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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.

• 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

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林宜悉 posted on 2020/03/27
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