My apologies for the late post — I was stuck in a whirlwind of APs. But, I hope you enjoy the following piece!
Whoosh — you hit the ball and it sails across the net and onto the opponent’s side. Another point for your team to get to the tie, and your teammates — once again — clap you on the back in congratulations. Another win invigorates you more; you quickly stretch before getting into the stance — crouched knees, hands in front of you, poised for action. Your team serves, and then the other team spikes the ball back. This is it. Perfect this hit, and you’ll beat that tie. You’ll send your team to states, maybe even nationals.
You leap… you slam the ball with your palm… Then, crunch! You collapse. The ceiling rotates above you in your last moments of consciousness and you think, “Oh, f***.”
That foot placement changed everything. The minute the athlete slammed her feet onto the court with brute force, she was in a path of destruction. But what if she jumped and positioned her feet flat, landing gently? Then she would have seen the ball hit the opponents’ floor, before her team screamed in glee and they all went to states. Maybe that’s what could have happened. Instead, because of that one moment of foot placement, she’s in the hospital inundated with pain meds and her volleyball career is in shreds.
At that point in time, she exhibited chaos — “sensitivity to the tiniest changes in initial conditions or seemingly random and unpredictable behavior that follows precise rules” (1). The smallest change in her foot placement changed the outcome of everything. Now, this system as a whole displays characteristics similar to those of a chaotic system.
A chaotic system is aperiodic and nonlinear. Aperiodic behavior means that the system variables never repeat any values regularly. In the case of the star athlete and her unfortunate foot, the stage of the system never repeats. The athlete is never in the exact same position as she was, nor will the applied force on her foot be the exact same. In fact, nothing will be exactly the same. This system is also nonlinear, where there is no proportional relationship between occurrences. For example, if she applied a force of one newton on her foot to jump, we would jump one inch (hypothetical, of course) and come back down. Two newtons? Two inches. Come back down More than five newtons, she will jump high and come back down five inches, but also break her foot. Breaking her foot is not a foreseen mathematical consequence, so this constitutes this relationship as nonlinear.
Now, after analyzing this example, can we argue that all of our lives are chaotic systems? I, innocuously traipsing down the sidewalk below a tiler on top a roof, am suddenly struck by a tile. The outcome of your walk would have been completely different had I had a different starting point: walking earlier/later or starting at a different location. This situation is common to everyday life, but can be completely different if there were different starting points.
Physically, a double rod pendulum is typically used to illustrate chaos.
The video above highlights that the smallest changes in initial conditions — a smaller angle at the edges, a bigger height from which the pendulum is released — leads to vastly different paths and locations. This applies to our lives, too. On the outside, we all appear to have similar lives — same classes, same school. But we all started at different points the minute we were born — different socioeconomic statuses, different families — and coupled with all the tiny decisions we make, have such distinct futures. So, looking at that pendulum, we started life at a different angle from our best friends, or had been let go at a lower height than others, making for a completely different path. Throughout, we are all the jumble of pendulums, bouncing around each other in a whirl of life changing events and decisions.
We always picture ourselves as chaos. My friends always groan, “My gosh, my life is a mess.” And in a beautiful way, it is. By the very literal definition, a mess is chaos. But by the mathematical definition, what is chaos? We are. Our lives are chaotic systems, our minds flurrying to carry out decisions that have a multitude of unforeseen consequences, from the shirt we wear to the college we choose to attend. We are messes, a jumble of decisions waiting to be executed to unleash even more havoc on the world. Beautiful, isn’t it?
1. This definition of chaos was given by the Stanford Encyclopedia.
2. This scenario was illustrated by Poincaré to be an example of a chaos where trajectories can drastically converge from each other