A post, in two parts.

Part 1: Strange Attractors, and their strangely attractive backstory

I recently had the pleasure of driving to Texas and back, and while on the road, partook in some podcasts. One of them was called, “Stuff You Should Know”, and dedicated an episode to Chaos Theory. They did an admirable job with a slippery idea, and while I took away some new insights, this topic was not new to me.

No, I’ve been a fan / advocate / evagnelist / lunatic / devotee of Chaos Theory since the heady days of 2001. Early in my undergraduate forays into math and physics, I convinced a professor to let me explore the interesting ramifications of nonlinear and fractal geometry. Before going any further, if you’ve searched for “fractal geometry” or even just “fractals” on the internet, you’ve gotten some websites that looked like they emerged from the 1990’s. You see, the people that love fractals have a need to share their insights. I was firmly in this camp.

I graphed henons in QBASIC, porting code from the appendix of – gasp – print books. I drew sierpinski triangles everywhere. I used “bifurcate” every chance I could. It was a wild high, and I chased that tempestuous beast into the darkest of nights.

I became fascinated by coastlines: did you know they are infinitely long? On a map, the circuitous circumference around an island nation like Iceland may look to be… ~1,000 miles? But in reality, we can dive into any fjord or inlet, crouch low and begin poking around in the rocks that roll gently in the lapping tides. Where is the coastline? Is it before or after that pebble? If that pebble is included, don’t we have to include it in the total distance of the island perimeter? Yes, you most certianly do, as a responsible member of society, a reasonable steward of information.

This leaves us with an uncomfortable truth that “edges” and “boundaries” are often much more complex than we anticipated. Also revealed from fractal geometery, the same complex gravitational forces that create a valley of flour in your bowl when an egg is cracked into it, are the same that shape the timeless Grand Canyon (at least, many are shared - in all likelihood, your flour valleys have not been whipped by wind and rain for millenia). But the point remains: brocoli looks like trees, and a bunch of geese flying look suspiciously like reflections of light on lazily undulating waves.

And yet there is so much more to this story, which again, I must commend the hosts of that podcate for nobly wading through, with patience and a singular sense of where they are in the discussion. Restating known things can be useful for honing intuition, and setting the stage. So to explore the strange attraction of Strange Attractors, we must go back to Sir Isaac Newton, King Oscar II of Sweden, and the “n-body-problem”.

Newton could predict apples falling, and was having good success predicting comets and cannonballs. So what about predicting the location of a handful of planets 10, 100, 1000 years out? Not so much:

“Knowing three orbital positions of a planet’s orbit – positions obtained by Sir Isaac Newton (1643-1727) from astronomer John Flamsteed[6] – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet’s motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity.[7] Having done so, he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits very well or even correctly.[8] Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits.”

And here, the badlands of Chaos Theory and nonlinear dynamics peeked forth. Scientists and mathematicians started to understand that the gravitational pull from each planet was simultanesouly pulling on one another. Each moment, or “iteration”, magnified any variances or inaccuracies in the initial measurements that were plugged into the equations. If a cue ball hitting three pool balls with speed x is turned into equations, we can predict where they will be after 1000 “moments”. However, if speed x is ever so slightly different, those pool balls will be in wildly different places just a few “moments” later. This is a gross over-simplification of the n-body-problem, but it gets at the heart of it.

King Oscar II held a contest for anyone who could solve this problem. As I didn’t intend to delve too deeply into the history, but instead muse on what it might mean in the present, I continue to muddy and gloss over the finer points of this. This article explores the contest and solution in much better detail. But for our purposes here, Poincaré pointed out that it was unsolvable, and won the prize. He proved that a perfect predition relied on infinitely accurate measurements, and we know that’s not possible. And Chaos Theory was born.

Fast forward lots of years, and we’re finally getting back to Strange Attractors, and the meteorologist Edward Lorenz stumbld on this very problem while trying to distill complex equations around thermal dynamics to a simple form. This passage from Wikipedia sums it up nicely,

“Minute variations in the initial values of variables in his twelve-variable computer weather model (c. 1960, running on an LGP-30 desk computer) would result in grossly divergent weather patterns.[2] This sensitive dependence on initial conditions came to be known as the butterfly effect (it also meant that weather predictions from more than about a week out are generally fairly inaccurate).[13]”

And here is what he graphed:

The hands grow cold, the coffee cup has tipped past halfway. Where are we in this discussion? Those dark spots in the graph above, those are the strange attractors we’ve danced around. Why does this matter? Why is interesting? How is this related to coastlines, pebbles, brocoli, and eggs? That, is fodder for another post…