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What do you think this is?
Like maybe seaweed? Or...
-It definitely looks like something from the sea, like...
Okay well it's interesting mathematical seaweed,
and I can explain exactly how we made that image,
because to me that image is one of the most mind-blowing pictures,
illustrations of very simple math ideas.
So it is - and we're going to get there - the Collatz Conjecture.
I know you've done videos on the Collatz Conjecture,
but here is your quick refresher.
When you take a number and we can do two things to this number depending on its parity,
that's whether it is odd or it is even.
If it's even, we're going to divide by two,
and if it's odd, we're going to multiply by three, so triple it, and add one.
-Which will then make it even. -Which will then make it even.
and then we know what we do even numbers, that we divide them by two.
Let's say start with say 13.
So 13 is odd, so x 3 and add 1
so 13 will go to 39+1, 40 - goes to 40.
Okay it's even, so divide that by 2, goes to 20.
Okay we divide that by two, goes to 10
again divided by two, goes to 5.
Uh-oh, odd, multiply by 3, 15, add 1, 16.
I get 16.
And then look, 16 to 8 to 4 to 2 to 1.
The conjecture made by Collatz is that every number that you choose
is eventually going to go down to 1.
So some of them are going to go all over the place,
but eventually everything's going to 1 as a conjecture,
which is, we think is true but we can't prove it
and one of the reasons why the conjecture is one of the most famous unsolved problems in math
is that we appear to be nowhere close to a solution, nowhere close at all.
People believe that whole new bits of math need to be invented before we can do this,
and is one of the few things that is so simple to state, yet still so complicated.
It's got the sort of mythical status now among advanced mathematicians,
and what I want to do is show how Edmund Harriss, who is a mathematician who I work with,
has created this amazing image that illustrates what's going on with Collatz
and when he first showed it to me, I was like blown away.
I've started at 13 and then a line like that.
Actually already what people do is you've built this big tree,
because eventually things are going to have to get to one.
So if they get to one they're going to have to come from two,
then they're going to have to come down from four,
and so you can build this big tree.
What you see with the tree is that you've got all these branches coming down like this,
you know, and going into it.
And normally when you see this done,
they're done to try and maximize the sort of logic of how it's done,
so the distance will be the same, they'll be done so it like kind of looks good on the page,
and it looks like something in a math textbook.
But actually what Edmund has done is to make it so it's...
interesting to look at,
but so that you get more of an insight about why this whole problem is interesting.
Imagine if you start at the bottom,
either we are going to go up to an odd number or an even number.
If we're getting to an even number, it's going to slightly go off in that direction.
If it's going to an odd number it's going slightly in that direction.
-Clockwise or anti-clockwise. -Yeah anti-clockwise for odd, and clockwise for even.
And then you get - the tree becomes this.
This is something already... startling to look a bit interesting.
And then obviously we're not that interested in the individual numbers,
we're interested in sort of the overall feel.
He gets rid of the numbers,
and sort of he just looked at the lines between them, how the lines relate,
and sort of thickens the lines to make them look like branches,
and then carried it on until he includes every number less than 10, 000.
You end up with hundreds of branches,
and then it looks like that.
It is so messy.
It's so disorderly.
We've seen this incredibly simple thing:
divided by 2 or multiply by 3, add 1, and then divided by 2.
Who would have thought that you would get this thing here?
But it looks organic.
Brady you were saying it looks like something definitely floating at the bottom of the sea.
So you've got two things going on there:
you've got the nature of the growth, that you've also got the windiness of the waves.
And to think that something so natural looking, something so organic,
something so complicated, comes from something so simple,
this to me is the best way to explain the kind of complicated nature of the Collatz Conjecture,
because what it's doing is you're looking and you're thinking,
oh my god that looks complicated,
and you think, oh now I understand why no one's got a clue.
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and I'm bit of an Egypt nut, that's a topic that fascinates me somewhere.
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Collatz Conjecture in Color - Numberphile

19 Folder Collection
林宜悉 published on March 28, 2020
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