What is the Matrix?
a movie and hydrodynamics
“This movie will change my life,” I thought, and with goosebump on my arms I left the cinema. Deep in thoughts, I began walking home in the direction I believed was right. The city of Lund was new to me; I was a university freshman at the beginning of the first semester.
I halted in front of the white university building, which was accentuated by the twilight. The building excited and frightened me; it stood there as a token of knowledge, yet it was bewildering, as the night was. That evening I had watched The Matrix, and it did, in a way, change my life.
When it premiered, The Matrix was celebrated for its special effects, which undeniably boosts it. But the special effects aren’t the reason for its later cult. No, the entrancement is in the theme, combining the ancient philosophical question, “Is the world different from what it seems?”, with the awe over internet in the era of 56k dial-up modems.
The game-changing scene in The Matrix is when Neo, the computer hacker protagonist, first meets Trinity, at an underground nightclub. As the music turns edgy, the alluring storyline becomes irresistible when Trinity whispers to Neo:
“I know why you’re here Neo. I know why you hardly sleep, why you live alone, and why night after night you sit at your computer. I know, because I was once looking for the same thing. […] It’s the question that drives us, Neo. You know the question, just as I did.”
“What is the Matrix?”
From Neo’s reply, I vaguely recalled the number tables called matrices we’d learned about in high-school mathematics, but I didn’t see a connection to the movie. Yet my curiosity was awoken.
~
In high-school, I had mixed feelings about mathematics. It was unsatisfactory to memorize algorithms and apply them to variations of poorly motivated equations. But programming was fun. I’d been hooked since my father had bought a Macintosh Plus in the late ’80s and had taught me to write simple programs in ZBasic.
During high-school lessons we competed to write demo-programs on our TI-82 calculators. The king of the hill was a classmate nicknamed “Hacker-Björn”. One morning he kept energetically tapping his calculator while refusing to reveal anything. As he finally and triumphantly announced his demo, we gathered over his calculator like a flock of hungry gulls. He unveiled a smooth animation resembling the twisting of DNA. It was beautiful, and the way he had programmed it was remarkable. The rest of us had relied on the TI-82’s lagging graphics engine, but Hacker-Björn used simple ASCII output to produce fast, animated graphics. I was so impressed and jealous. It was an extraordinary lesson of thinking outside the box.
After high-school, I moved to the south of Sweden to study engineering at Lund University. During the first year, soon after I had watched The Matrix, I began a course in linear algebra. Our teacher was Magnus Fontes and he was brilliant. With wits, he filled our minds with vectors and linear transformations, represented by matrices.
Once he gave a motivation lecture, where he illustrated how his own research (in harmonic analysis) applies to the brewing of beer and to synthesizers in electronic music. Suddenly, in-between sentences, he said:
“It’s a well-hidden secret among math professors that most problems can’t be solved.”
These words grew in me over the years. They evolved into a mantra, applicable to mathematics, science, and life in general: “The more I learn, the more I realize I don’t know.” There’s beauty in this awareness; it’s a whisper of unlimited possibilities.
The course in Linear Algebra changed my view of mathematics. From there on, it wasn’t about memorizing algorithms. Linear Algebra is a self-contained theory. It’s abstract, yet concrete because it’s geometric and therefore evokes mental pictures. Soon after the course ended, I decided to quit engineering for full-time studies in mathematics.
~
Twenty years later, Milo Viviani and I began to learn how geometry manifests itself in the equations of hydrodynamics. These equations were formulated by Leonhard Euler in 1757 but are still far from understood. Somehow, the more mathematicians learn about them, the more peculiar their solutions appear to be. Yet, there are ways to approach them.
In 1854, Bernard Riemann developed a framework for describing generalized, higher-dimensional surfaces. Today we call this framework Riemannian geometry, and it’s a cornerstone of modern mathematics. Vladimir Arnold realized in 1966 that Riemannian geometry also describes Euler’s equations of hydrodynamics. The discovery is important because it maps mental pictures of geometry to intuition about hydrodynamics. A surface has curvature. What’s the curvature of hydrodynamics? What does it mean that hydrodynamics has curvature?
Consider a mountaineer walking along geodesics. That is, she walks along a path such that whenever she pauses to look back at a point she passed before, she couldn’t have made it from that point to where she is via a shorter path. Imagine now two such mountaineers walking next to each other along a mountain ridge shaped like a saddle. The ridge has negative curvature, which implies that the mountaineers eventually diverge from each other, ending up on different sides of the mountain. On the other hand, if they were walking in a crater, or near the top of a hill, where curvature is positive, they’d converge towards each other and eventually cross paths. Thus, geodesic motion is stable (i.e., converging) where curvature is positive and unstable (i.e., diverging) where it’s negative.
Now take a leap of thought: just as the curvature of a two-dimensional surface signify stability, the curvature of hydrodynamics reveals stability of fluid motion. This line of reasoning exemplifies the power of mathematics.
A mathematical equation is blind to its applications. It’s a blueprint of a perfect machine, potentially useful but with no predestined purpose.
Arnold’s discovery showed that two previously distinct areas of mathematics were connected. (i) Euler derived the hydrodynamic equations, (ii) Riemann generalized geometry, (iii) Arnold saw the former in the latter. And these events happened one century apart. Fascinating!
We cannot write down a general formula that solves Euler’s equations, so we need other ways to latch on to their solutions. One possibility is to use “approximate” equations which we can solve via computer algorithms. But Euler’s equations are evasive, as though designed to keep their solutions secret. There are computer algorithms that approximate solutions on short time intervals, but they fail on longer time intervals. And “longer” is typically still quite short. A different strategy is needed.
In 1890, in his work on the stability of the solar system, Henri Poincaré laid down a new strategy to understand dynamical systems: stop focusing on individual solutions and instead study the qualitative behavior of generic solutions. And here Arnold’s geometry enters, because it unravels qualitative features and thereby enables Poincaré’s approach for the study of Euler’s equations. But for computer generated approximations there is still a problem: standard computer algorithms fail to preserve the geometry, so on long time intervals the qualitative properties disintegrate. Viviani and I asked if there are computer algorithms that approximate solutions of Euler’s equations in such a way that Arnold’s geometric description remains intact. I eventually remembered from my post-doc time in New Zealand that Robert McLachlan had showed me a curious way to approximate the 2-D Euler equations via something called the sine-bracket. I returned to it and learned that it’s a method developed by Vladimir Zeitlin in 1991 which indeed preserves the geometric structure, and which at heart uses quantization theory to replace the continuous vector field in Euler’s equations with a matrix. My excitement went off the charts.
We plowed the literature for follow-up studies on Zeitlin’s model, and we found – not so much. While the method preserves Arnold’s geometry, it approximates individual solutions worse than standard algorithms and therefore didn’t catch on in the computational mathematics community. But we weren’t bothered, since individual solutions can’t be approximated for long times anyway, regardless of the choice of algorithm. Instead, we set out to further study Zeitlin’s model, which clearly was underexplored (and still is).
The beauty of Zeitlin’s model, or matrix hydrodynamics, is that it gives a mathematical dictionary between matrix theory and hydrodynamics, which in turn enables new theory about matrices for addressing old questions about hydrodynamics. Furthermore, computers are superb at matrix calculations, so we can explore theoretical ideas via computer experiments. At the same time, the map from a fluid velocity field to a matrix is hard to grasp. What do the matrix elements really mean? Indeed, these days I often find myself late at night in front of the computer, half asleep, while thinking: “What is the matrix?”