I was a good math student in junior high, and I’m sure I did well on all my pi-related homework and test questions. I understood that pi was an irrational number, and knew, theoretically, that it had an endless number of non-repeating decimals. However, I had no real grasp of why it’s irrational. It was just another abstract, memorized math fact. But a few years back a realization came to me. So, in belated celebration of Pi Day on March 14, I offer my pictorial explanation of why pi is irrational.
The realization came when thinking about the formula for an area of a circle, Π r2 (pi * r2) where r is the circle’s radius. It dawned on me that the r2 referred to a real geometric square with sides and corners and all, not just a letter with a number above it, and that:
Pi represents the number of squares with a side of length r that would fill a circle with a radius of length r.
Therefore, my explanation of why pi’s digits are infinite is that no matter how many squares you put inside a circle, there will always be a smaller “corner” into which you can cram a smaller square. This leaves an even smaller corner for an even smaller square, and on and on without end.
I don’t know whether this is a mathematically valid explanation. But transforming this abstract concept into something I could draw with a pencil pleased me no end.
Pi and VBA
A pencil is one thing, but on Pi Day I decided to illustrate my explanation with Excel and VBA. It took me a really long time! First I tried using shapes, a reasonable, but really bad, idea. I ended up using just one shape, a circle, and a bunch of tiny cells, which are, after all, squares:
The circle above has a radius of 50. When all the squares that fit are filled in, their total area is about 3.08 times that of a square with a radius of 50. In other words, it’s fairly close to pi. If I change the settings to a radius of 100, the number climbs above 3.1, but still short of pi.
Programming this was challenging and fun. Because of the nature of the project I coded some things more loosely than normal. There’s a bunch of global variables, and even a “Select Case True” statement.
I did a lot of Unioning and Intersecting in the code. One thing I rediscovered is a major glitch with the Union statement, which gives you an incorrect cell count for the Union of two overlapping ranges:
Two Miscellaneous Things
My search for a screen-capture-to-animated-GIF program continues. I had been using Cropper, but it was unable to keep up with the circle’s “exit stage left” in the animation above. As you can see, I’m now trying ScreenCast-o-Matic, which has a nice interface, and both desktop and web versions. For $15 I can upgrade and remove their logo from the video, but when I tried this morning their website refused to take my money.
Finally, on Pi Day’s website it claims to be celebrated all around the world. It must be a fairly abstract reference in countries that use the, more logical, DMY date system, e.g., 14/3/2013.
Here’s a workbook with the code.