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The Great Tale of the Endless Pi(e)

(Cover Artist: Sophie Cheng)

Happy Pi Day! If you look around on March 14th, you’re very likely to find those words on least one graphic t-shirt with this symbol: 𝝅, most likely along with an illustration of a cartoon pie. But what makes pi so special?

Pi can’t be expressed as a fraction like other decimals can, which is what makes it irrational. It is also what mathematicians call an infinite decimal; pi is roughly equal to 3.14 (which is why March 14 is pi day), but goes on to infinity and beyond.

However, pi isn’t just good for memorizing as many digits as you can and showing off to your friends. Pi measures the ratio of the circumference of the circle — the length of the line that goes around the circle — and its diameter, which is the line drawn between two opposite points on the circle that crosses the center. In other words, it's the number you get when you divide the circumference of any circle by its diameter.

Pi has been observed since the time of the ancient Egyptians and Babylonians, both of whom calculated pi to be somewhere around 3. More than 4,000 years ago, the ancient Babylonians had almost accurately calculated the true value of pi, estimating it to be 3 and one eighth (3.125, though decimal measurements didn’t exist yet), which was quite impressive, considering humans at that time were still getting the hang of using iron tools.

But how do you estimate pi? By definition you would need to measure a curved object, which can be very difficult to do accurately and to a precise decimal. In 212 BC, around 2200 years ago, a Greek mathematician, Archimedes, known for many widely used theorems and ideas in mathematics today, devised a relatively simple method to accurately estimate the value of pi.

Straight objects like polygons were much easier to measure when not having to account for curvature. If you put a square inside a circle, with all four corners touching a point on the circle, and another square outside a circle with each side of the square touching a point on the circle, you could come up with two rough estimates of pi by measuring the diameter and perimeter of each square and dividing. The true value of pi would be somewhere in between those two measurements.

But a square isn't that much like a circle, is it? The range you produce using a four sided polygon is wide and not very accurate. But, what if you kept adding sides? Look at the following shapes:

From the left, we have a square (4 sides), a pentagon (5 sides), a hexagon (6 sides), a heptagon (7 sides), an octagon (8 sides) and a nonagon (9 sides). Observe how each time we add a side, the polygon gets closer and closer to looking like a circle. Another way of looking at a circle is as a shape with infinite sides. So, if we used the same method as before but used a nonagon instead of a square we would have a much more accurate range for pi than with a square. If you added another side and used a decagon, you could have an even more accurate estimate.

You could go on forever using this method (called the method of exhaustion, because, as you can imagine, it got tiring after a while to keep drawing many sided shapes). Archimedes eventually got up to a 96-sided shape and narrowed the range down to somewhere between 3.1408 and 3.1429, which was accurate to two decimal places. This was likely his last contribution to the world of mathematics, as he was reported to have still been working on this when he died in 212 BC.

For the next thousand years after Archimedes, exploration into pi seemed to have died down in Europe, but the rest of the world was still searching. Chinese mathematician Liu Hui used a similar method of exhaustion as Archimedes, and estimated pi to 4 decimal places using a shape with 3,072 sides. 200 years later, Zu Chongzhi and Zu Gengzhi, father and son, built on the previous ideas and estimated pi to 5 decimal places using a shape with 12,288 sides.

As you can probably tell, the more precise you wanted to get, the more sides you had to keep adding — it just got too exhausting to continue using the method of exhaustion. It took 200 years and adding almost ten thousand more sides to a polygon to narrow the estimate down by a single decimal place — no one achieved a more accurate estimate until 1630, when Austrian astronomer Christoph Greinberger determined pi to 38 decimal places using a shape with 10,000,000,000,000,000,000,000,000,000,000,000,000,000 sides. That number, for reference, has forty zeroes. Mathematicians everywhere needed a better, more efficient, way to calculate pi.

To do that, there was one other key ingredient they were missing. In the 800s CE, Persian mathematician Al-Khwarizmi created algebra, which uses a variable, often x, to represent up a missing section of a mathematical equation for us to find. It took a few hundred years for Al-Khwarizmi’s work to spread across Europe, but when it did, the Scientific Revolution in Europe was in full swing. Europeans used this newly-obtained method of algebra as a basis for a concept called infinite series, which is the idea of a list of infinite values resulting from specific parameters. If the numbers get smaller as the series continues, they add up to a set value.

For example, if you add up all the fractions in this infinite series, which results from halving every item in the series, you get 1. Mathematicians competed for finding the fastest, most efficient way to calculate the most digits of pi by finding new infinite series or combining infinite series that converged on, or added up to, pi. Using this method, mathematicians could find pi up to hundreds of decimal places.

Then came the invention of computers, which supercharged the process. In 1949, the first calculation of pi by an electronic computer was done to estimate pi to 2,036 decimal places. Now, the world record for digits of pi calculated is to 62.8 trillion decimal places.

Okay, so what’s the point of it all? No one is measuring the area of a circle to the 62 trillionth place, so why keep calculating pi? Well for one, pi is a great random number generator, and each of the 10 digits in our base 10 number system appear in nearly equal percentages in a long enough estimate of pi. This is very helpful for people working in fields like cryptography, where they have to encrypt data and, in basic terms, hide important information in a sea of random, useless information. Pi is also a great way to check that your supercomputer actually works, by running difficult calculations that can be very easily checked.

But most importantly, pi is still being calculated because it’s interesting. It’s why we memorize digits of it, and why we wear puns about it on our graphic tees and eat pie on March 14, even though the dessert has nothing to do with the mathematical concept except for sounding suspiciously the same. There’s always going to be someone around who, even knowing they’ll never reach infinity, will want to take just a few more steps closer anyway, just because they can.

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