Two substances share a dish.
A is fed in from everywhere at rate
f.
B is drained away at rate
f+k. And
B is autocatalytic — it eats
A to make more of itself, two molecules at a time:
A + 2B → 3B
That's the whole story. Both diffuse, but
A spreads twice as fast as
B —
and that single asymmetry is the crack the universe pries open to make structure.
∂ₜA = Dₐ∇²A − AB² + f(1−A)
∂ₜB = D_b∇²B + AB² − (f+k)B
Turing proposed this shape of equation in 1952 as an answer to how a featureless ball of cells
becomes a leopard. He was right, and he never got to see it run.
The curve on the phase map is
k = √f⁄2 − f, the exact place where the two nontrivial
steady states of the reaction collide and annihilate — below it the dish has three possible
resting states, above it only one, the dead one where
B is gone.
What's strange, and worth staring at: the interesting patterns don't hide safely below that
line. They
straddle it. Mitosis, worms, solitons all sit just above, in territory
where the equations promise nothing can persist. They persist anyway — not by resting, but by
never stopping. A spot there can't settle, so it splits. That's the whole trick of the
self-replicating regime, and it only happens because the resting state was taken away.
Scale sets how big the pattern wants to be.
Feature size goes as √D, so the slider is just
D = scale² — but raising diffusion
shrinks the stable timestep, so the clock slows and the step count climbs to compensate.
Nothing about the chemistry changes; only the grain does.
Glow is not brightness — it's |∂B/∂t|, the rate
the reaction is running, kept as a short decaying trace. Settled structure goes dark; only
what is currently becoming something lights up. Pause, and the picture cools.
Keys
space pause
r reseed
c clear
a atlas
d drift
p palette
h hide UI
s save
1–
9 regimes
Mouse drag to seed
B ·
right-drag to erase · wheel to size the brush