I thought that we could talk about fractions. So , for instance, one over two. Also known as one half. Okay, so another one would be two over five, or two fifths. And then there are more complicated ones,
like eighteen over forty-two, and, well, you know that there is something a little wrong here, which is that the top and the bottom, you can both divide them by two, and then you get that this is nine over twenty-one. And, oh, I’m not done. And so it’s three sevenths. I wanted to do this example because all the fractions I’m going to consider are going to be reduced like this. Namely, it’s going to be of the form p over q,
where p and q are integers, and they are what we call coprime,
namely that they have no common divisor. [Brady]: We’ve cancelled them
down as far as they can go. [Dr. Bonahon]: That’s completely correct, yes. And now, if you remember, with fractions you can do things: you can add them up, or you can multiply them. And multiplication is fairly easy because if I want to take, for instance, two over five, two fifths,
and I want to multiply it with three over seven, I take the two numbers on the top, I multiply them, and I take the two numbers on the bottom and I multiply them, and so that gives me six over thirty-five.
That’s pretty simple. However, I don’t know if you’ll remember this, but if you want to add them, I’m going to tell you
what it is NOT equal to. I have to put a big “not” like this. So it is NOT two plus three divided by five plus seven,
five over twelve. I hope you remember that
you’re NOT supposed to do that. So let me remind you of what you are supposed to do. You’re going to reduce them to a common denominator, which in this case will be five times seven. I just wanted to remind you of
what people told you you should always do. But we could ask ourselves, what if we decide
that this WAS the correct rule? [Brady]: We’re going to break the rules, are we? [Dr. Bonahon]: We’re going to break the rules, yes, and we’re going to venture in the brand new world. Because I don’t want to go against the other people, we’re going to create a new addition, and to remember it’s new, this plus,
I’m going to put a circle around it. Five over twelve. It’s a symbol that we mathematicians, we use a lot, every time we mean, “Well, it’s not quite what you think it is.” We could put a hat on top, we do things like that,
or a tilde. Okay, and so, let me do it with letters like this p and q. That will mean that p over q, funny addition, p-prime over q-prime is p plus p-prime divided by q plus q-prime. Probably you never heard of it, but it’s actually well-known among mathematicians, and it has a name, and that’s called, so I’m going to write, the Farey addition, named after John Farey, who was mostly known as a geologist in the end of the eighteenth century. My intention is to show you that this funny addition actually comes up in two very different contexts, the original one discovered by Farey and another one, which would be called the Ford circles. What we are going to do is that we’re going to list all possible fractions. There are a lot of numbers so we’re going to only to focus on the ones between zero and one. That’s going to shrink. So here is the number line, and I’m going to be between zero and one. And by the way, one is a fraction: it’s one over one. Zero is also a fraction: zero over one, There is one half, one fourth, three fourths, and then there is one third which is probably somewhere like here, Two thirds, somewhere like this -and of course, I can keep going for a long time because there are infinitely many numbers. So what I’m gong to do is that I’m going to stop and I’m gong to take the numbers p, q, so first of all between zero (including zero) and one, and then where q, well, I cannot go infinitely so I’m going to assume q is going to be less than a certain number, so let’s say 234. There is nothing special about 234. It’s just 2, 3, 4. I don’t have much imagination when picking numbers. So I’m going to do it, and I started doing it, and there are quite a few numbers like that, probably between twenty and forty thousand. So perhaps I’m not going to do it in front of you. I have a computer. And so I did it. [Brady]: That’s not twenty thousand fractions!
[Dr. Bonahon]: I only got a little piece of it, because when I use my computer I got pages and pages and pages and pages… So I got all these numbers between  over 81 and 103 over 154. If you think about it, it’s about two thirds. So one is just a little below and one is a little after, and actually there should be two thirds somewhere…
Ah yes, here. It’s a very small piece of the whole list. So we get all the numbers in that very small interval and they are in order. They come up in this order exactly. And so, that looks like — there doesn’t seem to be anything special about this list, right? You remember that funny addition where you add up the two numbers at the top and the two bottom? You see it here. Look, for instance: you take these three numbers here; look at the one in the middle. 148, well that’s exactly the sum of 123 and of 25. And then 225, that’s exactly the sum of 187 and 38. And you see that in many, many places. So let’s see if we can find another one like that. Okay. Look, hundred — these three, I mean.
103 plus 35, 138. 156 plus 53, 209. 31 plus 95, that’s 126. 47 for 144, 191. And it’s pretty clear that there are examples where this is not one of those Farey additions. Like, you have the ones where in the middle the term is way, way smaller. Look at this one here. If I take the sum of the two — the Farey sum of the two on the edges, so that [it] gives me 212 divided by 320, and on the paper it’s 53 over 80. That’s not the same, right? Well, except, wait a minute. You remember how I can simplify fractions? Look, clearly the top and the bottom I can divide by two. I can cancel out. There is also two as a factor here. I divide by two the top and the bottom (I’m going a little fast) — over eighty — hey hey! [Brady]: So that was a Farey sum.
[Dr. Bonahon]: That’s a Farey sum. And so, let me show you how to pick a Farey sum in this piece of paper. What you do is.. this. Every time I take three consecutive numbers here, then the one in the middle is the Farey sum of the two on the sides. And what’s amazing is that when I change my number, 234 here, if I take something else, like for instance if I take 200, then I’m losing this number, it will not be on my list anymore; however, the next — the one that is closest, that’s going to be this one probably, will still have this property. It’s very surprising, to some extent unbelievable, and it took a while for mathematicians to be able to devise a complete proof of that result. The one who did that was Cauchy, one of the great mathematicians of the nineteenth century. When we find something really surprising,
we want to know: Is this a coincidence, or is there a deep fact behind that? Or at least, is this always true? I’m going to do some geometry and construct circles. I’m going to use the number line that I had here, but here is the following recipe that I want to use: Whenever I see the number p over q, again where p and q have no common divisors, then I’m going to draw the circle which is tangent to the number line, which is above the number line, and whose diameter is.. well, I could use any formula, but my pinky tells me I should use one over q squared. Let’s do it. So, one half.
At the bottom I have two, q is equal to two, so I’m going to want one over two squared, namely one over four. So it sort of looks roughly like this. Let’s do one third here, one ninth — oh, this one I don’t completely know, but my pinky just told me that it touches like this. And then one quarter — my pinky is
better than my drawing skills. And then here, by the way, is one over one, I had to draw the circle of radius [diameter] one over one squared, namely one, it’s this big circle here, and it turns out that it’s going to look the same with this one here. And then I can keep going, but as you — I think I convinced you of one thing, which is that my drawing skills are not very good. But fortunately, I have my trusted computer. So, it’s much smaller than this. This is the circle going at zero over one, diameter one. The one going over one, one over one, diameter one, one, one, so these big circles are all above the integers, and then above the half integers I’m going to get a circle of radius [diameter] one quarter, and then above the multiples of one third, I don’t know if you can see — one third, I have a circle of radius [diameter] one ninth, and then it’s beginning to be really hard to see. I can also use my computer and, look, here there is one third and one half. I’m zooming in: one third, one half, and then it’s what you have in between. And you have that. And then you have more. And then you could go, I don’t know, like this, or how about here, between five over eleven and six over thirteen. And it keeps going. Okay. So it is funny that with my funny rules like this, all these circles, you see, they are tangent to each other. They touch, but they don’t overlap, and there is no space in between, they do touch. This is called the Ford Circle Packing. And actually, there is some geometry here that you can show. When you have two of these circles which touch each other, and then you have the number line here, there is a unique circle which touches the number line and these two circles here. Okay? That’s a little bit of a geometrical fact. Look, look look look, at this. So I was taking this one: eleven over twenty-four, seventeen over thirty-seven, and the one that touches these two circles and the number line, that’s.. oh my God, that’s twenty-eight over sixty-one! Eleven plus seventeen, that’s twenty-eight; twenty-four plus thirty-seven, that’s sixty-one. [Brady]: It’s — gah, what have you done?! It’s back!
[Dr. Bonahon]: It’s coming back! So we have these Farey additions
everywhere in this picture. Again, this is true for any three circles that touch each other like that in this whole picture, like this one here. This pattern is closely related to non-euclidean geometry. I cannot completely explain the details, but there is something deep behind that. …origin point. So you can deductively reason what happens here Okay Brady. So this big circle is this one and this circle is this one. Now what happens to this circle here? Well, again, we don’t have to measure anything This circle here touches the big circle, and touches this circle…