The Geometry of Chaos

The notion of interdependence is universal in the arts. We do not try to understand a Beethoven symphony or a Shakespeare play simply by analyzing a tiny fragment of the symphony or play. Yet this is how scientists try to understand the world. We take a small sample of the world and show it is made of atoms. We show these atoms, in turn, are made of nuclei and electrons, and the nuclei are made of protons and neurons, which are made from quarks. We think this reductionist process stops there—but we can’t be sure.

We can get some sense of what matter is made of, but we have no idea why matter exists in the first place. Indeed, why the universe exists.

It is this seeming division between the holistic and the reductionist that characterizes for many the demarcation between arts and the sciences and is perhaps the reason why many people are deterred from a career in the latter. A molecular biologist can now probe down and study individual genes in a piece of DNA. This may help us find cures for certain illnesses. But it doesn’t seem to help in understanding what life actually is and why life exists.

However, this characterization in terms of holism and reductionism is a false one. Ideas from what is known as chaos theory are helping to break down the demarcation lines. Chaos theory reveals how one cannot always break a system into isolated parts and illustrates how understanding the interdependence between different parts is crucial.

Chaotic systems are often characterized by what is called the Butterfly Effect: the idea that the flap of a butterfly’s wings in Brazil could catalyze a tornado forming in Texas a week later—or not.

The Butterfly Effect can also be used to describe historical events. My favorite example is that if the doctors presiding over the birth of Princess Vicky’s baby boy had taken a little more care, neither World War I nor World War II would have happened. Vicky was Queen Victoria’s first child, and she married a German prince, Fritz. The baby was born in breech position, and during the delivery his left brachia plexus was torn, leading to a permanently paralyzed and disfigured arm. The anesthetic given to Vicky, who was in great pain, led to oxygen starvation in the baby’s brain, likely causing emotional and learning difficulties later in his life. The royal family kept the child out of public view, especially when his family came to England for the summer social season. As a result, the child grew into a young man with a visceral hatred of all things British. In time, Fritz became Kaiser of Germany, only to die shortly from cancer. The young man with the paralyzed arm and hatred of all things British became Kaiser Wilhelm II, who significantly contributed to the outbreak of World War I. The rest, as they say, is history.

And yet the grip of the Butterfly Effect extends further still. Imagine a game of billiards, with one ball colliding with a second ball, and then a third and so on. How many collisions would have to occur before the direction of travel of a billiard ball was completely changed by the gravitational effect of someone waving their arms a mile from the billiard table? Gravity is a very weak force, so you might expect it to be an astronomical number of collisions. But it’s not. It’s an extraordinarily small number—about fifteen collisions.

Let’s consider something where the answer is surely astronomical. How many collisions would have to occur before the motion of the billiard ball is affected by the gravitational influence of a single electron at the edge of the observable universe? The answer, as estimated by theoretical physicist Michael Berry, is a little more than fifty collisions.

And it’s not just billiard balls—it’s also collisions between atoms and molecules. Take the brain, for example. When we make a seemingly whimsical, on-the-spot decision, perhaps it’s because of the behavior of a single electron in a distant galaxy many years ago.

Through the Butterfly Effect we see the profound interconnectedness between all parts of the universe.

Chaos theory  also describes the phenomenon of exponential growth. Many of us became familiar with this during the COVID pandemic. If a person with COVID infects two people, and each of them infect two more people, and this continues, the number of people infected can grow to very large numbers. Indeed, after fifty doublings the number is a million billion (one followed by fifteen zeros).

The Butterfly Effect is a characteristic of chaos. But there is more to chaos than just the Butterfly Effect. Fractal geometry underpins chaotic systems. One of the most famous of these fractal geometries is called the Lorenz attractor. It is named after the MIT meteorologist Ed Lorenz who discovered this geometry while studying three relatively simple equations that describe a simplified type of weather system. Isaac Newton would have understood Lorenz’s equations, but he would have had no conception  of the type of geometry that is linked to them.

What does it mean to say that the geometry of chaos is fractal? As a simple illustrative example, imagine a length of rope. If we zoom into the rope, we see it is made of strands, and the strands are made of smaller yarns, and the yarns are made of twisted fibers. In principle, this could continue indefinitely, as it is characteristic of fractal geometry: it repeats as you zoom into smaller and smaller scales. The famous Mandelbrot set is an example of a fractal geometry.

The fractal geometry of chaos is a manifestation of the interdependence of the different parts of a chaotic system. It is a manifestation of the fact that these individual parts are not independent of each other—they act together in concert. In a chaotic system we cannot isolate individual parts. The whole is much more than the sum of the parts. 

The reductionist paradigm has been extraordinarily successful over the last few hundred years. It has given us the industrial revolution, medical advances, the electronics industry, and space travel—to name just a few. Yet for all that, contemporary physics is facing some profound challenges. Despite being one hundred years since Werner Heisenberg and Erwin Schrödinger formulated the quantum theory of atoms and elementary particles, physicists have been unable to describe the phenomenon of gravity in terms of quantum theory. Yet we know that at the time of the Big Bang, the whole universe was, in some sense, squeezed into the size of an atom. Moreover, despite being able to describe ordinary matter in terms of quarks and electrons, we now know that over ninety-five percent of all mass/energy in the universe —the two are equivalent according to Einstein—is of a form about which we have no understanding at all.

My own view is that the reductionist paradigm may have run its course, and we need to encode the interdependence of different parts of the universe, as described by the fractal geometry of chaos, into the primitive laws of physics. Everything is a microcosm of the universe itself, an assemblage of interconnected pieces. If you don’t understand interdependence, you will not understand the universe. The worlds of the arts and the sciences may finally be converging.