Attractlet driven pattern · supplied

Dendritic Growth

Particles wander in at random and stick where they first touch, and what grows is a branching fractal that looks like frost on a window or the scar left by lightning.

There is one rule. Release a particle far away, let it wander at random until it touches what is already there, and freeze it. Nothing decides where the arms should go. And yet what grows is not a blob: it is a branching thing with a definite, measurable density, because a tip that sticks out is more likely to be touched first than a hollow that is screened behind it, and that small advantage compounds into everything you see. The cluster shapes what arrives at it. It does not make it. Every particle came from outside, and the structure that is so good at catching them supplies not one.

One Rule, Applied Twenty Thousand Times

colour is arrival order · the core came first

particles stuck,

outer radius,

radius of gyration,

fractal dimension, fitted live,

particles released,

view,

settling

The dimension in that readout is not looked up. The panel records its own radius of gyration at a ladder of sizes as it grows and fits R_g ∝ N^1/D to whatever it has so far, so the number you are watching converge is a measurement of the cluster in front of you. It is also one sample. Press Start again a few times and watch it land somewhere different each run; the panel beside this one gives the spread across twelve clusters, which is the only form in which this number means anything. Particles are off-lattice discs, because a square grid would print its own four-fold symmetry onto the arms and the dimension would be partly the grid’s.

The Rule, and the One Number It Produces

Witten and Sander, 1981

Diffusion-limited aggregation has no parameters. A seed sits at the origin; particles are released one at a time from a circle just outside the cluster, wander at random, and stick on contact. That is the entire specification, and out of it comes a shape with a fractal dimension: the mass grows not as the square of the radius, the way a solid disc would, but as some power between 1 and 2. For two-dimensional off-lattice DLA the accepted value is 1.71.

So the whole verification of this page is one number, and the honest way to report it is with an error bar. Twelve independent clusters, each fitted separately, on exactly the ladder of sizes the panel itself uses:

estimator	measured	vs accepted 1.71
R_g ∝ N^1/D	1.707 ± 0.011	0.3 standard errors low
N(r) ∝ r^D	1.659 ± 0.009	6.0 standard errors low

The first lands within a third of a standard error of the accepted value. Individual clusters in that set ran from 1.655 to 1.766, so the agreement belongs to the ensemble and to no single run. The second estimator does not agree, and that is worth saying plainly rather than quoting only the flattering one. The mass-radius estimator, which counts how much of the cluster lies within each radius, runs about 3% low on clusters this size and the gap is far outside its own scatter. This is a known finite-size effect: the radial mass profile of a DLA cluster approaches its asymptotic slope slowly from below. This panel uses the R_g estimator, and it says so here rather than letting you assume the number was the only one available.

A Measurement This Page Nearly Got Wrong

recorded because it is the more useful half of the result

The first fit run for this page gave D = 1.6946 from a single cluster, which is 0.9% from the accepted value and was very nearly reported that way. Then a check of an unrelated setting, the radius at which a far-wandering particle is given up on, produced D values of 1.82, 1.69, 1.66 and 1.73 for four choices that should barely have mattered.

That scatter, about ±0.07, is seven times the agreement that was about to be claimed. The single fit had not been a measurement of anything; it had been one draw. The numbers in the table above come from twelve clusters precisely because of that, and the standard errors are what make the two rows comparable at all.

Why This Is an Attractlet

the plainest case in the set

The physical cluster is an authored reading, which is why the card tag carries no citation, and it is the easiest reading here to make. Press Cut off the supply.

Nothing happens. The cluster does not collapse like the ferrofluid, does not spin down like the vortex street, does not even freeze mid-motion like the Chladni figure. It simply stops growing, and sits there being exactly what it already was. There was never anything to maintain. Every atom of this structure arrived from outside, and the structure’s entire contribution was to be in the way.

Which is not nothing, and the framework should not pretend otherwise. The branching is real and it is the cluster’s own doing: a protruding tip screens the hollow behind it, catches more of what is drifting past, protrudes further, and screens more. That is a genuine positive feedback and it is why the thing is a fractal instead of a ball. But feedback that shapes an incoming supply is not the same as producing one, and this is the clearest case in the set for seeing the difference, because here the structure does nothing at all except exist and intercept.

The Shape of the Basin

a basin in shape, not in state

Press Start again. The cluster that grows is completely different: different arms, different gaps, a different silhouette. Measured across five independent clusters at 12,000 particles, the fraction of the plane they occupy in common is only about 0.15.

And yet their radii of gyration agree to 2.2%. The arrival order decides every detail of what you are looking at and almost nothing about what can be measured from it. This is the sense in which DLA has a basin at all: not a state that recurs, but a set of statistics that every run lands on regardless of how it got there.

What This Model Is Not

the limits

A walker that strays too far is given up on and a fresh one released, rather than followed forever. About a tenth of them do. That is a real bias on the distribution of where particles arrive, and the sweep described above is what bounds it: changing the give-up radius by a factor of thirteen moves D around within the run-to-run noise and not outside it.

Particles are released from just outside the cluster, not from infinity. For a circle enclosing the whole cluster the two are equivalent by symmetry; for the tight launch circle used here, not exactly.

One particle at a time. That is what DLA means, and it is also what real frost and real electrodeposits do not do. Many things arrive at once in a real system, and the density field of the stuff arriving is depleted near the cluster in a way this model does not carry.

Sticking is certain. Touch once, stay forever. Lower the sticking probability and the cluster gets denser, which is a whole family of models this panel does not explore.

The fast walk is exact, and is worth explaining because it looks like a cheat. When nothing is within a distance L of a wandering particle, it cannot touch anything before it has travelled L. Random motion has no preferred direction, so the point where it first reaches distance L is uniformly distributed around that circle. Jumping straight there draws from exactly the same distribution as taking ten thousand small steps. The only thing discarded is the waiting.

Witten, T. A. & Sander, L. M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Physical Review Letters, 47(19), 1400–1403.
Diffusion-limited aggregation, Wikipedia. Source of the accepted dimension of about 1.71 for particles unrestricted by a lattice, and of the statement that simulating on a lattice shifts it.
Meakin, P. (1998). Fractals, Scaling and Growth Far from Equilibrium. Cambridge University Press.

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