The Pi of Rivers

I met pi in school. You probably met pi that way too. It is the number used to calculate the circumference of a circle. Pi is shown symbolically as: π. It is the ratio of the circumference of a circle to its diameter. It is an “irrational number” which means its exact value is inherently unknowable.

Using computers, we have calculated billions of digits of pi, starting with 3.14159265358979323…   –  but no recognizable pattern emerges. So strange. The digits of pi continue to infinity. Does anyone really understand infinity?

Ancient mathematicians did not like irrationality because it didn’t work with the concept of an omniscient God.

Recently I read about another pi connection which is also strange. In 1996, the UK earth scientist Hans-Henrik Stølum published a paper announcing that pi explains the seemingly chaotic paths of rivers in a mathematically predictable pattern.

This is called a river’s sinuosity. By dividing the river’s actual meandering length by the length of the direct line drawn from source to sea. Albert Einstein used fluid dynamics and chaos theory to show that rivers tend to bend into loops.

I am sailing on a river today, so all this seems in some way very relevant. I would have said that the course of a river is controlled by things like mountains, valleys, tributaries, rock formations – in other words, geography and geology.

Some rivers flow in a fairly straight path from source to mouth, so they have small meandering ratios. Some rivers wander all over the place and have high meandering ratios. But the average meandering ratio of rivers seems to be good old 3.14.

If a river has a curve that will generate faster currents on the outer side of the curve. Those currents will cause erosion and so a sharper bend. That will eventually make the loop tighten. I have read that then chaos will eventually cause the river to double back on itself and form a loop in the other direction. I did some more research on this river connection and found that this claim may not be accurate.

Someone put up a website at one point to crowdsource river data. The site at PiMeARiver.com seems to be dead now. People could put in the coordinates of the mouth and the source of a river, and the length of the river (from Google Maps and Wikipedia probably) to calculate the sinuosity of a river. That study looked at 258 rivers and found an average sinuosity of an un-Pi-like 1.94.

Hmmm. Maybe it is another mathematical constant, like the golden ratio (phi) which we often find in nature. That value is 1.618. Nope.

What about if you look at pi/phi? You get 1.94. Okay, that’s a strange “coincidence.”  Or something more than coincidence?

I need to be careful with all this, because I saw the film titled Pi when it was released in 1998 and it changed the way I thought about numbers. It is a difficult film to label. It is a surrealist, psychological, thriller, that delves into religion, mysticism, the relationship of the universe to mathematics, and number theory. It was written and directed by Darren Aronofsky in his directorial debut.

I view it as a cautionary tale. It is about a genius oddball mathematician, Max, who has been working for a decade trying to decode the numerical pattern beneath ordered chaos. The ordered chaos he studies is the stock market. Max’s belief that there is some mathematical “code” underlying everything compares in my mind with Einstein trying to find that theory that explains it all. That quest frustrated Einstein through the end of his life.

Beware of that quest.