Times Tables, Mandelbrot and the Heart of Mathematics

by birtanpublished on September 10, 2020

okay so to start off I've got four four images for you so look at them one two three four now quick question what do all of them have in common well I would be really surprised if you guess this one here they're full of lines and

They're in a circle and these are things but actually pictures of times tables okay fixed all the time so I better explain this all right so to get these pictures what you do is you start with a circle

And then you pick a random number okay and I will just choose something nice to start with so 10 okay so then was 10 we put 10 points on the on the perimeter here equally spaced and then we label them with one two three four five six

Seven okay let's go action star zero zero one two three four five six seven eight nine who then 10 we also want so we kind of put that right there again so that's also 10 that's also 11 12 13 14 15 and so on okay and then

Again and so on right so this guy has tens for one 11 21 31 and so on yeah and now we're going to do the two times table okay so two times table and we'll start with 0 so 2 times 0 is so 0 is the same thing so we don't do anything okay

2 times 1 is 2 okay so we connect the 1 to the 2 then 2 times 2 is okay we do this and we keep on going pretty obvious now 5 2 times 5 is 10 10 remember tens also over there so we connect that guy up and then 2 times 6 is 12 which is

Also the 2 so we connect that one and then kind of just keep on going like this ok and then you're at 9 and you've pretty much gone all the way around now you could go on so for example we could not do 2 times 10 and 2 times 11 and 2

Times 12 and and draw in those connections but actually the connections are going to be exactly the ones that we've already drawn with who can actually stop there just to illustrate let us go up 2 to 2

Times 12 is 24 which corresponds to the 4 and that connections already there at and two times 13s not quite 2 times 1326 and before you got to 6 of course we made a choice there at the beginning that 10 right if you change to

Attend to anything else well the picture changes but pretty much everything I said stays the same ok so for example if you'd switch from 10 to 11 we get this guy here you know and then 12 13 14 15 and let's just see

What happens that's right yes yeah there's a reason and actually let's just give this as homework for our guys here to figure out so in the comments tell us why there's a symmetry

Of course what's much more interesting here is that curve that somehow magically materializes when you kind of up the number that you choose at the beginning okay and actually I'll just highlight a little bit so it's this

Strange curve here just actually got a name pops up all over mathematics it's called a cardioid like in Cardiology's means heart okay so it's like a hard curve mathematical hard curve and it comes up with all kinds of places now to

Show you a few okay so for example you can have it as a rolling curve you roll one circle around another circle and just see what happens to one of the points on the boundary of the rolling circle okay so from here and what it

Does is it traces the car job okay now where else can you see it well sometimes you cannot see it in your coffee cup if you've got a conical coffee cup like this and the Sun is in the direction of one of those green lines in the coffee

Cup then you couldn't also see cardioid in there where else well they're right there in the Mandelbrot set so the biggest bulb in the Mandelbrot set is cardioid the second one is actually a perfect circle

Now this would be a good one to figure out why why is this guy cardiologist to cart race this down somewhere now just for those people who know something about this stuff that's the equation that we use to make up the

Mandelbrot set and there's a two in there okay so that's important we just had a two times table let's just keep going okay now yeah one more thing about the cardioid shape actually as video producer I know that microphones some

There is a category of microphones called cardioid microphones and that's because they pick up the sound in that area from that direction okay right well maybe we do another video tonight okay now we can also instead of doing the two

Times table we can do the three times table and if you do a three times table what we get is this pattern here for initial choice 10 and then I'll just you know up up the numbers there and what do we get whoo another curve another very

Famous curve it's called the nephrite which is kidney you know okay whereas this one's also as a rolling cough here the circle that rolls is half the size of the one that it's rolling around it's also in a generalized Mandelbrot set so

Remember before we had an exponent to here now we're talking about three times table we change to exponent 2 3 and you actually get the main bulb here being the nephron okay we also get in a coffee cup this time you need a like a straight

Coffee cup and the light rays kind of come in like from from one side okay so they're kind of coming straight like this and then they kind of bounce around inside the cup and and make up this curve actually you know this way of

Making up a curve it's called a catacaustic in mathematics so there we go what else have you got four times you know you can see you kind of see a pattern now you know four times gets you three three petals before we had you

Know three times two petals okay and then all these other things that we've been highlighting here they work to their you know exponents gone up to for just a generalized Mandelbrot set here that we've got the drawling curve

Happening again right and they're just for good measure that's one more guy here that's the five so it gets four petals and we've got the Mandelbrot set here and we've got the Roland curve so our obey very neat yeah in minus one and

It keeps on going like this so there's a pattern like this okay so let's start again with two okay so there was two times table now one really nice thing to try is actually make the increments not one two three – four – five but actually

Do smaller increments okay or actually dude it did transition here continuously so let's show you if I just do the multiplication times two point one what you get 2.1 2.2 2.3 2.4 2.5 2.6 two point seven two point eight two point

Nine and three and you can kind of see how the card turns into this net freud which is kind of cool and now i just let it go and the numbers are going up and you can see the petals going more and more and more and more right so that's

Continuously and well you think well that's probably what what it's going to be yeah you could also look at this in the in demand abroad set and you can actually see exactly this sort of pattern here happening in the middle and

If you kind of push the exponent to infinity the whole thing kind of turns into one big circle you know that's pretty cool okay and it's you know I think that's the end of it but no I don't know let's just keep keep going

Here so the petals are pretty much going to go away now but if you just look at what's happening here you see there's a lot more no structure somehow being hinted at right I mean I'm moving very fast at the moment right

Been moving very fast but if you actually went through this frame by frame you would see a lot of really really nice structure happening and I've highlighted a bit here so just a few of the frames that you get you know some

Really amazing stuff happening here so if you stop at 33 you get this one here you stop at 34 you get that one here and there's a couple of other highlights here there's a lot more so that's this is how many I could fit here

Okay and maybe have a close look at some of them okay so here we've got four now the number that all these correspond to is actually 200 okay 200 what we're really doing he has a special name in mathematics okay so what we're really

Doing here is we're doing the times table modulo 200 times but a special name it's actually incredibly important in all sorts of applications to do this sort of modular arithmetic okay but anyway for our purposes you know you

Understand how it's build up and if you know a little bit more you know a little bit more okay so for two times table we get the card out for the 34 times table modulo 200 we get this guy here and then some other things interesting things

Happen so for example for if you multiply by 51 we get this guy here and if you multiply by 99 we'd get that guy here and obviously I mean if you just look at the two numbers here below you know and a 51 is pretty close to you

Know one force of 200 so must have something to do with that 99 is pretty close to one-half of 200 must have something to do with that much okay now I really wanted to explain something I really want to explain where this one

Here comes from and well I mean the first time I saw this I was pretty surprised like pretty much everybody else sees it but then I remembered something I had seen the cardioid before and you know many many different guises

And in one particular one that I was familiar with I could see where the connection is I just want to tell you about that one okay and it's got to do with this light rays bouncing around it's a little bit different from what we

Had before okay so basically we've got a circle here and then we've got a light source right over on that side right and that circle kind of reflects the light right so we switch on the light over there and then well what happens well

Right light rays kind of emanate in all kinds of directions get reflected off the circle right bounce around and you kind of see this this pattern here emerging the cardioid so I knew this one I had actually calculated the curve you

Know from this prescription at some point in University okay then they can actually really see very easily why the cardioid comes in the two times table I just want to show you okay so let us highlight like

One of those rays coming out it kind of hits the wall here okay and then it gets reflected at the same angle as it comes in right so this angle here that you see here is the same as their margin can see it kind of goes like that okay so what

Does that tell us now well if we if we put this line here in the middle means that this side really flips over to that side here which means that this segment in particular is exactly as long as that one here aren't so if I travel from from

This light bulb to this point and then from this point to that point it's exactly the same distance okay so that's going to be important okay so let's make this zero let's call this n so what's to end then right so to get to N what you

Have to do is we have to measure this distance here and then you have to measure it again where do we get right up there and so it's pretty clear that when you do it two times think you get that reflected ray you know in in the

Picture just to finish off I'm going to show you a little bit of a movie of all the stuff evolving a little bit slower than what we had before

Oh

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