INTERACTIVE COMPANION · GROWTH BENCH · STATISTICAL PHYSICS

The Growth Bench

A live model accompanying “Why Frost, Coral and Copper Deposits All Grow the Same Shape”

One particle at a time. Each one is released far from the seed, wanders at random until it touches what is already there, and freezes. Nothing in the rule says branch. The branching is what the rule produces anyway, because a tip that sticks out a little catches the next walker a little more often, and the advantage compounds until the arms are long and the bays between them never see a visitor.

Two benches below. The first grows the thing and lets you change the rule. The second measures it while it grows, and shows the fitted exponent walking toward a value you can compare against the published one. Both run the same code the article ran, transcribed from the Python line for line, with the same seeded generator, so the defaults land on the article's numbers rather than near them.

Model 1. Growing it

The rule, in full. Pick a random angle, drop a walker on a circle of radius \(R_{\max} + g\) where \(g\) is the release gap you set. Step north, south, east or west with probability \(1/4\) each. If the site you tried to step into is already occupied, freeze where you stand with probability \(p\); otherwise stay put and try again. If you drift outside the release circle, you are put back on it at the angle a two dimensional random walk would actually have returned at, sampled from the Poisson kernel, so nothing is ever thrown away.

Sticking probability 1.00 Leave it at 1 first and watch the open, feathery cluster. Then drag it to 0.10 and restart. The walkers now refuse most first contacts, wander deeper, and fill the bays. By 0.02 the thing is nearly a blob with a rough edge. Nothing about the diffusion changed; only the willingness to commit. Release gap 5 lattice units This one is a test, not a knob. Escaped walkers are put back at the correct angle rather than deleted, so where you release them should not matter. It does change which random numbers get used, so at 60 you get a different cluster rather than the same one: D moves from 1.7736 to 1.8052, well inside the seed-to-seed spread below. A release radius that shifted the answer further than that would mean a bug. Cluster size 8,000 particles 8,000 is the size the article used for its sticking-probability sweep. The published production clusters were 50,000, which is more than a browser tab should be asked to do while you watch. Particles per frame 180 Drawing speed only. It changes nothing about the growth: the random numbers are consumed in the same order whatever the frame rate, so the same seed always gives the same cluster. Seed Five different clusters. The shapes look nothing like each other, and the exponents run 1.6878, 1.7556, 1.7625, 1.7736 and 1.7866. That spread, about a tenth, is the real precision of a single 8,000-particle measurement, and it is why the article averaged six clusters of 50,000.

particles: – Rg: – R_max: – four-fold anisotropy A₄: – lattice steps per particle: –

Model 2. Measuring it while it grows

A mass fractal obeys \(N \propto R_g^{D}\). Take logs and that is a straight line with slope \(1/D\). Every time the cluster passes one of seventy log-spaced particle counts, the bench records \(\ln N\) and \(\ln R_g\), refits the line by ordinary least squares over the window you choose, and redraws. Watch the slope readout. It is noisy at first and then it walks, slowly, in one direction.

Fit from N = 1,000 The article's pre-registered window starts at 1,000, giving D = 1.7736 here. Slide it up to 5,000 and only eight points survive and D falls to 1.7205. Slide it down to 200 and fifty-nine points give 1.7238. Three defensible windows, three answers, spread over five hundredths. The window is a choice, and it has to be made before anyone looks. Reference line 1.71 is the consensus from off-lattice simulations of up to fifty million particles. 5/3 is the closed-form mean-field answer, close enough to be quoted in the 1980s and wrong. 2 is what you get when the walkers stop screening each other, which is where low sticking probabilities head.

slope 1/D: – D: – points in fit: – distance from reference: –

–

Defaults: p = 1, gap = 5, 8,000 particles, fit from N = 1,000, seed 20250704
What the bench lands on: D = 1.7736 ± 0.0075, slope 1/D = 0.56381, 33 points in the fit
The cluster itself: Rg = 71.2294, R_max = 139.3592, A₄ = 0.2956, 346.5 lattice steps per particle
Across the five seeds: 1.6878 to 1.7866, a spread of 0.0988
Article, six clusters of 50,000: 1.6992 ± 0.0172 by the same method
Checked against: a second, separately written Python transcription of this page, which returns the same Rg, R_max, A₄, step count, slope and D to every digit printed above

Three things worth trying

Set the sticking probability to 0.05 and restart. The cluster is denser, its radius of gyration falls from 71.2 to 43.1 lattice units, and the fitted D climbs to 1.8891. That is not a new universality class. It is a crossover: below some length scale the growth is compact, above it the ordinary branching resumes, and at 8,000 particles a reluctant cluster is still mostly inside its own compact regime. Grow it bigger and the exponent comes back down. Meakin reported in 1983 that the radius-of-gyration exponent is insensitive to sticking coefficient anywhere between 0.1 and 1, and that lowering it simply makes denser clusters.

Put the sticking probability back to 1 and move the release gap from 5 to 60. Each walker now has further to travel, so the build is slower. What comes out is a different cluster rather than a shifted one, because the extra travel consumes different random numbers, and its D of 1.8052 sits inside the seed-to-seed spread. That is the reinjection working. A walker that wanders past the release circle is put back on it at an angle drawn from the Poisson kernel instead of being thrown away, so the arrival statistics do not care where you started it.

Then change only the seed, five times, and read the five values of D: 1.6878, 1.7556, 1.7625, 1.7736, 1.7866. They disagree in the second decimal place. That width is the real precision of a single cluster this size. Anyone quoting a fractal dimension from one cluster to three decimals is quoting noise, and we include ourselves in that.