Who dreams of a world where nothing touches anything?

I wanted to write about something beautiful.

My writings are filled with tough topics: politics, society, who we are, the ways we make and break our worlds. It’s not the subjects that wear me down. But one evening, I felt worn out. I needed a break from arguing.

A memory came back: from my time making art with code, a design born from a short two-line rule, run over and over. It looked like looking into a kaleidoscope, or the walls of the old mosque in Córdoba. a perfect mix, like a machine dreaming of balance. I’d never quite understood it. That night, I stayed with it, lost in its quiet wonder.

This is what I found.

The rule is easy, like something you’d scribble on a napkin. New speed equals old speed plus K times the sine of where you are. New position equals old position plus that new speed. K is a number you pick to set how strong the push is. The sine ties the push to your place. Start somewhere, use the rule, get a new place. Do it again. Simple enough for anyone.

Picture it: a stick spinning in empty space, getting a soft kick now and then. Between kicks, it turns freely. Each kick looks at the angle and adjusts. The place shapes the kick; the kick moves the place. A back-and-forth.

Boris Chirikov wrote this down in 1969, trying to understand why particles in accelerators go wild. He didn’t know what it would draw.

To see it, you do something Henri Poincaré figured out in the 1880s: stop trying to watch everything, just look at the right moments. Take a snapshot after each kick. Mark the position and speed as a dot. Plot it. Kick again, plot more. A strobe light on a continuous system, a beautiful idea in itself. From many starting points, across thousands of kicks, the dots build up.

Not scattered points, but a lovely weave: rings inside shapes, shapes inside patterns, like ancient floor designs. At period-4 resonances you get soft squares. Period-6, hexagons. Period-7, something special: orbits that turn perfectly but never repeat, a tidy mess, full of order yet one-of-a-kind.

I watched it for a long time.

The rule doesn’t say “make patterns” or “add balance.” You can’t see the picture in the words. No one could. It’s not a plan with the end hidden inside.

But it’s no accident either. Run it anywhere, any time, same design. Always there, set in stone.

Where does it come from?

From repeating, I think. The wonder comes from the pile-up. It is the rule feeding on its own results, again and again. Each kick carries the memory of the one before. It is a system with a history.

I thought I’d stop here, in this peaceful find.

But the material pulled me back in.

Thinking about harmonographs, those beautiful instruments where pendulums swing over a page, each one tracing its own arc, I realized they are almost the kicked rotator. Almost. They have the oscillation, the periodicity, the nested structure. What they don’t have are the kicks. Each pendulum swings on its own, untouched by the others. They share the page but never interfere. Perfect lines, regulated to the millimeter so that nothing ever touches anything else. Signals combined, but no histories. No friction, no contact, no feedback, no life.

And I recognized this. It’s the same dream that gets sold as organizational wisdom. Autonomous agents, minimal rules, smooth emergence. Watch birds flock: no leader, simple protocols, collective intelligence. But where is the intelligence in the swarm? It can be reduced to one rule. One variable: direction. A kind of soldiers walking to the beat of a drum. Don’t ask why. Maybe it’s important. Maybe it’s just beautiful for the orchestrator. Maybe just because I told you so. The pitch is beautiful. And we know the script by now. It starts as a project management tool in a startup. A few years later it’s a TED talk. Then a bestselling book. Then it’s taught in schools as a way of life. One movement decided by someone, followed by everyone else to perfection, without friction, without disturbance, without any of the mess that makes things real.

Who dreams of a world where nothing touches anything?

Coupled kicked rotors are something else entirely. Not separate things swinging in parallel, but pendulums that kick each other, each one’s motion shaped by the state of the others. Every swing carries the history of the whole system. No clean breaks. Recursive, self-referential, knotted by nature. And from that knot come richer things: orbits that survive chaos, stable islands in rough seas. The mathematics shows the most resilient structures are linked to the golden ratio, a number that resists being simplified into a clean fraction, making it maximally resistant to resonance. Robustness born from irreducibility.

The harmonograph says: no friction, and beauty comes out. The coupled kicked rotors say: the friction is what builds the beauty.

I’m not saying a math rule is a model for how people work together. Phase space is not a meeting room. But I don’t want to live in the harmonograph. I don’t want the world where every interaction is pre-regulated to avoid contact, where people orbit alongside each other without ever leaving a mark. We’re not separate pendulums. We’re more like the coupled rotors: every encounter is a perturbation that carries the full weight of what came before into the present. That’s not a problem to fix. It’s what makes depth possible. The structures that endure, the ones that survive chaos, are the ones most deeply entangled with the irreducible, the parts that resist being made simple.

It’s not a clean metaphor. But the mathematics gestures at something we struggle to say in plain words: that real resilience comes from entanglement, not from isolation. From kicks and their consequences, not from the absence of friction.

And then I remembered driving in Kinshasa. Where the only rule that existed was to keep as close as possible to the next car. Horns, heat, no lanes, everyone reading everyone else in real time. It was a beautiful time there, in its own way.

If you want to see what kicked rotators and harmonographs actually look like or play with them a bit visit: github.com/skacem/Algorithmic-Art