Accrete
One seed, and then particles that wander in from far away, one at a time, sticking wherever they first touch what is already there. Brightness is recency: an arrival is lit at the moment it lands and cools from there, so the light is always the last few hundred particles and everything else is the record of arrivals that already happened. Press anywhere to change the weather the walkers come in on. It moves where they set out from, not where they land.
Mass against size across four cluster sizes and ten seeds, which is the estimate that does not run into the edge of the cluster. The literature value for two-dimensional aggregation is 1.71. Over the largest interval alone it reads 1.737.
The mean radius of the last tenth of arrivals, as a fraction of the outermost reach. Tips grow and fjords starve, because a wandering particle almost never survives the trip into a bay it would have to pass a hundred branches to enter.
One connected component, every arrival touching an earlier particle at exactly the contact distance, and not one particle inserted into ground that was already occupied.
what this is
This is diffusion-limited aggregation, off lattice, from a single seed. A walker is released far out on a circle that encloses everything grown so far, and it moves by Brownian motion until either it drifts so far away that it is abandoned, or it comes within one particle width of the structure, at which point it stops and becomes part of it. Then another walker. Nothing arrives except by wandering, and nothing stays except by touching what came before.
What comes out is not a disc. A particle trying to reach a hollow between two branches has to survive a gauntlet of everything that already sticks out, so the tips take almost all of the arrivals and the bays are shadowed and stay empty. That screening is the whole shape. It is why the aggregate is mostly hole, and why its mass grows as roughly the 1.71 power of its radius rather than the square, which is what a solid thing would do.
The page shows this the only way it is honestly visible: as time. Brightness is recency. A particle is at its brightest the instant it lands and fades over the next several hundred arrivals until it is the same near black as the buried core. So the lit part of the picture is the growth zone, and the dark part is memory. When the aggregate is finished it is held for a moment, entirely cold, and then a new seed is dropped and it begins again.
the gates, and the two that did not hold
Five numbers were fixed in the project notes before any code existed, with tolerances and the direction an honest error would lean. Three held. Two did not, and the interesting part is what happened in between.
- 1 · fractal dimension in 1.66 to 1.76
Missed, low: 1.6335. The pre-registration declared that an honest error would lean low, and it did, but by more than the tolerance allowed. - 2 · the fit is a line, R squared at least 0.995 over 1.2 decades
Held: 0.99884 over 1.742 decades. - 3 · growth integrity, one component, no interior insertions
Held: 20,000 of 20,000 reached from the seed, adjacency 100 percent, zero insertions. - 4 · seed invariance, two runs within 0.04
Held: 0.0160. Whatever the dimension is, it is not a property of the random seed. - 5 · tip dominance above 0.8
Missed: 0.6930. Late arrivals do land near the outside, but not as near as the number I wrote down.
Gate five failed first, and it failed against a wrong engine. The first version took a fixed step near the structure and tested the straight line of that step for a collision. That is a real bug and a quiet one: a walker with only a fraction of a particle width of room would slide past a branch that a genuine Brownian path would have touched, so walkers reached into the bays that screening is supposed to protect, and the aggregate came out too filled in. Rewriting the walk so that every step is exactly the walker's clearance, which makes contact impossible to miss and tunneling impossible by construction, fixed the physics and made both marginal gates worse.
That is the useful part. The corrected engine measures 1.7056 when mass is fitted against size across four cluster sizes and ten seeds, which is the textbook value and is the estimator without a cutoff in it. The pre-registered gate instead fits one cluster's density profile out to eight tenths of its furthest particle, and that window runs straight into the crossover at the edge, where the aggregate thins out and the slope falls. So the gate reads about six hundredths low and drifts upward as the clusters get bigger, exactly as a finite-size artifact should. The same is true of gate five, whose denominator is the single outermost particle, a quantity with a long tail.
The band could be moved a hundredth and the piece would show five passes. It is not going to be. The gate was written down first so that it could say no, and what it is saying no to is a measurement I chose badly, not a physics I got wrong. Both of those are worth knowing, and only one of them is fixable after the fact.
The pointer bends the odds on where the next walkers set out from. It cannot place a particle, and it cannot make one stick. Aim it into a bay for a while and watch nothing much happen there: the screening does not care where the weather comes from.
statement
An aggregate is a history that kept its shape. Nothing here arrives except by wandering, and nothing stays except by touching what came before: twenty thousand landings, each lit for a breath, cooling into the structure it extends. Two of the five gates failed as written, and they failed the way the pre-registration said an honest error would fail: the ruler shortens before the object does. Measured across sizes the aggregate is textbook. The practice keeps the misses on the page, because the misses are where the learning was. Growth is arrival, arrival is contact, and what glows is simply what came last.