 Turing’s model of morphogenesis has several striking features. First, it is based on physics and chemistry rather than biology: reaction-diffusion equations incorporating Fourier series and the law of mass action. Reaction-diffusion equations are a class of non-linear partial differential equations. (Hodgkin and Huxley’s model of neural propagation is also of this class—at the time, one of the few applications of these equations to biology; see the entry on  mechanisms in science.)  Turing’s use of these equations to model the transition from homogeneity to spatial patterning is rather counterintuitive, as diffusion is a characteristically homogenizing process. So his model was a theoretical advance, extending the insight that reaction-diffusion equations can model transitions and “waves of advance” (e.g., in neurons) to morphogenesis (Berestycki 2013). Second, the model is strikingly abstract—“…a simplification and an idealization, and consequently a falsification” (Turing 1952: 37). Yet it has many biological applications. Turing (1952) explored mainly simple cases of chemical diffusion through a tissue of fixed size. But he also applied the model to chemical waves on spheres to represent gastrulation. That more complex case, with three interacting morphogens, can produce traveling waves and out-of-phase oscillations. Turing also applied his model to plant development, specifically phyllotaxis (arrangement of leaves on a stem; see S. B. Cooper &amp; van Leeuwen 2013). Subsequent generalizations of Turing’s equations have been used to model a wide variety of developmental phenomena, including fish pigmentation patterns, human mesenchymal cell differentiation, butterfly wing patterns, mouse hair follicles/coat color patterns, limb and tooth development, and bacterial colony growth dynamics (Meinhardt 1982, Murray 1989). However, its explanatory role is controversial, primarily because it is hard to show that Turing’s model describes how development actually proceeds in real organisms.