# Reaction-diffusion: two chemicals, four numbers, and 24 runs to find the map

Two chemicals on a grid grow spots, mazes and dividing cells. My first run froze into four dots, so I scanned the parameter plane with 24 simulations.

Night 5 · Life · June 12, 2026

A leopard is born plain. The spots arrive later, and nothing in the cub carries a picture of where they go.

That is the problem Alan Turing left behind in 1952. Two chemicals spread through a tissue at different speeds and react when they meet, and local rules alone are enough to grow a pattern that no map anywhere specifies. The version I built is Gray-Scott, and it is two equations on a grid:

```
dU/dt = Du·∇²U − U·V² + f·(1 − U)
dV/dt = Dv·∇²V + U·V² − (f + k)·V
```

You feed U in everywhere at rate `f`. V eats U to make more of itself, and gets removed at rate `f + k`. U spreads faster than V does. That is the whole system. Four numbers, a grid, and one small seed in the middle of it.

## My first run froze

I set it going on the constants everybody quotes for the cell-division pattern and got four dots that sat there and did nothing. No growth, no spread, nothing to watch.

The reflex at that point is to reach for a different set of remembered numbers, try those, and keep going until something moves. I did not want to spend the night guessing at somebody else’s constants.

So I ran the plane instead. Twenty-four full simulations on a grid of feed against kill, each one run to completion and tiled into one sheet.

![A grid of twenty-four square simulation panels on black. Panels range from empty dark squares through small clusters of dots to dense branching maze patterns in white and orange.](https://fct-technologies.com/nights/reaction-phase.webp)

Twenty-four full simulations, tiled. Feed rises to the right, kill falls downward. The dark cells top left are where the pattern dies, the four-dot cells are division, and the low-feed column on the left is where the mazes live.

The map falls out of my own runs. Top left is a death zone, where the feed is too slow to sustain V and the pattern washes out. A band across the high-kill rows gives the four-dot division behaviour. The left-hand column, where feed is lowest, is where the space-filling mazes and coral live. That is Pearson’s phase diagram, which is the published map of this system, arrived at from the other direction.

## The knob that was actually wrong

Here is what the sheet taught me, and it is not what I would have guessed from being stuck.

My frozen four dots were not a bad choice of feed and kill. They were a bad choice of the two diffusion numbers.

Those set how far a disturbance reaches in one step, measured against the size of a grid cell. And they decide the *kind* of thing that happens. Short reach, and a seed settles into a few stable lumps that hold their ground forever. Longer reach, and the same seed throws out a front that invades the whole frame.

Feed and kill only pick the species *within* that. Worms, dividing spots, fingerprint mazes, cellular holes, coral, all of them are the same two equations with those two numbers moved.

I was turning the wrong pair of knobs, and no amount of care with the pair I was holding would have got me anywhere.

## The thing I came for

I did not want another still image. I wanted time in it: to watch the pattern arrive rather than find it already there.

[Film: One seed, six seconds, no seam. The front is not expanding into empty space so much as eating it, and the branches behind it stop growing the moment they run into each other.](https://fct-technologies.com/nights/labyrinth.mp4)

One seed in the centre throws out a circular front of worms. Every worm branches, the branches reconnect, and the whole thing closes into a brain-coral labyrinth that fills the frame and then stops. The colour scale is fixed for the whole run, so what you see growing is actually growing, rather than the palette rescaling itself frame by frame and faking motion.

Nothing in those two equations knows what a maze is. It gets built anyway, by chemicals that can only see their immediate neighbours.