Everything Everywhere Daily: History, Science, Geography & More - Pi Day (Encore)

Starting point is 00:00:00 The following is an encore presentation of Everything Everywhere Daily. Every year on March 14th, the world celebrates one of the most important mathematical constants, Pi. It's a number which appears all over nature, even in places you wouldn't expect it. It's also a number that's been known, or at least has been approximated, by civilizations for thousands of years. Today, there's still more we're discovering about this number with the help of supercomputers. Learn more about Pi and how our knowledge of it has advanced over time on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong? ThruLine is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed.

Starting point is 00:00:58 It effectively turned day into night. And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. We might as well start the discussion of Pi with its definition. Pie is nothing more than the ratio of the search. circumference of a circle to its diameter. On one hand, it's extremely simple, but at the same time, it's devilishly complicated. The problem is that the circumference of a circle isn't divided evenly by its diameter. Finding out exactly what this ratio was has been the subject of inquiry

Starting point is 00:01:37 for centuries. Almost every civilization knew about the importance of this ratio. How they differed is in how they approximated the number and the methods they used to figure it out. A Babylonian tablet that dates back to about 1700 BC, has an approximation of Pi of 3.125. Earlier Babylonian approximations just used the number three. An Egyptian text known as the Rhin Papyrus has a value that works out to 3.1605. Some people who vandalized the Great Pyramid have determined that the Egyptians used the ratio of 22 to 7 as an approximation for Pi. Ancient Indian mathematicians wrote in the Shahatapatha Brahmata that Pi was approximately 339 over 108. which works out to 3.139.

Starting point is 00:02:21 The techniques used to find these approximations in the ancient world were primarily geometric and physical. They would use either a compass in a straight edge or physically create a circle and measure it. Believe it or not, these ancient measurements weren't that bad. Given the type of engineering that they were doing and the level of precision involved, they were able to get Pi to within 1%. However, what's good enough for hand construction doesn't cut it for pure mathematics. Much of the story of Pi from here on out is all about finding better ways to calculate the number and ever greater precision in the number of digits that can be

Starting point is 00:02:52 calculated. The first big step in calculating pi was independently discovered by both Chinese and Greek mathematicians. The ancient Chinese mathematician Liu Wei and the Greek mathematician Archimedes both realized that you could approximate the circumference of a circle by creating ever larger polygons inside of it. For example, a hexagon inside of a circle is clearly smaller. But an octagon is slightly bigger, and a decagon would even be closer to a circle, and so on and so on. In the year 245, Lu Wei eventually calculated a polygon with 3,072 sides, and came up with a value of pi of 3.1416. These early techniques developed by the Greeks and the Chinese were both early forms of integral calculus. In the year 480, another Chinese mathematician by the name of Zhu Shangji

Starting point is 00:03:41 used the same technique as Lou Way and calculated a 12,288-sided polygon. His value of Pi was correct down to seven decimals. This was a huge leap in calculating Pi, and one which would stand for over 800 years. Louie's algorithm worked, but there was a practical limit to how many sides of a polygon you could measure, but there were still a few more decimal places to be had using this method. In 1424, the Persian astronomer Jamshed Al-Kashi calculated Pi to 16. digits by calculating the equivalent of a polygon with 30 octillion sides. And it's amazing that something so large can only get you 16 digits, but that's the reality of pie. Here I should mention just how good

Starting point is 00:04:25 16 digits is. The head engineer at NASA has publicly stated that they only need to use 15 decimal points of pie when they're doing calculations. With that level of precision, you could calculate a circle with a circumference of 78 billion miles and have a margin of error the length of your little finger. In 1596, Dutch mathematician Ludolph von Kulin managed to calculate Pi to 20 digits and later 35 digits using the polygon technique. And this was pretty much the limit for using polygons, as it was simply too hard to calculate that much further. The next big innovation in calculating Pi was the use of infinite series.

Starting point is 00:05:02 Infinite series are, as the name suggests, adding up an infinite number of fractions. the more you add up, the closer it converges to the number you're trying to approximate. This technique is much easier than trying to determine the area of an ever larger polygon. For example, the codeventor of calculus, Godfried-Libniz, came up with an infinite series that converges to pi. The series is 4 over 1 minus 4 over 3, plus 4 over 5 minus 4 over 7, plus 4 over 9 minus 4 over 11, etc, etc. basically four times one over every odd number with alternating adding and subtracting each term. There are actually many different infinite series that converge to some multiple of pi. The way they differ is in how quickly they converge.

Starting point is 00:05:50 The development of calculus led to an explosion of these infinite series. In 1699, English mathematician Abraham Sharp was able to calculate Pi using a modification of the Leibnit series out to 71 digits. I should note that at this point, nobody was calling this number pi. The first use of the Greek letter pie to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. 71 digits of pie are far more than anyone could ever possibly use. To put it into perspective, if you use just 40 digits of pie, you could calculate a circle, the size of the observable universe with a margin of error less than the same.

Starting point is 00:06:32 size of a hydrogen atom. As pi was getting calculated to ever more precise values, mathematics advanced and mathematician started asking questions about it. For example, did the digits of pi ever repeat, and could pi be represented in some sort of polynomial equation? There are several attributes of pi which have subsequently been proven. One is the fact that pi is a transcendental number. This means that it's an irrational number that is not algebraic. For example, the square root of two is irrational, but it's not transcendental. The fact that Pi is transcendental was proven in 1882 by the German mathematician Ferdinand von Lindmann. This actually resolved what was perhaps the longest unsolved problem in the history of mathematics, squaring the circle. The very earliest mathematics was geometry

Starting point is 00:07:20 done with a straight edge in a compass. There was a surprisingly large amount of mathematical proofs that could be done using such simple tools. One problem that confounded everyone from Archimedes to Leonardo da Vinci, was trying to create a circle with the exact same area as a square using a compass and a straight edge. No one ever found a way to do it, and it turned out that it was impossible, and the reason is because pie is transcendental. Likewise, it was proven that the numbers of pie never repeat, although there may be short segments of numbers that do repeat themselves. The numbers also meet the criteria of being random. Calculations of pie kept getting better. It passed 100 digits in 1706, 200 digits in 1806, 200 digits in 18, and over 400 by the end of the 19th century.

Starting point is 00:08:06 The next big breakthrough came from the great Indian mathematician Srinivasa Ramanujan, who created several rapidly converging infinite series, which could increase the number of decimal points by eight at a time with every addition to the series. And this radically changed the ability to calculate Pi. In 1946, Pi was calculated to 460 digits by hand, which was the end of hand calculation records. After that, computers started to take over. In fact, calculating Pi was a way to test the performance of new computers.

Starting point is 00:08:37 In 1949, Levi-Smith and John Wrench calculated Pi to 1,100 digits using a desktop calculating device. Just months later, one of the first electrical computers, ENAC, calculated Pi to 2037 digits in just 70 hours. In 1961, an IBM-70-90 computer calculated Pi to 100,000 digits in nine hours. With ever more powerful computers and improved algorithms, the ability to calculate Pi exploded. The 1 million digit threshold was crossed at 1973, 10 million in 1983, 100 million in 1987, and a billion in 1989. As of the time of this recording, the record for calculating the digits of pie was set in 2021 by a team at the University of Applied Sciences of Eastern Switzerland,

Starting point is 00:09:25 and they've calculated Pi out to 66.8 trillion digits. With so many random digits, memorizing Pi has become a competitive activity. Memorizing Pi even has its own name, Pi philology. Most people can easily memorize Pi out to 10 or 20 digits, as it isn't that much harder than a phone or a credit card number. The current Guinness World Record for Pi Memorization is 70,000 digits. The feat was accomplished by Indian Rajviramina in 9 hours and 27 minutes in 2015. Pie is a universal constant. If we should ever encounter an alien intelligence, they should be just aware of Pi as we are. However, it's entirely possible that if they contacted us, they wouldn't let us know that they were there by sharing Pi

Starting point is 00:10:09 with us. Some mathematicians claim that's because Pi isn't the number we should be using. The reason is that you almost never encounter the diameter of a circle in mathematics. What defines a circle is its radius. If the radius is the important measurement, then why do we use the diameter? The real number that we should care about, as some suggest, would be the ratio of the circumference to the radius, which would be the same as two times pi. This number, 6.2838318, etc., has been dubbed tau. If you've ever studied a sign or cosine function, you know that it makes a complete cycle once every 2 pi. Likewise, if you've ever worked in radians, one complete circle is measured as 2 pi. If aliens send us a number that shows some certain universal mathematical mathematics,

Starting point is 00:10:57 mathematical constant, they may be sending us tau instead of Pi. Likewise, I should really be doing an episode on Tao Day, which takes place on June 28th. Even if the Tao advocates are right, pie is still an important component of it, and pie will still be used as it has for over a thousand years as one of the most significant mathematical constants. So, to everyone out there, happy Pi Day. The executive producer of Everything Everywhere Daily is Charles Daniel. The Associate producers are Peter Bennett and Cameron Kiefer. I wanted to give a big thanks to everyone who supports the show on Patreon. Your support helps me put out a new show every day.

Starting point is 00:11:38 And if you're interested in Everything Everywhere daily merchandise, Patreon is currently the only place where it's available. And if you'd like to talk to other listeners of the show and get notified of future episodes and projects, please join my Facebook group or Discord server. Links to everything are in the show notes.

Everything Everywhere Daily: History, Science, Geography & More - Pi Day (Encore)

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