Reaction-Diffusion Pattern

By the end of this lab you'll be able to:

• Store a grid of numbers in a flat Python list and index it with i * w + j
• Update every cell from its neighbors using a Laplacian (diffusion) stencil
• Explain how spots and stripes on animal skins emerge from two simple local rules

Two invisible chemicals, U and V, spread across a 32×32 grid. U feeds V, V eats U, and both diffuse to their neighbors. From four small seed squares, 300 simulation steps grow organic blobs and worm-like stripes — the same Gray-Scott mathematics believed to paint leopard spots and zebra stripes.

Welcome to Reaction-Diffusion!

No shapes are drawn in this program — the pattern grows itself from chemistry rules, one neighbor at a time. This is real computational science! Be patient when you click Run: 300 steps of simulation take a little while.

How It Works

Each cell holds two numbers: u (food) and v (creature). Every step, each cell looks at its 8 neighbors to compute a Laplacian — a measure of "how different am I from my surroundings" — which makes both chemicals spread. Then the reaction happens: u * v * v converts food into creature, feed tops up the food, and kill removes creature. After all the steps, each cell is painted as one dot colored by how much v it contains.

Sample Code

w = 32
u = [1.0] * (w * w)
v = [0.0] * (w * w)

for step in range(300):
    u2 = u[:]
    v2 = v[:]
    for i in range(1, w - 1):
        for j in range(1, w - 1):
            p = i * w + j
            lapu = neighbor_average(u, p) - u[p]
            lapv = neighbor_average(v, p) - v[p]
            uvv = u[p] * v[p] * v[p]
            u2[p] = u[p] + du * lapu - uvv + feed * (1 - u[p])
            v2[p] = v[p] + dv * lapv + uvv - (feed + kill) * v[p]
    u = u2
    v = v2

What Do You Think Will Happen?

The simulation starts with just 4 small squares of chemical V. After 300 steps, will the picture still show 4 separate squares, or something else? Make your guess, then click Run (and give it a moment)!

Try It Now

▶ Run ↻ Reset

The four squares are long gone — they melted, spread, and merged into curling organic blobs. That's emergence: no line of code says "draw a worm shape", yet worms appear. Were you right?

How It Works

The Laplacian stencil weights the 4 edge neighbors by 0.2 and the 4 corner neighbors by 0.05 — a smoothed average of the surroundings minus the cell itself. Two details matter enormously: the update writes into copies (u2, v2) so every cell sees the same "before" picture, and V diffuses at half the speed of U (dv = 0.25 vs du = 0.5). That difference in diffusion speed is exactly what Alan Turing showed in 1952 could destabilize a uniform mixture into spots and stripes.

Explanation Table

Line	What it does
p = i * w + j	Finds cell (i, j) inside the flat list
lapu = (...) * 0.2 + (...) * 0.05 - uc	Diffusion: 8-neighbor weighted average minus self
uvv = uc * vc * vc	The reaction: U + 2V → 3V
u2[p] = uc + ...	Writes to a copy so all cells update together

Learning Check

Your Turn — Catch the Pattern Mid-Growth

This copy is set to stop after only steps = 60. Predict what you'll see: finished blobs, four fuzzy squares, or nothing at all? Run it, compare with the 300-step picture above, then try 150 and 600!

▶ Run ↻ Reset

At 60 steps the four seeds have only just started dissolving into rings — the pattern is still recognizably "four squares melting". Emergence takes time: the interesting structure appears somewhere between 150 and 300 steps.

Experiments

1. Change the recipe. Try kill = 0.065 (coral growth) or feed = 0.035, kill = 0.060 (solitons). You'll know it worked when the blob shapes change character completely.

2. One seed only. Delete three of the four seeds. You'll know it worked when a single blob grows rings outward like a tiny coral colony.

3. New color scheme. Replace the color list with reds and oranges for a lava-skin look. You'll know it worked when the pattern glows warm instead of ocean-cool.

4. Bigger canvas, coarser grid. Try w = 24 with cell = 14 and steps = 400. You'll know it worked when the pattern is chunkier but evolves further in the same run time.

You Simulated Nature!

A grid, two chemicals, and neighbor math — and your program grew patterns no one drew. Alan Turing dreamed this up in 1952; you just ran it in a browser. You've reached the end of the gallery — browse the Turtle Graphics Gallery for your next challenge!