ok so to start off I've got 4 images for you.
Quick question what do all of them have in common. Well, I would be
very surprised if you guess this one here. (Giuseppe) They are full of lines. (M) They are full of lines, they are in a circle.
But actually they are pictures of times tables. I better explain
this. So to get these pictures what you do is you start with a circle and
then you pick a random number. And we just choose something nice to start with, a 10.
Then we put 10 points on the on the perimeter here
equally spaced and then we label them with 1,2,3,4,5,6,7. Actually, let's start with 0--0123456789 10
10 we also want, so we kind of put it right there again. So that's also 10, that's also 11
12 13 14 15 and so on. And then again, and so on. So this guy here stands for 1, 11, 21 and so on.
And now we're going to do the two times table and we'll
start with 0. So 2 times 0 is 0, same thing, so we don't do anything.
OK, two times 1 is 2. So we connect the 1 to the 2. Then 2 times 2 is 4, connect the 2 to the 4 and we keep on going, pretty
obvious. Now 2 times 5 is 10 and remember 10 is also over there so we connect that guy up.
And then 2 times 6 is 12 which is also the 2 and we just
keep on going like this
okay and then at 9 you pretty much gone all the way around now you could
go on so for example could now do 2 times 10 and 2 times 11 and 2 times 12 and draw in those
connections but actually the connections are going to be exactly the ones that we've drawn
before. Just to illustrate, let's just go up to 2
times 12 is 24 which corresponds to the 4 and that connection is already there.
2 times 13 is 26, and we've already got that. Of course we made a choice at the beginning that 10
If you change that 10 to anything else
well the picture changes but pretty much everything I said stays true.
So for example if you switch from 10 to 11 we get this guy here here. And then 12
13 14 15 and let's just see what happens.
(Giuseppe) Is there a reason why the diagrams have a horizontal symmetry axis. (M) yep yep yep yep yep there's a reason and let us give this as homework.
Guys figure it out in the comments, tell us why there is a horizontal symmetry axis. Of course, what's much
more interesting is that curve that somehow magically materializes when you
up the number that we chose at the beginning and actually just highlight it a bit.
This strange curve got a name and pops up all over mathematics, it's called the cardioid
like in cardiology and means heart. So it's like a mathematical heart curve. It comes up in all kinds of places. I'll just show you
a few. Ok so for example you can have it as a rolling curve rule one
circle around another circle and just see what happens to one of the points on
the boundary of the rolling circle and what it does is it traces the cardioid.
Now where else can you see it? Well sometimes you can see it in your coffee cup. If you've got a conical
coffee cup like this and the Sun is in the directions of one of those green lines
in the coffee cup
then we can also see it. Where else? Right there in the Mandelbrot set so the
biggest bulb in the Mandelbrot set is a cardioid and the second one is a perfect circle.
Now this would be a good one
to figure out why why is this guy a cardioid. Just chase this down
somewhere. Now just for those people who know something about the stuff, that's the equation that we use
to make up the Mandelbrot set and there is a 2 in there. That's important
We just had the two times table. Let's keep going now
(Giuseppe) One more thing about the cardioid shape. As a video producer I know that there is a microphone called cardioid microphoes
and that is because they pick up the sound in that area, from that direction. (M) Okay maybe we do another video on this.
Now instead of doing the two times table we can do the the three times table. What we
get is this pattern here for initial our choice 10 and then I just up up the
numbers and what do we get? Another curve, another very famous curve. This is called the nephroid which means kidney.
It's also a rolling curve, here the circle that is rolling is half the size of the other one.
It's also in a generalized Mandelbrot set. Remember before we
head an exponent of 2. Now we're talking about the three times table we
change the exponent to 3 and you actually get the main bulb here being the nephroid.
We also get it in the coffee cup. This time we need a straight
straight coffee cup and the light rays come in from one side.
So they come in straight like this and then they of bounce around inside
the cup and and make up this curve. Actually this way of making up a curve is called a catacaustic in mathematics.
What else have we got. Four times you can see a pattern now. Four times gets you three petals.
Before we had three times and so two petals and all these other
things that we've highlighted they work too. Here the the exponent has gone up to to 4
in the generalized Mandelbrot set. Here we've got the rolling curve happening again.
And then just for good measure just one more guy here. That's the 5. It gets four
petals and we've got the Mandelbrot set here and we've got the rolling curve.
All very neat. (Giuseppe) So the number of petals is n-1?
Yep and it keeps on going like this. Ok so let's start again with 2 times table. Now one
one really nice thing to try is to actually make increments not one ... to 3 to 4 to 5 but
actually do smaller increments
actually do the transition here continuously. I'll just show you what it looks like when I do
the multiplication times 2.1. What you get would you get 2.1, 2.2, 2.3, 2.4, 2.5, 2.6,
2.7, 2.8, 2.9 and 3 and you can see how this cardioid turns into the nephroid.
Now I just let it go.
The numbers are going up and you can see the petals going more and more and more.
You think well that's probably what it's going to be. You can also look at
this in the Mandelbrot set and you can actually see exactly this sort of
pattern happening in the middle.
As we push the exponent to infinity the whole thing kind of turns into one big
circle. Okay and now you think that's the end of it but no, no, no.
Let's keep going here. Now the petals are pretty much going away but if
you just look at what's happening here you see there's a lot more structure
somehow being hinted at. I mean I'm moving very fast at the moment 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.
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. If you stop at 34
and there are a couple of other highlights here. There are a lot more but that is how many I could fit here.
Ok maybe let's have a close look at some of them. So here we've got four. The
number that all these correspond to is 200.
What we're really doing he has a special name in mathematics
So what we're doing here is the times table modulo 200. So it's got a
special name. It's actually incredibly important in all sorts of applications
to do this sort of modular arithmetic. But paper anyway for our purposes
you know you understand how it's built up and if you know a little bit more you
know a little bit more. So for the 2 times table we get the cardioid. For the 34 times table modulo 200
we get this guy here and then some other interesting things happen.
So, for example, if you multiply by 51 we get this guy here
and if you multiply by 99 you get that guy here. And obviously if you just look
at the two numbers here below so 51 is pretty close to 1/4 of 200 so
must have something to do is that. 99 is pretty close to one-half of 200, must
have something to do with that. Now I really want to explain something I
really want to explain where this one comes from. And well I mean the first time I
saw it I was pretty surprised like everybody else who sees it but then I remembered
something I had seen the cardioid before in 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 and it's got to do with these light rays
bouncing around. It's a little bit different from what we had before. So basically we've got a
circle here and then we've got a light source right over on that side. And that circle
reflects the light right, so we switch on the light over there and then well what
happens?
Well light rays emanate in all kinds of directions and get reflected off the
circle bounce around and you can see this pattern emerging, the cardioid. So I
knew this one I had actually calculated this curve from this description at some point in
university. And starting with this you can actually see very easily why the cardioid comes up
in the two times table and I just want to show you.Let's just highlight like one of those rays
coming out here. It hits the wall here and then it gets reflected at the same
angle as it comes in, so this angle that you see here is the same as
it comes in. So what does that tell us now? Well,
if we if we put this line in the middle
that means that this side really flips over to that side, which means that this
segment in particular is exactly as long as that one here. If I travel from
this light bulb to this point and then from this point to that point it's exactly the
same distance, that's going to be important. So let's make this 0 let's call this
N. What's 2N then? To get 2N we have to measure this
distance here and then we have to measure it again and where do we get? Right up there !
And so it's pretty clear that when you do the 2 times thing you get that reflected ray, you
know, in the picture. Just to finish off I'll show you a little bit of a movie of all the stuff
evolving a little bit slower than what we had before.