WEBVTT 00:00:00.000 --> 00:00:05.248 Pretend you're me and you're in math class. Actually... nevermind, I'm sick so I'm staying home today 00:00:05.248 --> 00:00:10.449 so pretend you are Stanislaw Ulam instead. What I am about to tell you is a true story. 00:00:10.449 --> 00:00:13.785 So you are Stan Ulam and you're at a meeting but there's this really boring presentation so 00:00:13.785 --> 00:00:17.578 of course you're doodling and, because you're Ulam and not me, you really like numbers... 00:00:17.578 --> 00:00:20.828 I mean super like them. So much that what you're doodling is numbers, just counting 00:00:20.828 --> 00:00:25.379 starting with one and spiralling them around. I'm not too fluent in mathematical notation so 00:00:25.379 --> 00:00:29.489 so i find things like numbers to be distracting, but you're a number theorist and if you love numbers 00:00:29.489 --> 00:00:32.926 who am I to judge? Thing is, because you know numbers so intimately, 00:00:32.926 --> 00:00:37.593 you can see beyond the confusing, squiggly lines you're drawing right into the heart of numbers. 00:00:37.593 --> 00:00:40.821 And, because you're a number theorist, and everyone knows that number theorists are 00:00:40.821 --> 00:00:44.327 enamoured with prime numbers( which is probably why they named them "prime numbers"), 00:00:44.327 --> 00:00:50.132 the primes you've doodled suddenly jump out at you like the exotic indivisible beasts they are... 00:00:50.132 --> 00:00:54.822 So you start drawing a heart around each prime. Well... it was actually boxes but in my version 00:00:54.822 --> 00:00:59.002 of the story it's hearts because you're not afraid to express your true feelings about prime numbers. 00:00:59.002 --> 00:01:02.833 You can probably do this instantly but it's going to take me a little longer... I'm all like - 00:01:02.833 --> 00:01:08.174 "Does 27 have factors besides one and itself? ... o.0 ... Oh yeah, it's 3 times 9, not prime." 00:01:08.174 --> 00:01:10.380 "Hmmm what about 29...? pretty sure it's prime." 00:01:10.380 --> 00:01:14.908 But as a number theorist, you'll be shocked to know it takes me a moment to figure these out. 00:01:14.908 --> 00:01:20.086 But, even though you have your primes memorised up to at least 1000 that doesn't change that 00:01:20.086 --> 00:01:22.547 primes, in general, are difficult to find. 00:01:22.547 --> 00:01:26.889 I mean if I ask you to find the highest even number, you'd say, "that's silly, just give me 00:01:26.889 --> 00:01:30.465 the number you think is the highest and i'll just add 2.... BAM!!" 00:01:30.465 --> 00:01:39.706 But guess what the highest prime number we know is? 2 to the power of 43,112,609 - 1. 00:01:39.706 --> 00:01:44.510 Just to give you an idea about how big a deal primes are, the guy that found this one won 00:01:44.510 --> 00:01:47.026 a $100,000 prize for it! 00:01:47.026 --> 00:01:50.821 We even sent our largest known prime number into space because scientists think 00:01:50.821 --> 00:01:54.683 aliens will recognise it as something important and not just some arbitrary number. 00:01:54.683 --> 00:01:57.423 So they will be able to figure out our alien space message... 00:01:57.423 --> 00:02:00.163 So if you ever think you don't care about prime numbers because they're 'not useful', 00:02:00.163 --> 00:02:04.807 remember that we use prime numbers to talk to aliens, I'm not even making this up! 00:02:04.807 --> 00:02:10.310 It makes sense, because mathematics is probably one of the only things all life has in common. 00:02:10.310 --> 00:02:14.342 Anyway, the point is you started doodling because you were bored but ended up 00:02:14.342 --> 00:02:18.205 discovering some neat patterns. See how the primes tend to line up on the diagonals? 00:02:18.205 --> 00:02:21.990 Why do they do that?... also this sort of skeletal structure reminds me of bones so 00:02:21.990 --> 00:02:25.821 lets call these diagonal runs of primes: Prime Ribs! 00:02:25.821 --> 00:02:29.769 But how do you predict when a Prime Rib will end? I mean, maybe this next number is prime... 00:02:29.769 --> 00:02:32.694 (but my head is too fuzzy for now this right now so you tell me.) 00:02:32.694 --> 00:02:35.922 Anyway...Congratulations, You've discovered the Ulam Spiral! 00:02:35.922 --> 00:02:38.035 So that's a little mathematical doodling history for you. 00:02:38.035 --> 00:02:41.216 Yyou can stop being Ulam now... or you can continue. Maybe you like being Ulam. (thats fine) 00:02:41.216 --> 00:02:46.719 However you could also be Blaise Pascal. Here's another number game you can do using 00:02:46.719 --> 00:02:50.063 Pascal's triangle.(I don't know why I'm so into numbers today but I have a cold so 00:02:50.063 --> 00:02:55.078 if you'll just indulge my sick predelections maybe I'll manage to infect you with my enthusiasm :D 00:02:55.078 --> 00:02:59.119 Pascal's Triangle is the one where you get the next row in the triangle by adding two adjacent 00:02:59.119 --> 00:03:03.781 numbers. Constructing Pascal's Triangle is, in itself a sort of number game because it's not just 00:03:03.781 --> 00:03:07.380 about adding, but about trying to find patterns and relationships in the numbers so you 00:03:07.380 --> 00:03:09.161 don't have to do all the adding. 00:03:09.161 --> 00:03:12.610 I don't know if this was discovered through doodling but it was discovered independantly in: 00:03:12.610 --> 00:03:16.603 France, Italy, Persia, China and probably other places too so it's possible someone did. 00:03:16.603 --> 00:03:19.436 Right... so I don't actually care about the individual numbers right now. 00:03:19.436 --> 00:03:24.754 So, if you still Ulam, you pick a property and highlight it(e.g. if it's even or odd) 00:03:24.754 --> 00:03:29.746 If you circle all the odd numbers you'll get a form which might be starting to look familiar. 00:03:29.746 --> 00:03:33.117 And it makes sense you'd get Sierpinski's Triangle because when you add 00:03:33.117 --> 00:03:35.412 an odd number and an even number, you get an odd number. 00:03:35.412 --> 00:03:38.244 (odd + odd) = even and (even + even) = even... So it's just like the 00:03:38.244 --> 00:03:42.935 crash and burn binary tree game. The best part about it is that, if you know these properties, you can 00:03:42.935 --> 00:03:45.698 forget about the details of the numbers 00:03:45.698 --> 00:03:50.481 You don't have to know that a space contains a 9 to know that it's going to be odd. 00:03:50.481 --> 00:03:55.799 Now, instead of two colours, let's try three. we'll colour them depending on what the remainder is 00:03:55.799 --> 00:03:57.935 when you divide them by three(instead of by two). 00:03:57.935 --> 00:03:59.299 Here's a chart! :) 00:03:59.299 --> 00:04:03.327 So, all the multiples of three are coloured red, remainder of one will be coloured black and 00:04:03.327 --> 00:04:07.863 remainder of two will be coloured green. The structure is a little different from Sierpinski's Triangle 00:04:07.863 --> 00:04:12.359 already but I'm tired of figuring out remainders based of individual numbers, so 00:04:12.359 --> 00:04:16.046 Let's figure out the rules... If you add up two multiples of three you always get 00:04:16.076 --> 00:04:21.201 another multiple of three( which is the sort of fact you use everday in math class) 00:04:21.201 --> 00:04:22.943 However, here this means (red + red) = red. 00:04:22.943 --> 00:04:26.403 and when you add a multiple of three to something else, it doesn't change it's remainder. 00:04:26.403 --> 00:04:30.884 So, (red + green) = green and (red + black) = black. 00:04:30.884 --> 00:04:34.713 (remainder 1 + remainder 1) = remainder 2, (remainder 2+ remainder 2) = remainder 4 00:04:34.713 --> 00:04:40.288 and the remainder of 4 divided by 3 is one and (1+2) = 3 remainder 0. (whew...) 00:04:40.288 --> 00:04:43.585 The bottom line is you're making up some rules as to what coloured dots combine to 00:04:43.585 --> 00:04:46.279 produce which other coloured dots and then you're following those rules to their 00:04:46.279 --> 00:04:49.344 mathematical and artistic conclusion... 00:04:49.344 --> 00:04:52.525 The numbers themselves were never necessary to get this picture. 00:04:52.525 --> 00:04:56.751 Anyway, those are just a couple of examples of number games that are out there but you should 00:04:56.751 --> 00:05:00.768 also try making up your own. For example, I have no idea what you'd get if you 00:05:00.768 --> 00:05:04.065 highlighted the prime numbers in Pascal's Triangle, maybe nothing interesting(who knows...) 00:05:04.065 --> 00:05:07.895 Or, what happens if, instead of adding to get the next row, you start with a two(and a sea of invisible ones) 00:05:07.895 --> 00:05:10.913 and multiple two adjacent numbers to get the next row. 00:05:10.913 --> 00:05:13.516 I've no idea what hapens there either or if it's already a 'thing' people do. 00:05:13.516 --> 00:05:16.316 (Hmmm? o.0 Powers of two...) 00:05:16.316 --> 00:05:19.205 I know another way to write this. Ok, that makes sense. 00:05:19.205 --> 00:05:24.243 Then there is also a thing called Floyd's Triangle where you put the numbers like this... 00:05:24.243 --> 00:05:25.604 Maybe you can do something with that as well. 00:05:25.604 --> 00:05:26.892 ... Man, it seems like everyone has a triangle these days... 00:05:26.907 --> 00:05:28.000 I'm going to take a nap... ZZZzzz...