I wanna tell you about a really lovely problem.
Suppose you have a circle,
how many ways can you draw it?
Well, there's only one way.
I mean, we could make it bigger, we could make it smaller,
we could move it around the paper, but there's only one way.
So, the answer for one circle
is 1.
What if you have two cicrles?
They could be inside each other.
They could cross each other.
Well, they could be separated.
So for two circles - 3 ways.
And I better tell you what the rules are.
We don't allow this kind of thing,
we don't allow two circles to just kiss.
They've gotta make up their mind, they've gotta overlap or they've gotta be separate.
No, that's not allowed.
And when we get to more circles, you're not allowed to have three going through a point.
That's not allowed either.
What's more, we're doing this on a piece of paper.
We're doing it in - technically we're doing it in the affine plane
we're not doing it on the projective plane.
If we were doing projective geometry
we could take a circle and move the centre
out to infinity and it would become a straight line.
That's not allowed. All you can do is move the circles around.
The radii don't matter. You can make them big circles, little circles,
I mean, obviously this is the same as that
or this.
You can move them around as long as they don't cross.
Mirror images count the same, they're the same as the original picture.
We know what the answer is for two circles - there are three.
What about three circles? The answer is 14 and here they are.
So first of all, obviously,
we could do three in a row.
Three circles separate from each other.
Or we could do three that are linked.
Or we could do a nested pair and a separated one.
In fact, we could take any of the three that we had for two circles and wrap the circle around them.
So we could do that one
and put a circle around it, so we'd get that.
Or I could take this one and put a circle around it, and we'd get that.
Or I could take this one and put a circle around it and we'd get that.
And then, we could
put a circle next to one of them.
So now we have seven.
1, 2, 3, 4, 5, 6, 7, we're halfway there!
This one here can be modified a bit.
A meets B meets C. C could meet A and we would get something that looks like this.
And we get Venn diagram.
A, B and C, and they intersect in all possible ways.
But there's another way that could happen.
We could have - don't forget it doesn't matter how big or small the circles are.
I've drawn it - maybe I should've made the overlap bigger.
This is almost the same as the Venn diagram, except there's no point
where all three meet.
So we're up to 9 and we need 5 more.
And I'll take another bit of paper.
I'll start off by drawing two that overlap
and then I can put a third one there
in, inside one, but not inside the other.
I could put a third one there, so it crosses
the line where this is inside that.
Or I could put it inside the overlap
or I could put it so the intersection looks like that.
And then there's one more, it's subtly different
from that one, it looks like that.
And you can see here, we had these two overlapping,
this one - the third circle was inside the points where they met.
Here, it enclosed them. So these two are strictly different.
Of course, you might think that I'm now gonna go and put the circle here, for example,
but that's the same as this.
So there' are just five more, and the total is 14.
That's pretty easy, but not trivial to do.
Easy to miss one, if you start doing it.
The question is what happens next?
Jonathan Wild, professor of music at McGill University,
he has continued this sequence out to five terms.
And for four there are 173 ways to do it.
And even for three, yeh, the arrangements are quite pretty,
but Jonathan Wild classified the four circles and five circles.
The pictures, some of them are really amazing,
they're just brilliant.
I mean, this is mathematics, this is geometry.
But the results, they're fantastic. And I'd like to show you.
This is the complete set of the arrangements of four circles.
You can see, you start of with four disjoint circles,
and then they can be nested in various, obvious ways.
But when you get to more complicated configurations
some of the arrangements are really delicate.
Remember, you can make the circles as big or as small as you like.
Look at that one.
And going down,
look at these.
This is a very delicate configuration.
So we don't know - we know the answer for five - even more complicated.
And we know nothing more.
We don't have an upper bound for the number, we don't have a lower bound.
I think it would be great to hang, to draw the pictures of all 17 thousand
of the first five things, and hang them along the Great Wall of China.
It's more than long enough.
What is it? 13 thousand miles or something? So
we'd have plenty of space. But it would look awesome.
(Brady: Neil, this seems like it should be easier. Seems like it's something there should be some equation for.)
(You know, when the number of circles is n, you have, you have this equation, but)
(it's not, you know.) - It's not that simple, no.
I mean, one thing you could do is look at the pattern of intersections.
Say we have three circles: A, B and C.
You could ask "Does A meet B?
Does A meets C? Does A meets B meets C?" You can write down what's called a truth table.
It's described by a Boolean function, which says which circles intersect.
The trouble is if you specify the intersections that doesn't tell you how to draw them.
And it doesn't tell you how many ways there are to draw them.
It's really tricky. There may be no ways.
Jonathan Wild found some arrangements of five circles that should exist, but don't.
Conceivably you could do the following.
You could have four circles which were linked in a sequence, in a chain.
We've got A, B, C and D. Four circles.
A meets B, B meets C, C meets D, D meets A.
And
the fifth circle would be like this - you can draw it as an ellipse
but you can't draw it as a circle. That's E...
So the intesection patterns are: the A,B,C,D I already told you about,
E meets A, and B and C and D,
and it contains the intersection of A and D, and it contains the intersection of B and C,
it does not contain the interesection of C and D, and A and B.
You cannot draw that. Jonathan Wild and his coauthor showed that this cannot be realised with circles.
It can be realised with ellipses.
So as I said, even if you specify the way the circles meet each other,
what you might call the Boolean - the truth table,
showing the intersection pattern.
And of course, as you know, if you have in this case five circles
you draw a column, then you mark it A, B, C, D and E,
and then here you put a zero or a one,
depending whether they meet or not. So
for instance, here A meets B - true.
So 1, 1, 0, 0, 0 would get a 1.
So for each of the possible, and here we've got 32 all binary intersections.
So it's a function of five boolean variables, whether they meet in that particular configuration or not.
So then it 's described by a binary vector of length 32.
And so the number of them is 2 to the 32. And for each of them, well maybe you can draw it,
maybe not, and maybe you can draw it in more than one way!
We don't know.
I always like having a look at the daily challenges here on Brilliant.
And as we scroll down, there's one that caught my eye for obvious reasons - cause it's circles!
Circles and Triangles and Hexagons, Oh My!
Let's have a look at this one. Now like many daily challenges, it's starts off with something pretty simple.
We've got an equilateral triangle and a regular hexagon there.
And it says those dark blue shaded regions are similar.
So how does the area of the shaded region in that hexagon compere to the one in the triangle?
What do you think, is that 1.5 times bigger? 2 times bigger? 3 times bigger?
Maybe even 4 times bigger?
This is just an example of some of the content you're gonna find on Brilliant. Besides the daily challenges,
there's quizzes and courses and puzzles, all sorts of great stuff.
A lot of it's free. You can go and have a look straight away or you can sign on for a premium membership.
That gives you acess to everything. And if you go to brilliant.org/numberphile
they'll know you came from here and you'll get 10% off a premium membership.
Not a bad deal that. brilliant.org - don't forget the /numberphile
and our thanks to them for supporting this episode.
[Preview:] It's a simple problem, beautiful pictures,
wide open.