In honor of the never-ending number pi, we have one of a few international holiday celebrations. Pi Day was first celebrated on March 14, 1988, which also happens to coincide with Einstein’s birthday, March 14, 1879. Pi Day was first celebrated as a part of an Exploratorium staff retreat in Monterey, Calif. In March 2009, Pi Day became an official U.S. national holiday.
As part of my recognition of Pi day, I would like to explore the history of the number, who first discovered it, how it was originally estimated, and simple ways you can estimate it yourself. For starters, Pi is the ratio of a circle’s circumference to its diameter, or the length all the way around a circle divided by the distance directly across the circle. No matter how large or small a circle is, its circumference is always Pi times its diameter. Pi = 3.14159265358979323846… (the digits go on forever, never repeating, and so far no one has found a repeating pattern in over 4,000 years of trying.)
One of the most ancient manuscripts from Egypt, an ancient collection of math puzzles, shows Pi to be 3.1. About a thousand years later, the book of 1 Kings in the Bible implies that pi equals 3 (1 Kings 7:23), and around 250 B.C. the greatest ancient mathematician, Archimedes, estimated pi to around 3.141. How did Archimedes attempt to calculated pi? It was really by doing a series of extremely accurate geometric drawings, sandwiching a circle between two straight-edged regular polygons and measuring the polygons. He simply made more and more sides and measured pi-like ratios until he could not draw any more sides to get closer to an actual circle.
Hundreds of years later, Gottfried Leibniz proved through his new processes of Integration that pi/4 was exactly equal to 1 – 1/3 + 1/5 – 1/7 + 1/9 – . . . going on forever, each calculation getting closer to the value of pi. The big problem with this method is that to get just 10 correct digits of pi, you have to follow the sequence for about 5 billion fractions.
It was not until the early 1900s that Srinivasa Ramanujan discovered a very complex formula for calculating pi, but his method adds eight correct digits for each term in his sum. Starting in 1949, calculating pi became a problem for computers and the only computer in the U.S., ENIAC, was used to calculate pi to over 2,000 digits, nearly doubling the pre-computer records.
In the 1990s the first Beowulf style “homebrew” supercomputers came on the scene. The technology was originally developed to calculate pi and other irrational numbers to as much accuracy as possible. Some of these systems ran over several years to reach 4-billion digits. Using the same techniques over the years, we currently are at 22-trillion digits. This is a little overkill considering that, using only 15 digits of pi, you can calculate the circumference of the Milky Way galaxy to within an error of less than the size of a proton. So why do it? President John F. Kennedy said we do things like this, “not because they are easy, but because they are hard; because that goal will serve to organize and measure the best of our energies and skills.”
Attempting to calculate pi to such high accuracy drove the SuperComputing industry, and as a result, we have the likes of Google’s search engine that indexes trillions of webpages every day, computers that can replace physics research labs by simulating the real world and artificial intelligence systems that can beat the world’s best chess players. Where would we be today without the history of this number?
Now as I promised, there is a way you can estimate pi with very simple math. You play a simple game called “Pi Toss.” You will need a sheet of paper, a pencil and a bunch of toothpicks; the more toothpicks, the closer your estimate will be. Step 1: Turn the paper landscape orientation. Draw two vertical lines on the paper, top to bottom, exactly twice the length of your toothpicks apart. Step 2: Randomly toss toothpicks, one at a time, onto the lined paper. Keep tossing them until you are out of toothpicks. Make sure to count them as you toss them on the paper. Don’t count any that miss or stick off the edge of the paper, those don’t count. Step 3: Count all the toothpicks that touch or cross one of your lines. Step 4: Divide the number of toothpicks you tossed by the number that touched a line and this will be approximately equal to pi. How close did you come? To find out how this works, read more about Pi Toss at https://www.exploratorium.edu/snacks/pi-toss.