The science
Turing's idea (1952). Two chemicals, an activator u that makes more of itself and an inhibitor v that shuts it down, can be stable when well mixed yet become unstable when the inhibitor spreads faster. The uniform state then breaks into spots or stripes with a built-in length scale. Alan Turing, Phil. Trans. R. Soc. B 237 (1952).
du/dt = r f(u,v) + D_u m lap(u)
dv/dt = r g(u,v) + D_v m lap(v)
Linear stability. Linearise about the homogeneous steady state with Jacobian J. A Fourier mode exp(i k x + sigma t) grows at the larger eigenvalue of J - diag(D_u,D_v) k^2:
sigma(k) = max Re eig( J - diag(D_u,D_v) k^2 )
Turing conditions (activator u, d = D_v/D_u):
f_u + g_v < 0 f_u g_v - f_v g_u > 0
d f_u + g_v > 0 (d f_u + g_v)^2 > 4 d (f_u g_v - f_v g_u)
k* = argmax sigma(k), lambda* = 2 pi / k*
The unstable band is (k-, k+). The page computes it numerically from the eigenvalues (not from a formula for k*), once with the continuum k^2 and once with the lattice symbol 2-2cos k of the 9-point stencil actually used. The diffusion ratio d must exceed a critical value d_c (the page prints it); for activator-inhibitor kinetics it is typically several to a few tens, which is why a fast inhibitor is the rule.
Models. Gray-Scott (1984; Pearson 1993): f = -u v^2 + F(1-u), g = u v^2 - (F+k)v. Most of its spots, stripes and mitosis patterns are finite-amplitude structures that grow from the stable trivial state (u=1, v=0), not linear Turing patterns; only a narrow F-k sliver has a Turing-unstable nontrivial state, so the dispersion curve is shown only where it exists. Gierer-Meinhardt type (1972), with saturation: f = a^2/(h(1+kappa a^2)) - mu a + rho_a, g = a^2 - nu h; its homogeneous state is found by bisection and it is linearly Turing-unstable for a large enough D_h/D_a.
Body size matters (Murray 1981). The pattern wavelength is set by the chemistry, so the number of stripes or spots depends on how large the domain is compared with lambda*. A small embryo supports a few wide features; as it grows, stripes split and new ones are inserted (slider S, "Grow"). A thin tail has a circumference of only about one wavelength, so only variation along its length survives and spots turn into rings. Murray's corollary: a spotted animal can have a striped tail, but a striped animal cannot have a spotted tail. Angelfish stripes that drift and re-arrange as the fish grows (Kondo and Asai 1995; Painter, Maini and Othmer 1999) are the classic living example; the angelfish preset only imitates that qualitatively.
How the pattern is put on a 3-D body. Each animal was built in Blender from lofted tubes with a hand-laid UV atlas. The reaction-diffusion runs in that UV texture on the GPU (ping-pong float textures, one fragment-shader step per iteration). Because UV space is stretched compared with the real surface, Blender also wrote a per-texel metric map: the local surface length of one texel along u and v. The shader multiplies the Laplacian by c_u and c_v (anisotropic, from those lengths) so that diffusion is the same in real centimetres everywhere on the body, not in texels. UV seams are treated as no-flux walls (the pattern does not continue across a seam), texel anisotropy is clamped to keep the explicit scheme stable, and the poles of a tube are oversampled; these are visible artefacts and are not hidden.
Numerics. Explicit Euler with an adaptive step dt so that D_max m dt <= 0.32. Diffusion uses the isotropic 9-point stencil (weights 2/3, 1/6, -10/3), whose error is k^4/12 relative to the continuum. At about 13 cells per wavelength this is a few percent, which is why the lattice symbol is used for the prediction; the experiments below measure it.
Simplifications. The real morphogens of coat patterns are not identified; pigment cells, gene expression and growth are far richer than two scalar fields. Rosettes are drawn as an iso-level band of the spot profile, not from a true two-stage mechanism. Real wavelengths are approximate (zebra stripe period about 6 cm, leopard spot spacing about 4-5 cm, giraffe patch spacing about 20 cm, angelfish stripes about 2 cm); the body scale slider and the diffusion constants are tuned so the default preset reproduces them roughly. Not a prediction of any individual animal.
Key sources: Turing 1952; Gierer and Meinhardt 1972; Murray 1981 (Phil. Trans. R. Soc. B 295:473) and Mathematical Biology II; Gray and Scott 1984; Pearson 1993 (Science 261:189); Kondo and Asai 1995 (Nature 376:765); Crampin, Gaffney and Maini 1999 (Bull. Math. Biol. 61:1093); Kondo and Miura 2010 (Science 329:1616).