Now that we’ve explored pi’s footprint throughout history, let’s get into the good stuff: how to do these estimations on our very own!
We’ll start with the tried and true geometric route, used by many throughout time. By inscribing and circumscribing many-sided polygons about a circle, and comparing the ratio of their perimeters to their diameters, we can get a pretty darn good bound for the ratio of the circle’s circumference to its diameter, i.e. pi. Using the GeoGebra applet below, explore this method, and don’t forget to click over to page two in the applet for some bonus content!
This second method may seem a little out-of-pocket, but with the power of geometric probability, it works! In 1777, a mathematician named Georges-Louis Leclerc, Comte de Buffon introduced this fascinating discovery: if you drop some needles onto a flat horizontal plane with a series of parallel lines drawn on it such that the distance between the lines is twice the length of the needles, the ratio of the total number of needles dropped to the number that land crossing one of the lines estimates pi. How wild is that! The more needles you drop, the more accurate your estimation. There are other factors that may affect accuracy, such as grid shape, grid density, and needle length, but for the sake of simplicity we won’t get too nit picky (Siniksaran, 2009). Watch the following animation to see Buffon’s Needle in action.
As a fun side note, this process involving Buffon’s Needle is the same principle used by a type of ant called Leptothorax Albipennis to determine if a potential nesting site has enough room to house their entire colony (Mugford 2001). The more you know!
Another seemingly random estimation can be done using the Monte Carlo method. This method shows that if you take a square with a circle inscribed, and randomly plot points all over within it, the ratio of points within the circle to the total number of points within the square, multiplied by four, approximates pi (Strbac-Savic, 2015). How in the world could this possibly work?? Watch this video to find out!
Now that you understand why this works, try giving it a go yourself with this GeoGebra Applet.