Well I want to tell you about some numbers.
We'll start with 2 and go on to 8, pass by 7776 and come finally to 3264. We have a tour to make.
So how about 2? First of all this is a story about conics. Conics are curves in the plane that are
very familiar to students in high school already. If I have y equals x squared and I graph that it
looks something like this, that's a parabola. Parabolas are important in all sorts of ways
but they were also studied, along with the other so-called conic sections, by the Greeks already
for a long time and were a big subject in the 19th century geometry. So-called conic sections because
you can get all of them by slicing a cone. If you slice it right across straight you get a circle,
if you tilt your slice a little bit you get an ellipse, and if you slice it you get a parabola.
Tilt it a little more, just parallel to the center, you get part of a hyperbola. If you think
of the cone going both ways you get both parts of the hyperbola. In the 19th century people began to
think about these in a different way; uh they're given, all of them, by quadratic equations.
So here's an equation which is quadratic, has degree 2, an exponent of 2 and nothing higher.
And of course a circle can be written as x squared plus y squared equals 1, for example.
Ellipse if you put an x coefficient in front of the x you might get an ellipse. Hyperbola is x
times y equals 1. So all of these are quadratic equations, they have a term of degree 2. All
quadratic equations give conics; if you really include all of them then you get funny things
that you might not think of as conics at first. For example if you just take x squared equals
zero - what does that look like? Well here's x equals zero and x equals zero is the y-axis. X
squared equals zero is sort of twice the y-axis, and that will play an important part in our story.
Anyway those are conics. They're degree 2 things for a different reason as well. So one way that
mathematicians measure the degree of these things or of other curves is to draw a straight line and
see how many times it meets them. So let's try that with a circle: so we agree that a circle
is degree 2, it's given by a degree 2 equation, we drew it. If I draw this straight line it obviously
meets it in two points. You might object that well I cheated a little bit because I didn't draw this
line which seems to meet it at one point. But if you think of moving this line towards that
one then these two points just come together so I should really count that as two points - and
then that's two points. But now you'll say, okay that was fine but how about this line? It doesn't
seem to meet it at all. However you may or may not remember that I did a video about the fundamental
theorem of algebra which says that any polynomial in one variable of degree d has d roots if you
count it with multiplicity. That's the case here but the roots have become imaginary. One way to
see that, the equation of the circle we drew before is x squared plus y squared equals 1,
and if I now substitute y equals ax plus b, the equation of a line, then I can write this
as a quadratic equation in x and y. This is then y squared, would be a squared x squared
plus 2 axb plus b squared. If I add an x squared plus x squared and it's x squared
plus one x squared and I've set that equal to 1 - or subtract 1 and look for the roots. Then
that will have to have two roots because it's a polynomial of degree 2. So there are two points,
they just happen to have imaginary coefficients. - (Brady: So that line is touching the
circle) (I just have to imagine it) - That line touches the circle twice
but you imagine it that's right. So that's nice. Now in the 19th century this was well understood,
Gauss proved the fundamental theory of algebra, they knew that this should be true,
they worked in the complex numbers. But they went a step further; if you think about this,
what happens with this vertical line? Where does it touch the hyperbola? And the answer
is it touches it at the point of infinity. If you continued these lines out they would get closer
and closer together; they would never meet but you can imagine that there's a point out there
in infinity. We have a tangency at infinity of these two lines so that's a double intersection
again. And what about this one? Here these two go out and if we had a line at infinity - as they
imagined already in the mid 19th century - this thing would be tangent to the line of infinity.
That's what happens when you move that line all the way up. In fact when you look at equations
and allow yourself to make complex changes of coordinates, all these figures look the same.
There's no difference between a circle, hyperbola, parabola in terms of just the equation - they're
all degree 2 equations. And as I say they're degree 2 because they meet a line in two points.
Now I want to talk about tangencies of figures, this was a big deal in the 19th
century too. So what do I mean by tangency? Here's the simplest tangency. You can see
if I think of a line in the space of lines, what would that mean? So let's take a point
out here and think of all the lines through the point, so that's what I would mean by a
line in the space of lines. And you can see that as I do this there are two places where
it's tangent. So that's a condition of degree 2 also. In the space of lines this is a line
,this family of lines is a line, and that line meets the condition of tangency also in two
points. So condition of tangency is an equation of degree 2 on the coordinates of this line. So
that's another way that you can have degree 2. So now let's talk about the first number,
and another 2. Suppose I look at tangency of two circles. So two circles can be tangent, for
example this way - here are two tangent circles. - (They're just kissing each other)
Kissing each other, they could kiss on the inside too like this. And a line in the space of circles,
what would that mean? Well here's- here's how I draw a circle but I could have other radii right?
I could put an r in there, that would be the radius of the circle. Turns out if I
make r bigger and smaller that's a line in the space of circles. Here is one circle,
I'm going to fix this circle and I'm going to take another circle and change its radius. So
I'm going to fix the center, the center here is (0,0), and I'm going to change the radius and
watch it and it expands and at a certain point it becomes tangent to this circle and then it stops
being tangent afterwards. After a while it'll be a circle which has this as its center which
looks like this and is tangent up here. It has a second tangency right? So in the space of circles,
being tangent is a degree 2 condition. Right? That happens twice.
- (And it can't happen a third time?) - That can't happen a third time,
even over the complex numbers, you have to take my word for that. But we could work it
out in equations and then it would be easy. Here's another way the degree plays a role:
we saw that a line meets something of degree 2 in two points - how about two things of degree
2? An ellipse and a circle both have a degree 2 and they can meet in four points, 2 times 2.
And it's something called Bézout's Theorem, a very important theorem in
geometry, Étienne Bézout I think. - (But didn't your two circles
both) (have degree 2?) - Yeah these both have degree
2 but they- and they look like they're meeting only in two points at a time
- (Oh they're meeting, sorry, that's meeting not tangenting yeah)
That's right. But even so these circles look like they're meeting only in two points right but this
is a case like this- these lines, if we do it in equations then we could parameterize
the circle and substitute the equation of the other circle in it then we'd get an equation
of degree 4 and it would have 4 roots. So just as we computed here the degree 2 we'd have 4 roots.
So over the complex numbers they always have four intersections counted with multiplicity. Two
conics. And that's a case of Bézout's theorem: if you have a bunch of conditions of degree d
where you expect d solutions from a line, but if you have two of them then they meet in d
squared points. If you have three of them meeting in a 3-dimensional space like three
spheres would meet in eight points over the complex numbers. So that's a general fact. So
how many circles are there? What is the space of circles like? So if you think of circles,
there are really three numbers that are important: there's a center, let's say the point (a,b),
and there's the radius, the equation x minus a squared plus x minus b squared equals r squared.
So that's the equation of a general circle and there are three numbers involved. So the
space of circles is a 3-dimensional space. Now if you have a 3-dimensional space and
you have conditions which are satisfied by some- somewhere on a line, let's say two in a line, those
are quadratic conditions, then Bézout's theorem tells you if you have three such conditions there
ought to be three points of intersection. So Bézout interpreted that funny way says there should be
2 times 2 times 2 - 8 circles - that fulfill all of three degree 2 conditions, like being tangent to
a given circle. So we take three circles - so here's one, and here's one, and here's one. If you worked
over the complex numbers you could do it with any three circles in the plane you wanted. And
now let's see if we can draw eight circles tangent to all of them. And the trick is that a tangent circle
can be outside or tangent inside. So I have two different ways of being tangency and three
circles so 2 times 2 times 2 seems very reasonable. So here's a circle that's inside all of
them, here's a circle that's outside - whoops - outside all of them. These are Eisenbud circles you've- you've had
Eisenberg 17-gon so- - (We're used to your using a lack of precision yeah) - Yes so let's see if we can draw one
that's inside this and outside those two: well no problem, here's one that's inside that and outside
those two. And I can do the same thing over here clearly, something like this or something like this.
Let's see, so I have inside all of them, inside the three and outside the others, what am I missing? I'm
missing one that's- -(Gonna be one there isn't there?) - There's going to be one-
(Oh hang on, that's the same circle- ) - No that's right. So there's one here touching all three, so we now have the counting game. Let's
see there's one in here, we're missing this one, and this one- that's eight - ah we got it! So we got eight
circles tangent to the three given circles. Okay, so that's the beginning of the story, the 2 and the 8.
(Okay all right!) - Maybe this is two videos. Alright, so in the middle of the 19th century people
got more ambitious and they said, let's do the same thing with conics - why just circles? The circle
story is very old, I don't know who did it first, but maybe Apollonius or something like that. But
with conics it's more complicated, but this was a problem proposed by Jacob Steiner in 1848. He said
let's try this with conics but how many conditions do we need for conics? So we need to see how many
parameters a conic depends on. You can write it always as ax squared plus by squared plus cx
plus dy plus a constant e. One, two, three, four, five - so it's a 5-dimensional space of conics.
So we need five conditions. Well okay we could have tangent to a line, that might be interesting,
but why not tangent to conics? So if we- once again conics tangent to a conic is co-dimension 1, it's
it's given by one equation in the coefficients of the conic. If I fix a given conic like a circle
then there'll be a 4-dimensional family tangent to this conic. If I fix a second conic, maybe
that one, a parabola, there'll be a 3-dimensional family tangent to both. If I fix a third maybe it's
a hyperbola somewhere it'll be a 2-dimensional family tangent to all three. And if I fix two more
it should be just a finite set of conics tangent to all of them. So let's see if we can count how many.
So we have to see what the degree of the condition is of tangency. And with coni- with circles I could
draw it for you but with conics it's much harder and in fact it's a condition of degree 6. Each of
these five conditions is a degree 6 so Bézout's theorem says that five conditions of degree 6
should have solutions which are 6 to the fifth power and which is 7776 the last time
I looked. So Steiner said, okay this there should be that many conics tangent to five given conics.
(So just like you did the eight circles for me, you could do the same thing now with your conics)
(drawing other conics? And you could draw 7776?) - Well that's what Steiner thought but
he was wrong. And Bézout's theorem doesn't work so well in this case. This was discovered pretty
soon afterward and first published by Michel Chasles; the right answer is not 7776 but 3264.
So you could draw, in principle, 3264 conics tangent to five - these five - given conics. - (They- those those)
(new ones you would draw in in red pen for me - which I won't ask you to do -) - But I could draw it
just by filling the plane with red. You couldn't- you would never see the conics. - (But they would be)
(all different sorts of conics? There might be like 1000 circles and 20 parabolas?) - There might well, right.
(Right, and and you don't know what that mixture necessarily would be?) - I do not. But remember
in the- in the in the setting in which these guys worked there's no difference between parabolas
and hyperbolas or- they all look like equations like this, this coefficients just just they're
different coefficients. There are 3264 five tuples of numbers a, b, c, d, e which satisfy the condition,
that's what they- that's what Chasles proved. Or claimed! - (I mean as I- as I look at that now)
(it's almost hard for me to imagine one conic that would manage to tangent all of them-)
Well, that's because there's complex numbers involved right? - (So so you couldn't draw them- even if you)
(knew them you couldn't draw them?) - On my real plane that's right. So it was a question for a long time
whether there is any configuration of the five conics which allows all 3264 conics to be real.
And the answer is yes, uh could be real could be all real. - (Oh wow 'cos that would mean-) - And there are pictures.
- (There are pictures of it?) - Yes. Again you can draw 3264 it just fills the plane with black. But
you can draw a few hundred at a time and explain how they fit together. And this was done by a man
named Frank Sottile, that's quite recent. So sottile has specialized in this quest of realizing these
funny configurations over the real numbers in different ways. - (It was quite a come down from)
(the hope of 7776 to the reality of-) - That's right. So where did the other ones go, right? And there's a
good answer to that. So when you write down the the five equations that give the conditions of
tangency, unfortunately they have some spurious solutions. And the reason is very simple, and
I could even demonstrate it with the eight circles although it doesn't play a role there.
Circles are circles, you know what they are, they're round. But conics, as I said they come
in lots of shapes and if you let the equations degenerate a little bit; if you take b equals c
equals d equals e equals zero you get x squared equals zero, and that's the equation of a double line.
So the double lines could be in there; and if you think about it what is the condition
of tangency? It's that it where it touches it touches twice. But a double line touches twice
wherever it touches right? There's there's two lines there so it's it's tangent. And how many
double lines are there? Not just finitely many but a whole plane full. Because- you know you can draw
one parameter family of lines through a point, you can move the point so you have a two parameter
family of lines so there's a whole surface of lines. So when you intersected those five
conditions of degree 6 you got not finitely many solutions but a surface of double lines,
plus the solutions you really wanted. And Chasles had to figure out how to remove the surface and write
down the conditions without the surface. He got the right answer by methods which are not rigorous
by our standards; and probably the first rigorous demonstration of this number is due to Fulton and
MacPherson in the 1970s. So it took a long time to get this right. - (Well a long time to get it rigorous,)
(he got it right!) - That's true he got it right. Actually there were a lot of numbers computed by
by shady methods in the 19th century and they've all been right so they were pretty smart. - (Who needs rigour?)
Right, exactly. Nowadays we just compute on the computer we can see. And in fact numerically
you can find the 3264 lines in a second or so on a fast machine, but it's not quite a proof.
(Tell me about your special relationship with the number 3264 - you've used it) - Yes I have. Joe Harris
and I wrote a book on this subject which is called intersection theory and the book is called '3264
And All That' and there's the picture of the eight circles tangent to three given circles on the
cover. Let me explain a real proof of the case of the conics somewhere in here. Let's see, there are some
conics I wonder if I have a theorem somewhere in this chapter. We called it the five conic problem
and uh here it is as a keynote problem - how many conics are tangent to five? And another similar
problem given: 11 general points in the plane, how many rational cortic curves contain them all? And
there you can multiply these problems as long as you like. They're mostly good for sharpening
your tools for more serious things, but problems like this are all over the place in mathematical
physics today. In general this whole theory of what what are called excess intersections which Fulton
and MacPherson made precise in the mathematical sense play an enormously important role in
mathematics and physics. So that's the story of 8, 3264, and 7776.
...crude simple-minded way. And it turns out to be that you're really pretty close for big numbers. So let's let's do a little
calculation. One of the most famous theorems in number theory is the prime number theorem...