ふたつの化学物質が、どうやって斑点や縞を育てるのか
ふたつの化学物質：ひとつは補給され、もうひとつは広がっていきます。動物の皮膚のような斑点や縞が、ひとりでに育ちます。
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試してみよう：ルールを変える
0ステップ。シャーレをタップすると、Bのしずくを足せます。どのプリセットも同じふたつの方程式から生まれ、変わるのはfとkだけです。
なぜそうなるのか
ふたつの化学物質が、ひとつのシャーレに入っています。ひとつはエサで、もうひとつはそれを食べて自分を増やし、どちらも広がっていきます。たった一滴から、2つ目の物質はひとりでに斑点や縞、迷路に育ちます。動物の皮膚の模様のように。
∂A/∂t = D_A·∇²A − A·B² + f·(1 − A)∂B/∂t = D_B·∇²B + A·B² − (f + k)·B
自分で作る
Python 3 · numpy · matplotlib128 × 128のシャーレ、Bのしずく6滴、グレイ–スコットのルールで6,000ステップ。
"""Turing patterns: the Gray-Scott reaction-diffusion rule on a 128 x 128 grid.
Run: python turing-patterns.py (needs: pip install numpy matplotlib; takes a few seconds)
"""
import numpy as np
import matplotlib.pyplot as plt
plt.style.use("dark_background")
size, steps = 128, 6000
f, k = 0.0545, 0.062 # feed and kill rates: coral
DA, DB = 1.0, 0.5 # A spreads twice as fast as B
A, B = np.ones((size, size)), np.zeros((size, size))
rng = np.random.default_rng()
for y, x in rng.integers(10, size - 18, (6, 2)):
B[y : y + 8, x : x + 8] = 1.0 # six drops of B
def laplacian(Z):
"""Each cell against its neighbours: 0.2 for the four sides, 0.05 for the corners (Karl Sims)"""
sides = np.roll(Z, 1, 0) + np.roll(Z, -1, 0) + np.roll(Z, 1, 1) + np.roll(Z, -1, 1)
corners = (np.roll(np.roll(Z, 1, 0), 1, 1) + np.roll(np.roll(Z, 1, 0), -1, 1)
+ np.roll(np.roll(Z, -1, 0), 1, 1) + np.roll(np.roll(Z, -1, 0), -1, 1))
return 0.2 * sides + 0.05 * corners - Z
for _ in range(steps):
reaction = A * B * B # one A and two B make three B
A += DA * laplacian(A) - reaction + f * (1 - A)
B += DB * laplacian(B) + reaction - (f + k) * B
plt.figure(figsize=(6, 6))
plt.imshow(B, cmap="magma", vmin=0, vmax=0.5)
plt.axis("off")
plt.show()
# Try: f, k = 0.039, 0.058 (a cell net) or 0.034, 0.063 (spots) -> the same rule, another pattern
試してみよう： f, k = 0.039, 0.058 で細胞の網、0.034, 0.063 で斑点になります。同じルールで、別の種。
動画が語る数学
動画の解説を読む（YouTubeより、英語）
Two chemicals share a dish. A is food, fed in everywhere at a slow rate; B eats A to make more of itself (two B's plus one A give three B's) and is removed at a fixed rate. Both spread by diffusion, but A spreads twice as fast as B.
That one asymmetry is the whole secret: Alan Turing showed in 1952 that a "fast inhibitor, slow activator" pair turns a uniform dish unstable, so that the tiniest speck of B grows into a pattern with a built-in wavelength — spots, stripes, labyrinths, cells. The equations here are the Gray–Scott model, and only two numbers change during the film: the feed rate f and the kill rate k. Every "new species" you see is the same rule with those two dials moved a little: coral (f 0.029, k 0.057), solitons that split and travel (0.030, 0.060), a cell net (0.039, 0.058), turbulence (0.026, 0.051).
Colour is age: fresh growth fronts are pink-white, mature tissue violet, old regions indigo; the second seed carries a cyan tint so you can see where the two fronts meet. In the finale the dish is wiped and reseeded inside the shape of the channel's monogram — the same rule, confined — and then the walls are removed. These are the equations behind leopard spots, zebra stripes and the ridges on your fingertips.
次の動画の説明より： Turing Patterns: How Two Chemicals Grow Animal Spots（英語）.
