Pi Day

We at CatSynth once again join others in recognizing Pi Day today.  This time, Google is joining in as well with one of the special “Google Doodles” on their front page:

The image very nicely captures many of the well-known geometric and trigonometric properties of π in an abstract representation of the Google logo.

Of course, I tend to me be more curious about some of the more esoteric properties that interrelate to other parts of mathematics.  For example, consider  seemingly unrelated Gaussian Integral.

This is the area underneath the Gaussian function which is usually associated with normal distributions probability and statistics. The interrelation of π and e, which we have presented in previous years, is at play again here.

The approximation of the square root of π is 1.77245385…, and like π itself, is a transcendental number. In addition to its appearance in the Gaussian Integral and the Gamma Function (which we also presented on a past Pi Day), it plays an important part of the ancient mathematical problem of Squaring the Circle, that is constructing a square with the same area as a circle using only a compass and a straight edge. Because the square root of π is transcendental, it suggests that Squaring the Circle is in fact impossible. But that probably won’t stop some people from continuing to try.