Cómo dos sustancias químicas hacen crecer manchas y rayas
Dos sustancias químicas: una alimenta y la otra se extiende. Manchas y rayas como las de la piel de los animales crecen solas.
Se reproduce desde YouTube tras hacer clic.
Pruébalo: cambia la regla
0 pasos. Toca la placa para añadir una gota de B. Las mismas dos ecuaciones crean cada ajuste; solo cambian f y k.
Por qué funciona
Dos sustancias químicas comparten una placa. Una es alimento; la otra se lo come para hacer más de sí misma, y las dos se extienden. A partir de una sola gota, la segunda crece por sí sola en manchas, rayas o un laberinto, como las marcas de la piel de un animal.
∂A/∂t = D_A·∇²A − A·B² + f·(1 − A)∂B/∂t = D_B·∇²B + A·B² − (f + k)·B
Constrúyelo tú
Python 3 · numpy · matplotlibUna placa de 128 × 128, seis gotas de B, 6000 pasos de la regla de Gray-Scott.
"""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
Prueba: Prueba f, k = 0.039, 0.058 para una red de células o 0.034, 0.063 para manchas: la misma regla, otra especie.
Aprende este script línea a línea, con ejercicios y un cuestionario → · ejecútalo en Google Colab (en inglés) →
Las matemáticas, tal como las cuenta el vídeo
Lee las notas del propio vídeo (de YouTube, en inglés)
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.
De la descripción de Two Chemicals Grow a Living Pattern (en inglés).
