:01.511 --> 00:03:05.749 Then there's 11, which can't be made out of seperate parts at all, because 11 is prime. 00:03:05.749 --> 00:03:09.177 Though here you start to wonder how to predict how many times around the circle 00:03:09.177 --> 00:03:10.754 it will go before getting back to the start. 00:03:10.754 --> 00:03:14.208 But instead of exploring the exciting world of modular arithmetic, you move on to 12 00:03:14.208 --> 00:03:16.169 which is a really cool number 00:03:16.169 --> 00:03:17.562 because it has a whole bunch of factors. 00:03:17.562 --> 00:03:18.954 And then something starts to bother you: 00:03:18.954 --> 00:03:23.198 Is a 25 pointed star composite made of 5 5-pointed stars a Square Star? 00:03:23.198 --> 00:03:26.541 You've been thinking only of pentagons because the lower numbers didn't have this question. 00:03:26.541 --> 00:03:28.273 How could you have missed that? 00:03:28.273 --> 00:03:29.942 Maybe your teacher said something interesting about prime numbers 00:03:29.942 --> 00:03:31.831 and you accidentally lost focus for a moment. 00:03:31.831 --> 00:03:32.331 And, oh no. 00:03:32.831 --> 00:03:33.879 It gets even worse. 00:03:33.879 --> 00:03:36.619 6 squared would be a 36 pointed star, made of 6 hexagons. 00:03:36.619 --> 00:03:39.545 but if you allow use of 6 pointed stars, then it's the same as 00:03:39.545 --> 00:03:41.037 a composite of 12 triangles. 00:03:41.037 --> 00:03:44.049 And that doesn't seem in keeping with the spirit of square stars. 00:03:44.049 --> 00:03:46.176 You'll have to define square stars more strictly. 00:03:46.176 --> 00:03:49.813 But you do like the idea that there's three ways to make the 7th square star. 00:03:49.813 --> 00:03:53.196 Anyway, the whole theory of what kind of stars can be made with what numbers 00:03:53.196 --> 00:03:54.035 is quite interesting 00:03:54.035 --> 00:03:56.874 and I encourage you to explore this during your math class.