I’d like to inform you about some numbers:

  • Start with 2
  • Move on to 8
  • Pass by 7776
  • Finally, arrive at 32.64 the distance that they’re getting closer and closer to and that’s what mathematicians in the 19th century called the point of infinity

We have a tour to make, so how about two? 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 $y = x^2$ 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 or 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. They’re given all of them by quadratic equations. So here’s an equation which is quadratic, has degree two, an exponent of two, and nothing higher. And of course, a circle can be written as $x^2 + y^2 = 1$, for example. Ellipse, if you put an $x$ coefficient in front of the $x$, you might get an ellipse. Hyperbola is $xy = 1$. So all of these are quadratic equations; they have a term of degree two. 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 $x^2 = 0$, what does that look like? Well, here’s $x = 0$, and $x = 0$ is the y-axis. $x^2 = 0$ is sort of twice the y-axis, and that will play an important part in our story anyway.

Those iconics, they’re degree two 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 two; it’s given by a degree two 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 grew before is $x^2 + y^2 = 1$, and if I now substitute $y = ax + b$, the equation of a line, and I can write this as a quadratic equation in $x$ and $y$, then $y^2$ would be $a^2x^2 + 2axb + b^2$. If I add an $x^2 + x^2$ and it’s $a^2 + 1x^2$ and I’ve set that equal to one or subtract one and look for the roots, then that will have to have two roots because it’s a polynomial of degree two. So there are two points; they just happen to have imaginary coefficients. So that line is touching the circle twice.

Could you imagine that’s right? All right, so that’s nice. Now, in the 19th century, this was well understood. Gauss proved the fundamental theorem 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 rees and twos and sixes and fours come from this number so it’s it’s a special number for me

I have a special relationship with the number 3264. Joe Harris and I wrote a book about it, called Intersection Theory. The title comes from the fact that the number is composed of threes, twos, sixes, and fours. We discussed how it took a long time to prove that 3264 conics could be tangent to five given conics, with the first rigorous proof coming in the 1970s from Fulton and McPherson. Nowadays, we can compute it in a second or so on a fast machine.