Turing instability (1952): A spatially uniform steady state (u₀,v₀) of a reaction-diffusion system can be destabilized by diffusion when d=Dv/Du > 1 (short-range activation, long-range inhibition).
Schnakenberg kinetics: ∂u/∂t = Du·Δu + κ(a - u + u²v), ∂v/∂t = Dv·Δv + κ(b - u²v).
Steady state: u₀ = a+b, v₀ = b/(a+b)². Turing condition: d > (b+a)⁴/(b-a)².
On a sphere, unstable modes are spherical harmonics Y_l^m; the dominant mode number l determines spots (l small) vs stripes (l large). Visualized via cylindrical projection (Mercator-like).