Fractals and the Collatz Conjecture

hello people I'm making a new video because I'm a big fan of numberphile the duty of channel and last week they were talking about something called the Collatz conjecture and I found it super interesting and I've made some experiments and I found some little patterns but I also made lot of pie solicitations and graphs and I thought I would share all these visuals with you and some of the surprising things in them and you should go to numberphile and check what the Collatz conjecture is about but to recap if you take any number and you apply a set of rules to it then there are some behaviors that happen and if the number is even then you divide it by two otherwise you multiply it by three and you add one and that gives you another number and then you apply the same rules to this number and you get a sequence of numbers and there is a nice way to visualize this by drawing the real line and drawing dots and connecting them by lines for example if you start with the point two and you apply the rule then you get the number one and then four and then two and then one and then four and so on so here you see the cycle but if you start with the number three then you get three ten five 16 8 4 2 1 & 4 & 2 & 1 & 4 & 2 & 1 again all the numbers have more complex dynamics for example number 9 which is somewhere here will also bounce to the right and to the left well it does some big jumps as well but eventually - land in the 4 to 1 cycle so this is number 4 this is 2 this is 1 this is 4 this is 2 this is 1 and so on and so on so the conjecture is that any number you start with will eventually fall into these 4 - 1 4 - 1 cycle one way to try to tackle this problem is to see what happens when you apply these rules to real numbers but in that case the definition of odd and even is not possible so you have to do something about that so the first step is to express the same formulas that we had before but without any branching so f of n if you write it as 1 divided by 4 times 7 n plus 2 minus K times 5n plus 2 this is still expressing the same behavior we had before if K takes the value plus 1 when n is even and if K takes the value negative 1 when n is odd so what we can do is to replace K by something that accepts and not only even an odd numbers that real numbers as well and we can use a cosine wave which indeed goes through 1 and negative 1 continuously so we can kind of map it properly by using cosine of pi N and that will still give us plus 1 and negative 1 as we input even and odd numbers to it so f of x equals 1/4 of 7x plus 2 minus cosine of pi x times 5x plus 2 and now we can see what happens when you use these formulas with non natural numbers for example if we use 1.5 then we see that 1.5 moves to the right and then more to the right and then to the left so it behaves a bit like 9 but every now and then is really performing these huge jumps eventually it really explodes to infinity very quickly diverge and every real number I have tried diverges to infinity so the next obvious thing to do is to expand the conjecture not only to real numbers but the complex numbers as well and I have seen these done on the internet and I think they did maybe not the best way perhaps I will explain later why but I'm still going to spend a little bit of time in that approach they talk so that's simply to take the formula we had and replace X by C where C now presents a complex number and instead of visualizing the dynamics of each point with lines and arcs I'm going to just visualize the end result of the dynamics like either the point is divergent or it converges to a point or to a cycle or maybe it's chaotic but it doesn't go to infinity so if a point in the plane under this formula doesn't explode to infinity it becomes white otherwise it stays and you can see that it's one of these pockets of non divergent points have little islands of such points around it and then there are even more islands inside so I'll really lose like it's a fractal here and of course you cannot resist the temptation to use fancy colors to study the orbits of all these points when you are talking of fractals so this is the point a this was seven this is the point Z equals 6 and 5 and 3 2 and 1 and 0 this is the origin here this big black blob and of course it's a fractal right so if you zoom in into the graph you get more and more structure and patterns showing up which is really fun to watch but also distracting I must say so let's move on and try to use complex numbers in a different way and really think that using the cosine isn't the best way to generalize this and once you use complex numbers I would embrace the whole complexity of the problem and use Exponential's instead and I think this is better because the exponential doesn't have an opinion on whether the real part or the imaginary part of things are more important the cosine wave is kind of opinionated is I'm the real part of an exponential exponential doesn't have an opinion on its own so instead of making K equals the cosine of pi see if you make K of C equals the exponential of J pi C we can get still the same behavior where if we input odd natural numbers we still get negative 1 and if we input even numbers to the exponential we still get +1 but now we are in a better place to see what happens graphically when we use these complex numbers in the Collatz conjecture so this is what you get you get again a fractal don't get misled by the black areas in the graph these are not converging points this is still mostly divergent points actually most of the plane that's diverged under the formula that we are using except for a few points like we know the natural numbers don't diverge there is also many other points which don't average which are not natural numbers there is the point 0 but there are many other fixed points for example which when applied on the formula you get the same point again and there are infinitely many of them I found a little formula that expresses where they lay in the plane it's easy to derive this formula you can try to do it as an exercise let's try to swim in into the zero zero or C equals zero fix point to enjoy a little bit of the fractal structure and you can see there are all these spirals shopping app which is really nice now let's zoom in in one of the numbers that we do know don't diverge like the C equals three three is a natural number and we saw before the three becomes ten and ten becomes five and five becomes 16 and so on eventually falling into the four to one cycle so let's see what happens in this fractal when we zoom in into the C equals three point so this is assumed in this C equals three point and as you can see the whole structure of the fractals appears over and over again trésor which this kind of finger which branches into smaller fingers and so on and so on but I'm going to go back and swimming again to the same point but before doing that I'm going to mark some points in the plane in yellow and I should explain this properly but I'm going to make it short basically these points are the pre images of zero which means that these yellow points when I apply the Collatz conjecture formulas to them they all land into the fixed point C equals zero and I'm not going to explain why I'm doing this but I would say that this yellow points serve as anchoring points or reference points that will help us locate ourselves geographically in the different copies of the fractal so let me explain what this means this case we are all the way zoomed out again and this point here is zero and this is Z equals three this is the point we are going to zoom in it to as you can see if I count how many edges we have in these fingers between the C equals zero point and the C equals three point we have exactly three edges now let's zoom in until we see the next copy of the whole fractal all right let's stop here now let's see how many border just are between the anchoring point of the preimage of zero and the c equals three point that we are swimming into so we have one two three four five six seven eight nine and ten ten steps to go from the anchoring point to the C equals three point it would keep zooming in and I'm going to speed it up a little bit now we stop here and we get we can't again the amount of edges between the anchoring point which is the second preimage of zero and C equals three and now we get one two three four five steps to go from one to the other so I bet you can by now guess how many steps we will need to take to go from the anchoring point to three when we swim in one more level into the fractal and yeah that's going to be 16 so somehow the shape of the fractal its encoding the dynamics of the number three when we try three under the Collatz formulas you get a sequence of numbers which was three ten five and this is exactly the amount of branches that we have to skip over in the physical structure of the fractal to go from the anchoring points to the point we are submitting which is three indeed so it's very cool it's like the fractal is encoding the dynamics of the formulas or actually is the other way around actually the dynamics of the formula are defining the shapes and the structure of the fractal you know and there is some explanations for this is not truly magic it has to do with the derivatives of the function at the points we are swimming in and it has to do with self similarity and things like that that I think it's pretty cool to have a visual a beautiful visual outcome in the shape of a fractal for some otherwise very dry and weird and difficult to visualize problem which involves only numbers and sequences so that's the video for today and I hope you enjoyed it so bye

Channel: Inigo Quilez

Views: 49,288

Rating: 4.9598541 out of 5

Keywords: mathematics, maths, collatz, fractal, quilez, tutorial, collatz conjecture, fractals

Id: GJDz4kQqTV4

Length: 10min 53sec (653 seconds)

Published: Tue Sep 13 2016

Reddit Comments

[u/_blub]

This was way too cool

[u/Nimphious]

Here is the link to the Numberphile video in case you were interested: https://www.youtube.com/watch?v=5mFpVDpKX70

Also a great watch.

[u/vznvzn]

nice! there is not a lot of widespread recognition of the collatz fns as one of the earliest discovered fractals; its fractal structure was not really recognized at the time & is still not widely regarded. see also

theres a book by jeff lagarias that is a nice survey of knowledge. that ref & many others here, much more on visualization approaches, lots of experiments/ analysis/ refs etc

https://vzn1.wordpress.com/code/collatz-conjecture-experiments/

[u/AyeGill]

That part where the (integer) collatz orbit shows up in the fractal is the neatest thing I've seen all day.

[u/[deleted]]

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