The mysterious connections between the physical world and mathematics run deep. We all know the number Pi from geometry. The ratio between the circumference of the circle and its diameter and that its decimal digits go on forever without a repeating pattern. As in 2013 it had been calculated to 12.1 trillion digits. But somehow Pi is a whole lot more.
Pi appears in a whole host of other phenomenon which at least on the face of it, nothing to do with circles or anything. In particular, it appears in probability theory quite a bit. Suppose I take a piece of paper and divide it into six equal halves. And Suppose I take this needle so as if the length of the needle is equal to the distance between two lines on this piece of paper.
And suppose I drop this needle on the paper. Sometimes when you drop the needle it will cut a line and sometimes it drops between the lines. It turns out the probability that the needle lands so it cuts the line is exactly 2/π or about 64%. Now what that means is that in principle if I drop this needle millions of times, I would count the time when it crosses the line and when it doesn’t cross a line. And I would actually even calculate pi even though there are no circles here, no diameters of circle nothing like that.
Since pi relates a round object, a circle with a straight line as its diameter, it can show up in strangest of places. Some see it in the meandering path of the rivers. A river’s actual is its way that it makes from the source to its mouth compared to the direct distance on average seems to be about pi.
Models of just about anything involving waves will have pi in it. Like those for light and sound, pi tells us which color should appear in a rainbow and how Middle C should sound on a piano. Pi shows up in apples, in the way cells grow into spherical shapes or in the brightness of a Supernova.