In recent years, the date March 14 (3/14) has increasingly been recognized as “Pi” Day. Most people that have taken 10th grade geometry understand that the circumference of a circle is calculated as the product of Pi and the diameter of a circle. Pi is generally understood as 3.14 although there are those that boastingly reel off the number to 10 decimal places: 3.141592653589, etc. On this day, math geeks everywhere will celebrate this important number, sporting t-shirts bearing the Greek symbol for Pi. Undoubtedly, pie sales in local bakeries and groceries will spike up as well.
As a geometer by trade, I must take this opportunity to comment on this increasingly popular number. We generally think of Pi in a utilitarian sense -- we use it to accomplish the goal of calculating the circumference of a circle. We are also taught that this is a number expressed in decimal notation that never ends, repeating its sequence of decimal places on into infinity. As a young student in high school geometry, I remember that the first time I encountered that notion, I found it awe-inspiring. It tells us a lot about our wondrous cosmos that there is a singular number of infinite dimension that is at the basis of the most fundamental structure of the universe – the circle. Without Pi, well, we wouldn’t have circles. And it’s certainly impossible to imagine a universe without circles.
Much of geometry is based on the notion of proportions and ratios. I would like to make clear that geometry is explicitly NOT arithmetic. It is often misunderstood as “math” and spoken of as a bunch of boring proofs and theorems. However, it is not really math, it is the study of number in space. It is the study of patterns, proportions, and relationships in space. I have come to think of Pi more as a cause for the circle rather than as a tool for calculating the circumference of the circle. Without Pi, there would be no circle. While memorizing Pi out to 10 decimal places might amuse a 14-year-old, thinking of pi as a very long number rather misses the point. Understood as the ancient Greeks might have, Pi is seen as a proportion: it is the exact proportion between the circumference of a circle and its diameter. It expresses the relationship between the circumference and the diameter. It is the singular number that results from comparing the circumference of a circle to its diameter. It is always Pi, no matter if we never know exactly how many decimal places it has. It is the number that singularly holds the universe together, at least insofar as the universe is heavily dependent upon circles.
Pi is perhaps the most famous of what are known as the “transcendental numbers”. Pi is also both irrational (having infinite decimal places) as well as transcendental. [Transcendental simply means that there is no polynomial equation with integer coefficients for which the number is a root solution.] I, myself, am partial to the beautiful number “Phi” (1.618) but that’s a story for another day. It should be said, that, while Phi is extremely beautiful and widely used throughout the known universe, this number Phi is merely irrational and is not transcendental.
When I first heard about numbers that were both transcendental as well as irrational, I thought to myself, “now those are my kind of numbers! Transcendental comes from the Latin word “transcendere” which means to climb over or surmount. Transcendental numbers are named as such because they climb infinitely, unceasingly growing towards the infinite, numbers that climb to the sky. They just keep on going, forever and forever. At the same time, they are singular – completely unique. Completely unique and completely infinite at the same time. How could it be that something that is infinite and as such, completely unknowable, and yet be the basis of the finite known material universe? I will leave you to ponder that thought.