Doodling in Math: Sick Number Games I don't even know if this makes sense. Boo cold. http://en.wikipedia.org/wiki/Ulam_spiral Doodling in Math Class videos: http://vihart.com/doodling
Doodling in Math: Sick Number Games
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- Pretend you're me and you're in math class. Actually... nevermind, I'm sick so I'm staying home today
- so pretend you are 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 that what you're doodling is numbers, just counting
- starting with one and spiralling them around. I'm not too fluent in mathematical notation so
- 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 that number theorists are
- enamoured with 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. Well... 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? ... o.0 ... Oh yeah, it's 3 times 9, not prime."
- "Hmmm 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 out.
- But, even though you have your primes memorised up to at least 1000 that doesn't change that
- primes, in general, are difficult to find.
- I mean if I ask you to find the highest even number, you'd say, "that's silly, just give me
- the number you think is the highest and i'll just add 2.... BAM!!"
- But guess what the highest prime number we know is? 2 to the power of 43,112,609 - 1.
- Just to give you an idea about how big a deal primes are, the guy that found this one won
- a $100,000 prize for it!
- We even sent our largest known prime number into space because scientists think
- aliens will recognise it as something important and not just some arbitrary number.
- So they will 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 you started doodling 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
- lets 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 now 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.
- Yyou can stop being Ulam now... or you can continue. Maybe you like being Ulam. (thats fine)
- However 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 predelections maybe I'll manage to infect you with my enthusiasm :D
- Pascal's Triangle is the one where you get the next row in the triangle by adding two adjacent
- numbers. Constructing Pascal's Triangle is, in itself a sort of number game because it's not just
- about adding, but about trying to find patterns and relationships in the numbers so you
- don't have to do all the adding.
- I don't know if this was discovered through doodling but it was discovered independantly 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.
- So, if you still Ulam, you pick a property and highlight it(e.g. if it's even or odd)
- If you circle all the odd numbers you'll get a form which might be starting to look familiar.
- And it makes sense you'd get Sierpinski's Triangle because when you add
- an odd number and an even number, you get an odd number.
- (odd + odd) = even and (even + even) = even... So it's just like the
- crash and burn binary tree game. The best part about it is that, if you know these properties, you can
- forget about the details of the numbers
- You don't have to know that a space contains a 9 to know that it's going to be odd.
- Now, instead of two colours, let's try three. we'll colour them depending on what the remainder is
- when you divide them by three(instead of by two).
- Here's a chart! :)
- So, all the multiples of three are coloured red, remainder of one will be coloured black and
- remainder of two will be coloured green. The structure is a little different from Sierpinski's Triangle
- already but I'm tired of figuring out remainders based of 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 everday in math class)
- However, here this means (red + red) = red.
- and when you add a multiple of three to something else, it doesn't change it's remainder.
- So, (red + green) = green and (red + black) = black.
- (remainder 1 + remainder 1) = remainder 2, (remainder 2+ remainder 2) = remainder 4
- and the remainder of 4 divided by 3 is one and (1+2) = 3 remainder 0. (whew...)
- The bottom line is you're making up some rules as to what coloured dots combine to
- produce which 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 of 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
- highlighted 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 multiple two adjacent numbers to get the next row.
- I've no idea what hapens there either or if it's already a 'thing' people do.
- (Hmmm? o.0 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...
- Maybe you can do something with that as well.
- ... Man, it seems like everyone has a triangle these days...
- I'm going to take a nap... ZZZzzz...
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