0:00:00.000,0:00:03.276 Let's say you're me and you're in math class and you're supposed to be learning about factoring. 0:00:03.276,0:00:07.144 Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill 0:00:07.144,0:00:10.580 for the average person to know with real-world applications ranging from passing your state exams 0:00:10.580,0:00:12.026 all the way to getting a higher SAT score 0:00:12.026,0:00:14.376 and unfortunately does not have the time to show you 0:00:14.376,0:00:16.387 why factoring is actually interesting. 0:00:16.387,0:00:19.362 It's perfectly reasonable for you to get bored in this situation. 0:00:19.362,0:00:21.932 So like any reasonable person, you start doodling. 0:00:21.932,0:00:24.507 Maybe it's because your teacher's sophorific voice reminds you of a lullaby 0:00:24.522,0:00:27.007 but you're drawing stars. 0:00:27.007,0:00:29.373 And because you're me, you quickly get bored of the usual 5-pointed star 0:00:29.373,0:00:31.696 and get to wondering: why five? 0:00:31.696,0:00:33.478 So you start exploring. 0:00:33.478,0:00:35.846 It seems obvious that a 5-pointed star is the simplest one- 0:00:35.846,0:00:38.249 the one that takes the least number of strokes to draw. 0:00:38.249,0:00:41.000 Sure you can make a star with 4 points but that's not really a star 0:00:41.000,0:00:42.578 the way you're defining stars. 0:00:42.578,0:00:44.966 Then there's the 6-pointed star which is also familiar 0:00:44.981,0:00:46.898 but totally different from the 5-pointed star because 0:00:46.898,0:00:48.669 it takes 2 seperate lines to make. 0:00:48.669,0:00:50.944 And then you're thinking about how 0:00:50.944,0:00:52.973 much like you can put 2 triangles together to make a 6-pointed star, 0:00:52.973,0:00:55.380 you can put two squares together to make an 8-pointed star. 0:00:55.380,0:01:00.012 And any even numbered star with "p" points can be made of 2 "p over 2" gons. 0:01:00.014,0:01:03.878 It is at this point that you realize if you wanted to avoid thinking about factoring 0:01:03.878,0:01:06.505 maybe drawing stars was not the brightest idea. 0:01:06.505,0:01:09.341 But wait! 4 would be an even number of points 0:01:09.341,0:01:11.276 but that would mean you could make it out of 2 "2-gons" 0:01:11.276,0:01:13.778 Maybe you were taught polygons with only two sides can't exist 0:01:13.778,0:01:16.881 but for the purposes of drawing stars it works out rather well. 0:01:16.881,0:01:19.203 Sure, the 4-pointed star doesn't look too star-like 0:01:19.203,0:01:22.547 But then you realize that you can make a 6-pointed star out of 3 of these things 0:01:22.547,0:01:25.380 and you've got an asterisk, which is definitely a legitimate star. 0:01:25.380,0:01:28.478 In fact, for any star with a number of points that is divisible by 2 0:01:28.478,0:01:30.713 you can draw it asterisk style. 0:01:30.713,0:01:32.181 But that's not quite what you're looking for 0:01:32.181,0:01:34.335 what you want is a doodle game, and here it is: 0:01:34.335,0:01:36.337 draw "p" points in a circle, evenly spaced. 0:01:36.337,0:01:38.197 Pick a number "q". 0:01:38.197,0:01:41.075 Starting at one point, go around the circle and connect to the point q places over. 0:01:41.075,0:01:42.676 Repeat. 0:01:42.676,0:01:44.578 If you get to the starting place before you've covered all the points 0:01:44.578,0:01:46.046 jump to a lonely point and keep going. 0:01:46.046,0:01:47.381 That's how you draw stars. 0:01:47.381,0:01:49.835 And it's a successfull game in that previously you were considering 0:01:49.835,0:01:51.336 running, screaming, from the room 0:01:51.336,0:01:53.383 or the window is open so that's an option too. 0:01:53.383,0:01:55.049 But now you're not only entertained, 0:01:55.049,0:01:57.743 but beginning to become curious about the nature of this game. 0:01:57.743,0:01:59.444 The interesting thing is that the more points you have, 0:01:59.444,0:02:01.680 the more different ways there is to draw the star. 0:02:01.680,0:02:05.039 I happen to like 7-pointed stars because there's two really good ways to draw them. 0:02:05.039,0:02:06.536 but they're still simple. 0:02:06.536,0:02:09.814 I would like to note here that I have never actually left a math class via the window, 0:02:09.814,0:02:12.008 not that I can say the same for other subjects. 0:02:12.008,0:02:14.925 8 is interesting too, because not only are there a couple nice ways to draw it, 0:02:14.925,0:02:16.412 but one's a composite of two polygons 0:02:16.412,0:02:19.148 while another can be drawn without picking up the pencil. 0:02:19.148,0:02:20.573 Then there's 9, 0:02:20.573,0:02:22.872 which, in addition to a couple of other nice versions, you can make out of 3 triangles. 0:02:22.872,0:02:25.434 And, because you're me, and you're a nerd, and you like to amuse yourself, 0:02:25.434,0:02:27.842 you decide to call this kind of star a Square Star. 0:02:27.842,0:02:29.844 because that's kind of a funny name. 0:02:29.844,0:02:31.445 So you start drawing other square stars. 0:02:31.445,0:02:32.333 4 4-gons, 0:02:32.333,0:02:33.512 2 2-gons, 0:02:33.512,0:02:35.847 even the completely degenerate case of 1 1-gon. 0:02:35.847,0:02:38.754 Unfortunately 5 pentagons is already difficult to discern, 0:02:38.754,0:02:41.808 and beyond that it's very hard to see and appreciate the structure of square stars. 0:02:41.808,0:02:44.176 So you get bored and move on to 10 dots and a circle, 0:02:44.176,0:02:47.379 which is interesting because this is the first number where you can make a star 0:02:47.379,0:02:48.991 as a composite of smaller stars, 0:02:48.991,0:02:50.924 that is, 2 boring old 5-pointed stars. 0:02:50.924,0:02:54.261 Unless you count asterisk stars, in which case 8 was 2 4s, or 4 2's. or 2 2's and a 4. 0:02:54.261,0:02:57.749 But 10 is interesting, because you can make it as a composite in more than one way. 0:02:57.749,0:03:01.511 because it's divisible by five which itself can be made in 2 ways. 0:03:01.511,0:03:05.749 Then there's 11, which can't be made out of seperate parts at all, because 11 is prime. 0:03:05.749,0:03:09.177 Though here you start to wonder how to predict how many times around the circle 0:03:09.177,0:03:10.754 it will go before getting back to the start. 0:03:10.754,0:03:14.208 But instead of exploring the exciting world of modular arithmetic, you move on to 12 0:03:14.208,0:03:16.169 which is a really cool number 0:03:16.169,0:03:17.562 because it has a whole bunch of factors. 0:03:17.562,0:03:18.954 And then something starts to bother you: 0:03:18.954,0:03:23.198 Is a 25 pointed star composite made of 5 5-pointed stars a Square Star? 0:03:23.198,0:03:26.541 You've been thinking only of pentagons because the lower numbers didn't have this question. 0:03:26.541,0:03:28.273 How could you have missed that? 0:03:28.273,0:03:29.942 Maybe your teacher said something interesting about prime numbers 0:03:29.942,0:03:31.831 and you accidentally lost focus for a moment. 0:03:31.831,0:03:32.331 And, oh no. 0:03:32.831,0:03:33.879 It gets even worse. 0:03:33.879,0:03:36.619 6 squared would be a 36 pointed star, made of 6 hexagons. 0:03:36.619,0:03:39.545 but if you allow use of 6 pointed stars, then it's the same as 0:03:39.545,0:03:41.037 a composite of 12 triangles. 0:03:41.037,0:03:44.049 And that doesn't seem in keeping with the spirit of square stars. 0:03:44.049,0:03:46.176 You'll have to define square stars more strictly. 0:03:46.176,0:03:49.813 But you do like the idea that there's three ways to make the 7th square star. 0:03:49.813,0:03:53.196 Anyway, the whole theory of what kind of stars can be made with what numbers 0:03:53.196,0:03:54.035 is quite interesting 0:03:54.035,0:03:56.874 and I encourage you to explore this during your math class.