Dev Genes Evol (2011) 221:255–279 The main body of Turing’s paper is an in-depth mathematical analysis of generalised reaction-diffusion equations for two components acting in a ring of cells. To make the problem analytically tractable, Turing focuses on the emergence of the instability, which allows him to assume linearity for the reaction rates. He shows that depending on diffusion and reaction constants the system shows six different behaviours. (1) A stationary case, in which all cells tend to acquire the same temporally stable concentration. This is the most boring case. However, considering stochasticity effects, Turing shows that even this situation might be interesting for biology since it can produce dappled patterns in two dimensions and thus might account for phenomena like surface coloration. (2) Syn- chronous oscillation of all cells. Chemical oscillations are thus a subcategory within Turing’s theory. He was obviously not aware of Lotka’s or Volterra’s work and claims that he also does not know any experimental example. The BZ reaction was yet unknown outside Russia and the Bary oscillator was a real laboratory curiosity. (3) Two types of spatiotemporal patterns including travelling waves, which Turing thought require at least a three- component system. He did not analyse these cases in depth. Travelling chemical waves, too, would become of interest only after they had been experimentally observed in a BZ reaction. This lead to the rediscovery of mathematical papers which indeed had treated the problem of spatiotem- poral patterning even before Turing, albeit in the context of population genetics. Fisher had provided the partial solution of an equation describing the spreading of a beneficial gene within a population (Fisher 1937). The full solution was given in a paper by Kolmogoroff, Petrovsky and Piscounoff (1937). Exactly the same formalism can be applied to the diffusion wave produced by an autocatalytic substance (an early suggestion along this line had already been published in 1906 by R. Luther). However, Turing was not aware of this early work. (4) Two types of stationary spatial patterns. These were the most interesting cases for Turing, especially the case which allows stationary waves with a different wave-length depending on the difference in diffusion and reaction rates of the two components. It is ironic that both for oscillations and travelling waves, excellent chemical examples became accessible for experimental studies in the 1960s while it took almost 40 years until a stable spatial wave pattern was observed in a chemical system and named after Turing (Castets et al. 1990). In recent times, systematic strategies have been developed to produce beautiful Turing patterns in purely chemical systems (Horvath, et al. 2009). In his paper, Turing spends most of his analysis and discussion on the stationary waves forming in a ring of cells. In particular, he uses the Manchester computer, which was partially built with his help, to perform numerical 263 calculations on a realistic, fully non-linear example. He chooses a ring of 20 cells with realistic single cell diameters. The diffusion rates are derived from permeabil- ity studies of biological membranes and the reaction scheme is justified with concrete chemical considerations. Stochasticity effects are introduced for both the transport and the reaction kinetics. The computer simulations show how a homogenous starting situation evolves via a transitory pattern into a stable series of concentration peaks. Turing’s analysis represents the world premier for the application of a digital computer to a reaction-diffusion system. He ends his paper with the visionary note that the use of digital computers will be indispensible for analysing non-linear reaction-diffusion systems. As biological examples for pattern formation on a ring- like structure, Turing considers certain leaf arrangements (Woodruff) and tentacle formation in hydra. He also provides a linear stability analysis for a reaction-diffusion system confined to the surface of a sphere because he is interested in how symmetry-breaking takes place during gastrulation. Further, he announces a future paper on phyllotaxis. The work on this paper occupied the last 2 years of his life and remained unfinished when he died in 1954 (now published in Saunders 1992; Swinton 2004). With his paper, Turing had clearly entered an uncharted terrain both in mathematics (at least with regard to numerical analysis) and biology. This becomes clear from his small list of references which comprises only six papers of which five were written by biologists. (1) C.M. Child’s Patterns and problems of development (Child 1941). In this work, Child had introduced the concept of gradients controlling development. However, he mainly assumed metabolic gradients (e.g. oxygen consumption). (2) A book on the permeability of biological membranes (Davson and Danielli 1943). (3) Michaelis and Menten’s paper on enzyme kinetics (Michaelis and Menten 1913). (4) D’Arcy Thompson’s On growth and form. (5) C. H. Waddington’s Organizers and genes which provided Turing with a state of the art developmental genetics background (Waddington 1940). One month after The chemical basis of morphogenesis had appeared in print, Waddington sent a letter to Turing containing some prophetic remarks. Waddington character- ises the work as ‘extremely interesting and suggestive’ (Waddington 1952). However, he mentions two problems. First, he believes that there are no homogenous starting situations in biology, ‘The newly fertilized egg is, I think, never homogenous in this way, but always possesses some element of pattern of its own…’ (ibid). Second, Waddington mentions an experiment with hydra to illustrate the problem of size regulation. He thinks that a chemical wave theory will be unable to explain scaling. Waddington suggests that Turing’s mechanism will be most relevant for patterns