GIRL
Pretend you're me and you're in math class. Actually, never mind. I'm sick, so I'm staying home today. So please pretend you're Stanislaw Ulam instead.
What I am about to tell you is a true story. So you are Stan Ulam and you're at a meeting, but there's this really boring presentation so, of course, you're doodling, and, because you're Ulam and not me, you really like numbers, I mean 'super' like them. So much so what you're doodling is numbers, just counting starting with one and spiralling them around. I'm not too fluent in mathematical notation so I find things like numbers to be distracting, but you're a number theorist and if you love numbers who am I to judge? Thing is, because you know numbers so intimately, you can see beyond the confusing, squiggly lines you're drawing right into the heart of numbers. And, because you're a number theorist, and everyone knows number theorists are enamoured of prime numbers, which is probably why they named them 'prime numbers', the primes you've doodled suddenly jump out at you like the exotic indivisible beasts they are. So you start drawing a heart around each prime. I mean, it was actually boxes, but in my version of the story it's hearts because you're not afraid to express your true feelings about prime numbers. You can probably do this instantly, but it's going to take me a little longer. I'm all like, 'Does 27 have factors besides one and itself? Oh yeah, it's 3 times 9, not prime. What about 29...? Pretty sure it's prime.' But as a number theorist, you'll be shocked to know it takes me a moment to figure these things out. But, even though you've got your primes memorised up to at least 1,000, that doesn't change that primes, in general, are difficult to find. I mean, if I ask you to find me the highest even number, you'd be like, 'That's silly. Give me the even number you think is the highest and I'll just add 2 to it. Bam!' But guess what the highest prime number we know is? 2 to the power of 43,112,609 minus 1. Just to give you an idea of what a big deal primes are, the guy that found this one won a $100,000 prize for it! I mean, we sent our largest known prime number into space because scientists think aliens will recognise it as something important, not just some arbitrary number. So they'll be able to figure out our alien space message. So if you ever think you don't care about prime numbers because they're 'not useful', remember that we use prime numbers to talk to aliens, I'm not even making this up! It makes sense, because mathematics is probably one of the only things all life has in common. Anyway, the point is that you started doodling this because you were bored, but ended up discovering some neat patterns. See how the primes tend to line up on the diagonals? Why do they do that? Also this sort of skeletal structure reminds me of bones, so let's call these diagonal runs of primes 'prime ribs'! But how do you predict when a prime rib will end? I mean, maybe this next number is prime, but my head is too fuzzy for this right now, so you tell me. Anyway, congratulations. You've discovered the Ulam Spiral! So that's a little mathematical doodling history for you. You can stop being Ulam now, or you could continue. Maybe you like being Ulam. That's fine. But you could also be Blaise Pascal. Here's another number game you can do using Pascal's Triangle. I don't know why I'm so into numbers today, but I have a cold, so if you'll just indulge my sick predilections maybe I'll manage to infect you with my enthusiasm. Pascal's Triangle is the one where you get the next row in the triangle by adding up two adjacent numbers. Constructing Pascal's Triangle is, in itself, a sort of number game, because it's not just about practising adding, but trying to find patterns and relationships in the numbers so that you don't have to do all the adding. I don't know if this was discovered through doodling, but it was discovered independently in France, Italy, Persia, China and probably other places too, so it's possible someone did. Right, so I don't actually care about the individual numbers right now. You're still Ulam, right? So like you would do, you pick a property and highlight it - for example, whether it's even or odd. If you circle all the odd numbers, you'll get a form which might be starting to look familiar to you. And it makes sense that you'd get Sierpinski's Triangle because an odd number plus an even number equals an odd number. Odd plus odd is even, and even plus even is even. So it's just like the crash and burn binary tree game. The best part about it is that, knowing these properties, you can forget about the details of the numbers. You don't have to know that this number will be nine to know that it's gonna be odd. But instead of two colours, let's try three. We'll colour them depending on what the remainder is when you divide the number by three instead of two. Here's a chart! So all the multiples of three I'm gonna colour red, remainder of one will be black and two will be green. The structure is a little different from Sierpinski's Triangle already, but I'm tired of figuring out remainders based on individual numbers, so let's figure out the rules. If you add up two multiples of three, you always get another multiple of three, which is the sort of fact you use every day in math class. But here it means red plus red is red. And when you add a multiple of three to something else, it doesn't change the remainder of that something else, So red plus green is green and red plus black is black. Remainder one plus remainder one equals remainder two. Two plus two is four, and the remainder of four divided by three is one, and one plus two is three, remainder zero. The bottom line is that you're making up some rules as to what coloured dots combine to get what other coloured dots, and then you're following those rules to their mathematical and artistic conclusion. The numbers themselves were never necessary to get this picture. Anyway, those are just a couple examples of number games that are out there, but you should also try making up your own. For example, I have no idea what you'd get if you highlight the prime numbers in Pascal's Triangle. Maybe nothing interesting, who knows. Or, what happens if, instead of adding to get the next row, you start with a two and a sea of invisible ones and multiply to get the next row. No idea what happens there either or if it's already a thing people do or what. Oh, wait, powers of two... I know another way to write this. OK, that makes sense. Then there is also a thing called Floyd's Triangle, where you put the numbers like this, so maybe you can do something with that too. Man, it seems like everyone has a triangle these days. I'm going to take a nap.