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- [Instructor] Consider a square. Let's say this square is one inch across. And so let's set up a scale. And if we unroll this square on the scale the tip will land round four which should make sense of squares one inch across it should be four inches around. Now let's do the same thing with a circle. If we unroll a one inch diameter circle the tip lands almost on three and historically three was actually a typical guess for the circumference of a circle. This is a passage from the Bible, which claims that a circular Lake is three times further around than it is across. But if we look closely we can see that's actually not quite right. This quantity is actually just a little extra bit more than three. So we call this length Pi and this presentation is called Pi the story of an extra little bit, because for 2000 years people have been coming up with really ingenious schemes for trying to describe this number. So let's see if we can take a closer look at this extra little bit. Here we have the number line and there we have Pi just a little bit above three. And if we wanted to make a first approximation we could say Pi is about three plus an extra little bit. And to get a better view of this extra little bit, we can zoom in and again, those white points are two, three, and four. If we take the space between those points and divide it up into 10 equal spaces we can see that Pi is just a little bit more than 10% of the way from three to four. So if we wanted to get more accurate, we could say Pi is 3.1, plus an extra extra little bit. You can see it's not right on one. If we zoom in, we can see that by dividing the space between 3.1 and 3.2 into equal tents that Pi is just a little bit past the four marker. So we can get even more accurate and say, Pi is about 3.14. But once again, it's not exactly on four. There's just an extra, extra, extra little bit more. So if we zoom in again we can see that Pi is to a greater accuracy about 3.141. But again, that's not exactly right. If we zoom in again, I think you know this is just gonna go on forever. We're never gonna land exactly on a mark Pi is this funny kind of number where you'd actually have to talk forever in order to say what it is specifically. So this is a talk about Pi. And what I wanna look at is not so much these digits but the question of how do we figure out these digits in the first place? Because for over 2000 years people have been coming up with really ingenious schemes for figuring out what these numbers are. And each of them is kind of lovely in its own way. So this talk is just an opportunity to present some of those. The oldest method for calculating the digits of Pi comes from Archimedes and it's on three observations. To see what they are, let's go back to our circle and our square. The circle has less distance around than the square. And it's easy to see why if you were walking around around the circle, you would get around in less time than walking around the square because the circle kind of cuts every corner. On the other hand if you imagine a square that fits inside the circle, the square will have a smaller perimeter than the circle for kind of the same reason. You can see if you were walking around it you'd still be cutting every corner. So if we unroll these on the number line the big square will have the greatest perimeter. The small square will have the smallest perimeter and Pi, whatever it is will be somewhere in between. So we can actually use the Pythagorean theorem to figure out the perimeters of these squares. And it tells us that Pi is a number somewhere between 2.82 and four. So we have a kind of a per and lower bound. It's not a very good estimate, but it's a start. So that was Archimedes' first observation. The second is that if instead of using a four-sided polygon we used a polygon with more sides, the inner and outer polygons would sort of hug the curve of the circle better. So when we unroll them, we're gonna get a tighter bound between the upper and lower possible limits for Pi. So with a six-sided polygon we can determine that Pi is somewhere between three and 3.46. So the more number of sides we use on our polygon the closer the bound we can get for the value of Pi. So this is our Archimedes' second observation. So if we wanted to get more digits of Pi we could start over with say a 10-sided polygon but it turns out there's a trick to this. Archimedes realized that if you have an estimate like this, which we got from a six-sided polygon you can go directly from these numbers to the numbers you'd get if you calculated the circumference of a polygon with twice as many sides. That is we can go directly from the estimates from a 6-agon to the estimates, we'd get from a 12-agon and this procedure can be applied over and over and over again. So we can go from a 12-agon to the estimate, we'd get from a 24-agon and from there to what we'd get from a 48-agon and from there to a 96-agon. And for some reason, this is as far as Archimedes went but in principle, you could keep getting more digits of Pi by just calculating a higher and higher number of sides polygons. And it it's interesting to remember that Archimedes was working with Roman numerals. So he didn't have any concept of zero. He didn't have a fully standardized set of methods for arithmetic, and yet he was still able to calculate Pi to what we would now say is an accuracy of two decimal places. So for about 2000 years this polygon method was the only game in town. Anytime we made progress on figuring out more digits of Pi it was usually 'cause someone had done a marathon calculating session with ridiculously many sided polygons. So for example, in 482 Chinese father and son team got Pi out to seven digits using a 12,288-agon. In 1615 Ludolph Van Ceulen got out to 35 digits with a two to the 62th power-agon. And this was considered such an accomplishment at the time that they actually had the numbers engraved on his tombstone. And for a while, Pi was known as the Lodolphine constant. He was beaten about 15 years later by Christopher Greenberger who got out to 38 digits with a 10 to the 40th-agon. And I believe Greenberger might've been the last person to calculate a record-breaking number of digits of Pi using the polygon method, because then we discovered calculus and people started coming up with all kinds of crazy ways to calculate Pi. So here's one method for calculating Pi. Start with four over one, and then we're gonna subtract something from that, which is four thirds. And then we're gonna add something to that four fifths and then we're going to subtract four sevenths. So you can see there's a pattern developing, basically it's a sequence of fractions that alternately get added or subtracted. They all have four across the top and they have each of the sequential odd numbers along the bottom. And this turns out to get closer and closer to Pi. This is known as the Madhava method. It was first discovered in the middle ages in India and then was rediscovered about 300 years later in Europe. So this is what this one looks like in practice. You can see, we have that series across the top. And if we look at the first term, it's four over one which gives us 4.0, which is too big but to get closer to Pi luckily we're going to subtract four thirds from that. And that takes us down to 2.66, which is too low but now we're gonna add four fifths and that brings us up to a 3.466. So I'm gonna let this run. But the thing to focus on is that number Pi in the middle. And to notice that with each step we get closer and closer to Pi and Pi is in fact the ultimate destination of this series. So here's another way of calculating Pi. Start with pairs of even numbers and pairs of odd numbers but with only a single one then zipper these together into a series of fractions and multiply all of those together. And then double that. And this turns out to get closer and closer to Pi. This is known as the Wallis method. It was discovered in 1655 by John Wallis. And here's what this one looks like in practice. You start with two, and then you multiply it by two over one to get four, and I'm gonna let this one run but we're actually gonna keep up with it as it descends towards Pi. And the thing to notice is just the slow and steady pace with which this approaches the quantity Pi. So after eight steps, we get to 3.3437. So we're actually, we're approaching Pi very slowly. The Madhava method also only got to 3.0171 after eight steps. So we're actually gonna have to do hundreds of steps before we start getting the first few digits of Pi. So these are like, it's kind of cute that they work technically but they're kind of useless from a practical perspective. We're not just looking for methods of calculating Pi. We're looking for methods that are reasonably fast. So here's a somewhat faster way of calculating Pi. Start with three and then we're gonna add and subtract a series of things. These things are all gonna be fractions with four at the top. The first one will have two times three times four on the bottom. The next one we'll have four times five times six. The next one we'll have six times seven times eight. And the pattern is basically that you get these runs of three sequential numbers across the bottom. This is known as the Nilakanthan method. This was discovered around 1500, also in India. And here's what this one looks like in practice. It starts out basically on top of Pi. And then after one step, we get to 3.166. So we're already making significantly more progress with this method than we were with the others. The Madhava method only got this close after eight steps, for example. So we're gonna run this one and keep up with it. And the thing to notice is I think how much more rapid it is than the Wallis method. We're now making significantly quicker progress towards Pi. So after eight steps, we get to 3.14207. And just for reference, these are the upper and lower bounds that Archimedes found with his 96-agon. So this method is roughly competitive with the polygon method. Here's a fun one. Start with two multiplied by two over the square root of two. Then multiply that by two, over the square root of two plus the square root of two, then multiply that by two over the square root of two plus the square root of two plus root of two. So with each step, we're just sneaking in another plus square root of two under the square root sign. And this product gets closer and closer to Pi. And also if you write it out after a while it starts to take on this kind of beautiful musical staff looking quality. So this is known as the Viete method and this is a particularly rapid way of calculating the digits of Pi. Here's what this one looks like. We start with two and then when we multiply by two over the square root of two, that takes us up to 2.8284. And this one's kind of interesting because instead of jumping back and forth it's actually sort of sneaking up on Pi from the left. So if we let this one run for eight steps we see we get Pi to four significant digits. So 3.14151. So that's two more significant digits than we got out of the Nilakanthan method at the same rate. So basically there's this whole bumper crop of new ways of calculating Pi that start to get developed. And often when someone breaks a record in terms of calculating digits of Pi it's often because they're showing off a new method. So one of my favorite stories from this time is about a man named William Shanks who spent about 20 years calculating the digits of Pi on and off. And he was able to calculate it out to 707 digits. But years later, it turned out someone discovered that he made a mistake at the 527th digit and every digit after that is wrong. So he spent years of his life, calculating incorrect digits of Pi, which is a hazard of this hobby, I suppose. So the next significant thing that happened in terms of our ability to calculate the digits of Pi was that this person existed. This is Srinivasa Ramanujan. He was a self-taught mathematical prodigy from Southern India. And in 1910, he traveled to Cambridge and did some of the most groundbreaking mind-bending number theory that's ever been done. Ramanujan had this kind of extraordinary numerical vision where he could conceive of a formula like this and he would claim that he could perceive somehow that this was equivalent to Pi. And if you've done much math this is actually a really psychedelic looking formula. There's a lot of stuff stuck together in ways that you don't normally see. And normally a mathematician will derive a formula like this or provide a proof, but Ramanujan a lot of his results were actually divinely inspired. He claimed that his goddess Namagiri would just reveal these to him in visions. So if you actually calculate this out it very much does seem to generate the digits of Pi but nobody including Ramanujan could explain why it worked. And it took until about the 1980s for our mathematics to become sufficiently sophisticated that we could prove that this was indeed an exact formula for Pi and not just a really good seeming approximation. So this is known as the Ramanujan formula. And one thing I learned about the Ramanujan formula when I was making this talk is that it is really hard to animate because it is very fast. You get an eight additional digits of Pi with each step. So if you try and animate that it just kind of looks like a blur but also I was trying to do these out to eight steps and it turned out that actually exeeded the zoom range of the animation program I was using to make these. So I was having kind of a comically hard time trying to animate this one. So in 1988, a couple of Ukrainian American brothers named the Chudnovsky's figured out a way to turbo-charge Ramanujan's formula and came up with this crazy thing. So this generates 14 additional digits of Pi with each step. And the Chudnovsky's are actually kind of interesting characters. Not only did they come up with this Chudnovsky's brothers algorithm but they were actually pioneers in using super computers to do extremely deep calculations of the digits of Pi. One of them had an apartment on the upper West side in Manhattan and little by little, they were bringing in a hard drives and pieces of super computer hardware that they could get their hands on. And eventually they completely filled the apartment with super computer and they had air conditioners running in every window trying to keep the thing cool. And they were trying to keep it a secret from the landlord and the Chudnovsky's eventually got so complex that they were afraid to shut it off because they didn't think they could ever get it turned back on again. But for a while these guys were consistently the record breakers. They were the first people to get to a billion digits of Pi. I think they got up to about 4 billion digits on their own and that Chudnovsky brother algorithm continues to be one of the main tools we use to this day in doing extremely deep calculations of Pi. And the continued relevance of this algorithm is actually kind of interesting because it's not actually the fastest method of calculating Pi that we know about. For instance, in the 1970s we began to discover a whole new class of methods for calculating Pi. An example of this is known as the Salamin-Brent algorithm and I don't actually have the formula for it here. You can look it up on Wikipedia. It's not actually that complicated and it ends up calculating the digits of Pi in a really interesting way. On the first step, you get one accurate digit of Pi. On the second step, you get up to four digits. On the third step, you get up to nine digits. So you start off pretty slow but with each step you're getting approximately twice as many digits as you had before. So if you go up to eight steps you get 347 accurate digits of Pi and you know how there's repeated doublings accumulate. If you go up to 24 steps that actually gets you to 45 million digits of Pi. So you can get very deep into Pi with a very few number of steps. The problem with this is if you're going to calculate Pi to 45 million places, you need to be working with 45 million digits of accuracy from the beginning. So in order to carry out this calculation, the amount of computer memory you need, the amount of like digital scratch paper starts to get really exorbitant. So this isn't often practical for super deep calculations of Pi. So there actually seems to be kind of a trade off, on the one hand, you have algorithms like this which are very slow, but trivially easy to compute step after step after step. And then at the opposite end of the spectrum you have things like the Salamin-Brent which are very fast but required enormous overhead in terms of computational resources in order to carry it out. And then ones on the middle I wanna say, they're just right. You can bite off a good chunk of Pi with each step, but your material resources required don't get too much out of hand. So maybe as a final thought, I just wanna say that, you know, you don't need more than about 40 digits of Pi to calculate the circumference of the universe to an accuracy of a hydrogen atom. So these digits are not especially important to us in empirical practical terms. And yet it seems to be very much a part of reality that they exist. They're really specific and they go on forever and we have access to them. Not because we're doing any kinds of measurements or experiments, but because we have the capacity to just think rationally and logically about the properties of numbers. And I think this is a surprising and beautiful thing about mathematics. And I think these various algorithms represent the diversity of human endeavor in trying to make sense of this number. So that's what these mean to me. And I'm grateful to have been able to share them with you. Thanks so much.