Slow Lightning
Particles wandering at random until they stick, growing frost, and two ways of measuring it that disagree by a quarter.
What it is
One particle sits in the middle. Others are released nearby and stagger about at random until they touch it, then freeze in place. Repeat five thousand times.
Nothing in the rules mentions branches. You get branches, because the tips stick out and catch wanderers before they can reach the sheltered interior, so tips grow faster than hollows and the advantage compounds. Frost on a window, copper in an electroplating bath and a lightning channel are all this.
How it works
Diffusion-limited aggregation, with one practical detail. Walkers start on a circle just outside the current cluster rather than at the edge of the grid, and get abandoned if they stray too far. Starting them far away is more faithful and spends almost all its time watching particles wander through empty space.
Colour is growth order, warm at the ancient core and pale at the newest tips.
What surprised me
I set out to measure the fractal dimension, which for two-dimensional DLA is famously about 1.71. My box counting said 1.44. I assumed my cluster was too small, grew it seven times bigger, and got 1.45.
The problem was the method, not the cluster.
Box counting needs box sizes well inside the scaling range, comfortably larger than one particle and comfortably smaller than the whole cluster. With a few thousand particles there’s barely a decade of room between those two crossovers, so the fit is dominated by them.
Measured on the same cluster, box counting reports anywhere from 1.29 with small boxes to 1.55 with large ones. The answer depends on a parameter that has nothing to do with the cluster.
Mass-radius scaling has no such knob. The number of particles grows as the radius to the power D, so D is just the slope of log N against log R over the cluster’s own history, and that lands at 1.67, 1.71 and 1.75 for clusters of 1,500, 4,000 and 9,700 particles. It converges, and it converges on the right answer.
Both are on screen while it grows, disagreeing in real time. The tests calibrate the estimator against shapes whose dimension is known exactly, so a filling disc has to read 2 and a growing line has to read 1. That’s the only reason I trust it over the one I tried first.
What I would do next
Bias the walkers with a drift and watch the branches comb over in one direction, which is roughly what an electric field does.