REACTION-DIFFUSION Turing's last paper, made of spots and stripes
Two chemicals sit on a grid: one diffuses fast, one slow, and they react. From an almost-uniform start, spots, stripes, and coral self-organize — no designer, just local rules. Alan Turing predicted this “diffusion-driven instability” in 1952, in his last paper on biology (he published on the Riemann zeta and computability after it, before his death in 1954); the Gray-Scott equations are one vivid instance.
THE TECHNIQUE diffuse, react, repeat, per cell
Each cell holds two concentrations u and v. Every tick, each is nudged toward its neighbours’ average (diffusion, u faster than v) and then the reaction runs: v eats u autocatalytically (u·v²), u is fed in at rate F, v is removed at rate F+k. Tiny changes in F and k flip the whole grid between spots, worms, and stripes. Watch it grow. live demo
HISTORY & CREDIT the famous pictures are not Gray & Scott's
“Gray-Scott patterns” — Gray & Scott studied a spatial grid well-mixed reactor with no space at all; the spots-and-worms images are Pearson’s. cited
1937 / 1940 · Kolmogorov-Petrovsky-Piskunov and Rashevsky (cell-polarity diffusion, 1940) write reaction-diffusion PDEs — Turing was not first to the equation. 1952 · Alan Turing, “The Chemical Basis of Morphogenesis” — his last paper in biology (he published on the Riemann zeta in 1953 and computability in 1954, then died) — predicts diffusion-driven instability: diffusion, normally smoothing, can create pattern. That counter-intuitive result is his genuine novelty. 1968 / 1983–85 · Sel’kov’s glycolysis model feeds Gray & Scott’s cubic-autocatalysis kinetics — studied only well-mixed, no grid. 1990 / 1993 · first real chemical Turing pattern (Castets, De Kepper et al.); John Pearson runs Gray-Scott on a grid and produces the now-iconic zoo of patterns.
A 38-year gap sat between Turing’s prediction and the first pattern grown in a dish. Turing 1952 / Pearson 1993
RECOMMEND FOR I-13 a 2-D grid of f64 updates
The per-cell update is add, multiply, and the reaction term u·v² — pure f64, which runs on the compiler:
$ i13 run gs.i13 # one cell: u=0.5, v=0.25
R = u*v*v = 0.03125 # the autocatalytic reaction term; the whole tick is add/multiply
Recommend: the field is a 2-D grid — PS-004 once more — and every cell’s update is f64 arithmetic (the reaction term R = u·v² grounds at 0.03125 above), exactly like Life (020) or the Mandelbrot escape (019). It runs today flattened into a 1-D array with y*W + x indexing, a second buffer for the next tick. Note: emergence for free — no wall, just a grid and a rule; the pattern is not stored anywhere, it is computed into being, which is the corpus’s whole thesis in a dish.