Fractal Fractions

Vi Hart

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Fractal Fractions How to make snazzy-lookin' fractal equations using simple algebra. For more abacabadabacaba: http://www.abacaba.org/ and http://books.google.com/books?id=QpPlxwSa8akC&pg=PA60&lpg=PA60&dq=abacabadabacaba+%22martin+gardner%22&source=bl&ots=92eAyvrZGV&sig=J2uvF2DAyn9kY8nSarSy-XIXW74&hl=en&sa=X&ei=Np45T4K9GYixiQK92NG2Bg&ved=0CCEQ6AEwAA#v=onepage&q&f=false

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Fractal Fractions

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Okay so fractal fractions.
5 equals 5 okay, bear with me, now lets explode this 5 into 5ths: 25 5ths to be percise
Now I want to split it into 2 parts, 17 fifths and 8 fifths.
Could have been 1 plus 24, I don't care. Ok now I'm ready for the fun part
since this five down here is just as much a five as any five is five, such as this whole thing,
which is equivalent to five, lets go ahead and replace with that five,
with this messier looking but still very fivily five
this one too, oh and now we've got more fives
so we can do it again and again and again and then you
can give someone this whole thing and be like, WHOA! IT'S 5!!!
making things look more confusing than they actually are is a delicate art but it speaks to the true heart of algebra,
which is that you can shuffle numbers around all day, and as long as you follow the rules, it all works out.
Ok, here's another one, so you wanna do something with 7, maybe you could use 7 over 7, which is 1,
so you need 6 more, eeeehhh- why not just add it? There.
7 equals 7 over 7 plus 6 now you could replace one, or the other but why not both?
7 equals 7 over 7 plus 6
over 7 over 7 plus 6 plus
6 instead of writing it all again why not just extend this way
there we go, this equals 7 and you can actually
take this all the way to infinity the sevens kind of disappear but then again it didn't really matter
what they were in the first place as long as they're the same.
All these 7's could have been 3, or a billion, or pi to the i,
and this would still equal 7. As long als it's numerator equals it's denominator,
this fraction equals 1 and whatever else you may think of algebra,
at least it has the courtesy to make one plus six be seven every time.
The fractal structure of this first fraction was like a binary tree,
each layer with twice as many turns as the one above it, growing exponentially.
And this one does too, but sideways. Bus awsomely enough, this is obviously
an ABACABADABACABA pattern. That's a fractal pattern that's actually found
in lots of places, but I'm not going into that right now
My point is, if you name this innermost layer A, and the next B,
and the next C and the next D
and then try to read it from top to bottom
you get ABACABADABACABA, and, if your fraction was infinite,
you'd get ABACABADABACABA-EABACABADABACABA-FABACABADABACABA-EABACABADABACABA-GABACABADABACABA
and so on. Anyway, a foolish algebra teacher would teach you that algebra
is about Solving Equations, as if the goal of life is to get X on one side and the rest on the other
As if every fiber of your being should cry out in protest when you see X on the left side
and yet more X on the right side.
But you could replace that X with what it equals
and then you could do it again, and again, and each time your equation would still be true
How's THAT for getting rid of the X on this side?
And you could make equations even more confusing by remembering special numbers and identities
Write whatever you want: as long as you can sneak in a 'multiplied by 0'
you don't even have to bother knowing what the rest is.
Or, knowing that all you need is the top and bottom of the equation to be the same to equal 1,
these sixes don't need to be sixes. They could be threes, or 8 square root 13.
Or you could even make each layer different: 7, 8, 9, 10, 11: now look how confusing this is. Awesome!
Say you wanted to actually solve one of these things, say, you started with this puzzel:
What is 1/(1 + 1) but each 1 is 1 over 1 plus 1, and so on, all the way to infinity?
You could try doing it by hand, thinking maybe it'll converge on something
1/(1+1) is a half. So the next layer we'll use our half added up to one so this is one over one: One.
Three layers and you're back to 1/2. Uh-oh. Any whole number of layers is going to give either 1 or 1/2.
What could this possibly be? Well, you could try doing algebra to it
Say, all this equals X. Look, you isolated X on one side, and everything else on the other
and it doesn't help a bit, so take that, math teacher!
OK but if all this is X, than all this, which is the same as this, equals X
You can write this as 1 over x plus x which completely works. You can generate it all again
by replacing X with 1 over X plus X.
And now that you've got those helpful x's on the wrong side of the equation, you can solve it
and get the boring way to write this number, if you want.
One last fraction, this one with a caution sign. Say you want something to equal one
Split 1 into 1/2 + 1/2. These ones could be replaced with 1/2+1/2. Each time you do this, it works.
What happens if you go to infinity? It's weird, because if you look at any layer of two's
to see if it converges to something, the result is always two for each fraction
which might make you think that at infinity, it's also 2 for each fraction and therefore, 1 = 4?
And just looking at this, and trying to think it backwards, you might see 'all this equals x'
'and all this equals x' so it's x+x over 2. Just try and solve that equation!
The problem is, half of something plus half of something always equals that something
no matter what the x is. So this could be anything, it's undefined.
Or, say you want to make something with all ones, like this. Now, x=x+x over 1, or: x = x+x
You can algebra your way into a contradiction, and as far as algebra's concerned, it's undefined.
Or you could think "well, there's 2 numbers I know that fit this description: zero and infinity...
...this I suppose could be either, or both at once, or nothing at all, I don't know"
why does it do that? Maybe because the numerator got lost up there and could have been anything.
Interesting though, that even here where the denominator got lost in infinity, you can still solve it back to 5
That, to me, is the cool part of algebra. Unlike the neat little problems they put in gradeschool textbooks
not all problems can be solved and it's not always obvious when there's an answer and when there's not
Weird stuff happens all the time, and most importantly, algebra isn't a dead ancient thing
there are things no one has ever done before that you can do with the simplest concepts
as simple as that x is, what x is

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