It was a small paperback. I don’t remember his exact words when he handed it to me. Something about noticing I was enthusiastic about math, maybe, or that I seemed like the kind of kid who’d get into this. He was right.

I got into it.

What Lauwerier gave me wasn’t just a book about Cantor dust and Koch curves and the Mandelbrot set. He gave me a way of seeing. Once you learn that a coastline has no definite length, that the answer depends entirely on the size of your ruler. Once that clicks, something shifts. You start noticing. The branching of a river delta mirrors the branching of your lungs. Turbulence in a creek has the same structure as turbulence in Jupiter’s atmosphere. The world becomes recursive. Self-similar. Strangely beautiful.

I was sixteen and I suddenly wondered why no one had told me this before.

Georg Cantor went mad, or so the story goes. He spent the late 1800s wrestling with infinity, not the lazy infinity of “it goes on forever” but the precise, terrible infinity of uncountable sets, of showing that some infinities are larger than others. His colleagues called his work a disease. He died in a sanatorium.

Benoit Mandelbrot, a century later, took the shapes that classical mathematics had dismissed as “pathological”, curves that were continuous but nowhere differentiable, dust-like sets with fractional dimensions, and said: these aren’t monsters. These are the shapes of the actual world. Mountains are not cones. Clouds are not spheres. Rivers are not lines. Lightning is not straight.

I loved this. I still love this. The assertion that the real world is not smooth.

In my twenties I did something that, to some people, may be slightly unhinged. I wanted to understand where chaos shows up in ordinary life. Not as metaphor but as physics, so I started reading university textbooks on meteorology, oceanography, and fluid dynamics. I had no university obligation to do so. I just needed to know how a weather front forms. Maybe being scared of lightning played a role here too. And I wanted to know how ocean currents become unstable. What actually happens when laminar flow breaks down into turbulence and the equations stop being politely solvable.

This is where we meet Edward Lorenz.

Lorenz was a meteorologist at MIT who, in 1963, was running a simple numerical weather model — twelve equations, nothing extravagant — and discovered that rounding a variable from 0.506127 to 0.506 produced a completely different weather pattern. Not a slightly different pattern. A completely different one.

He had stumbled onto sensitive dependence on initial conditions. The butterfly effect, as it would later be called, though Lorenz himself was more careful than the popular framing suggests. He wasn’t saying a butterfly causes a tornado. He was saying that in a deterministic system, prediction can become practically impossible because you can never measure initial conditions with infinite precision, and in certain systems, that matters enormously. This effect was noticed earlier, by other scientists. Jacques Hadamard for example, described the first mathematical system proven to be chaotic in 1898. It’s the reason I chose Hadamard as an artist name for my electro music.

But back to Lorentz. His simplified model, the one that generated the famous attractor you can explore on this page, is three coupled differential equations:

$$\frac{dx}{dt} = \sigma(y – x)$$

$$\frac{dy}{dt} = x(\rho – z) – y$$

$$\frac{dz}{dt} = xy – \beta z$$

Three variables. Three parameters ($\sigma = 10$, $\rho = 28$, $\beta = 8/3$ in the classical case). That’s it. And from this emerges the butterfly-shaped attractor — a structure in phase space that the system orbits forever without repeating. Deterministic but unpredictable. Bounded but never periodic. Zoom in on the attractor. Zoom in further. It never resolves into a clean loop. It can’t. That’s the point.

I should say what I actually believe, since this is supposed to be a blog post and not a textbook. I believe we live inside a transition. Not metaphorically, but cosmologically. Shortly after the Big Bang, the universe was in a state of extraordinarily low entropy. A hot, dense, nearly uniform plasma, which sounds chaotic but is, in the thermodynamic sense, about as ordered as things get. The far future of the universe, by contrast, will be a state of maximum entropy: a diffuse, cold, featureless void where nothing happens because there are no gradients left to drive anything.

We live in between.

The second law of thermodynamics says entropy increases. On average, over time, in closed systems. Fine. But it doesn’t say the path from low to high entropy is smooth or boring. The path is us. Stars, galaxies, weather systems, oceans, bacteria, civilizations, consciousness. All of it is the universe finding elaborate, temporary ways to dissipate energy as it moves from order toward disorder.

Ilya Prigogine won a Nobel Prize for showing that systems far from thermodynamic equilibrium can spontaneously self-organize. That you get more structure, not less, when you push energy through a system. A Bénard cell is the classic example: heat a thin layer of fluid from below, and above a critical temperature gradient, the random motion of molecules suddenly organizes into beautiful hexagonal convection cells. Order from chaos. But only because energy is flowing through the system. Stop the flow, and the cells disappear.

Life is a Bénard cell. A very complicated one, but the principle holds. We are dissipative structures. We maintain our organization by continuously processing energy, eating, breathing, metabolizing, and increasing the entropy of our environment. We are not exceptions to the second law. We are expressions of it.

This idea, once you take it seriously, rearranges everything.

The Lorenz system has a parameter, $\rho$, that controls the behavior of the flow. When $\rho < 1$, the origin is a stable fixed point — everything decays to stillness. When $1 < \rho < 24.74$, you get two stable fixed points — the system settles into one of two steady convection patterns. But when $\rho$ crosses approximately 24.74, those fixed points lose stability and the trajectory is thrown onto the strange attractor. The system never settles. It wanders between the two lobes forever, switching unpredictably.

There is something in this that I find genuinely moving. The idea that a tiny change in a parameter, a gradient, a forcing, can push a system from predictable behavior into a regime where prediction fails. And that this isn’t destruction. It’s the emergence of something richer. Phase transitions. The edge of chaos. That narrow band where a system is neither frozen nor random but something in between, complex, adaptive, alive.

Stuart Kauffman called it “the adjacent possible.” The space of what could happen next, given what exists now. In a frozen regime, the adjacent possible is small. In a chaotic regime, it’s vast but incoherent. At the edge, it’s vast and navigable. That’s where interesting things happen.

I think about this in daily life.

Every project I’ve worked on has this quality. You start with some structure. A plan, a framework, a set of constraints. And then you push energy through it. People, ideas, conversations, deadlines. And if the constraints are too tight, nothing happens, the system is frozen. If they’re too loose, everything dissipates, nobody converges on anything. But sometimes you get the parameters right and something self-organizes. An idea crystallizes that no one person had. A group finds a rhythm. Things connect in ways you didn’t design.

You can’t force this. You can only create the conditions. Set the gradient. Adjust $\rho$. And then you let go of the idea that you’re in control, because you’re not. You’re a participant in a dynamical system, and the attractor, if there is one, is not something you chose. It’s something that emerges.

There’s a poem I keep coming back to. Arseny Tarkovsky — And this I dreamt, and this I dream. It has these lines about waves, each one carrying a star, a person, a bird, dreams, reality, death — wave after wave. And then: Life is a wonder of wonders, and to wonder I dedicate myself, on my knees, like an orphan, alone – among mirrors – fenced in by reflections.

I think about that poem and I think about the Lorenz attractor, and they feel like the same intuition arrived at from different directions. The poet and the meteorologist both reaching toward the same thing. Wave follows wave. The trajectory orbits and never repeats. Everything will be re-embodied, but never identically. You are inside it, fenced in by reflections, and it is beautiful, and it is not yours.

The universe is in transition. It started ordered and it will end empty. In between, there is all this. Turbulence and weather and life and thought and the particular way light moves through clouds on a Tuesday afternoon. None of it is permanent. All of it is the transition.

I never returned that book. I think my teacher knew I wouldn’t.