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Systems
System · Reaction-Diffusion

Same maths. Different species.

Two diffusing substances — one activator, one inhibitor — give rise to every kind of pattern we know from animal coats. Tweak two numbers and the same algorithm draws a cheetah, a tiger, a zebra, a giraffe. Turing wrote this down in 1952; biology has been catching up ever since.

Presets
Custom — drag f and k freely
Equations

Two lines do all of this.

The Gray-Scott special case of the Turing system. Each pixel updates itself from its eight neighbours via a 9-tap Laplacian, eight to ten times per visible frame.

∂u/∂t = Du ∇²u − u·v² + f·(1 − u)
∂v/∂t = Dv ∇²v + u·v² − (f + k)·v
u · Activator concentration
v · Inhibitor concentration
Du, Dv · Diffusion rates (Du = 1.0, Dv = 0.5)
f · Feed rate — replenishes u
k · Kill rate — removes v
∇² · Laplace operator, second spatial derivative
Story

Four moments in seventy years.

  1. 1952 — Turing writes it down.

    Alan Turing publishes 'The Chemical Basis of Morphogenesis' in Philosophical Transactions B. He proposes that biological patterns emerge from chemicals that interact and diffuse at different rates. The maths is sound; the biology, at the time, has no candidate molecules.

  2. 1993 — Pearson maps the territory.

    John Pearson explores the Gray-Scott parameter space numerically and finds fourteen distinct pattern regions — labelled α through ρ. The maps in his Science paper become the reference work for everyone who comes after.

  3. 1995 — Kondo finds Turing in fish.

    Shigeru Kondo and Rihito Asai photograph the same emperor angelfish over months. Its stripes shift and split exactly as a reaction-diffusion model predicts. The first hard evidence that real animal patterns follow Turing dynamics in vivo.

  4. 2017 — Lizard skin as a cellular automaton.

    Liana Manukyan and the Milinkovitch lab show that ocellated-lizard scales switch colours according to a discrete cellular automaton, which is itself a discretisation of the Turing-type partial differential equation. The bridge between Turing's continuous maths and discrete biology becomes explicit.

Sources