Swarm

One particle traces a curve. A second chases the first, a third chases the second, and so on down a chain of ninety. Nobody is told where the curve is except the leader; everyone else is only ever chasing the particle in front.

The picture is not where the particles are. It is where they have been — the canvas is never cleared, only washed over with a nearly transparent coat of the background each frame, so old strokes fade instead of vanishing. What you are looking at is a few thousand frames of accumulated wake.

The chain is what makes it fan

Each particle closes a fixed fraction of the gap to the one ahead of it every frame. That single rule, repeated down ninety links, is the whole mechanism.

The lag compounds. The leader is on the curve; the first follower is a little behind it; the ninetieth is a long way behind and still turning into the corner the leader left ages ago. When the leader whips through a tight part of the curve, the tail is thrown wide — and because every particle draws a segment from where it was to where it now is, that spread is written into the picture.

Turn slack up and the links go loose: each particle closes less of the gap per frame, the tail falls further behind, and the figure blurs into a broad smear. Turn it down and the chain tightens onto the curve until the followers are almost tracing it exactly.

The leader is chasing too. It never sits on the curve — it eases toward the moving point at the same tenth-of-the-way-per-frame as everyone else, so it cuts its own corners. The rose you see is a slightly smaller, softer version of the rose in the formula. That is the sketch’s look, and it is kept.

The curve underneath

The leader’s target is a rose:

Petals come from $k$. At $k = 3$ you get three petals, at $k = 2$ you get four — odd whole numbers give you $k$ petals, even ones give you $2k$, because the negative half of the cosine retraces petals the positive half already drew.

The default is $k = 1.5$, which is not a whole number at all, and that is where it gets interesting. Write $k$ as a fraction $p/q$ in lowest terms. The argument $k\theta$ grows by $2\pi p$ exactly when $\theta$ grows by $2\pi q$ — so the curve cannot repeat until $\theta$ has been round $q$ full turns. At $k = 3/2$ that is two laps before the figure closes. At $k = 9/4$ it is four.

Set $k$ to a quarter value and watch: the first lap draws an open, unfinished shape, and the pattern only resolves once the later laps have filled in the gaps the first one left.

Where it came from

A p5.js sketch of mine from 2025, seventy-six lines, with every number in it hard-coded — ninety followers, one fixed ease, one fixed fade, one fixed $k$. Those four constants are the four controls above. The sketch is the special case where they happen to be 90, 0.5, 3 and 1.5.