In 1951, when Turing was 39 years old, he turned to mathematical biology, finally publishing his masterpiece "The Chemical Basis of Morphogenesis" in January 1952. He was interested in morphogenesis, the development of patterns and shapes in biological organisms. Among other things, he wanted to understand Fibonacci phyllotaxis, the existence of Fibonacci numbers in plant structures.[107] He suggested that a system of chemicals reacting with each other and diffusing across space, termed a reaction-diffusion system, could account for "the main phenomena of morphogenesis".[108] He used systems of partial differential equations to model catalytic chemical reactions. For example, if a catalyst A is required for a certain chemical reaction to take place, and if the reaction produced more of the catalyst A, then we say that the reaction is autocatalytic, and there is positive feedback that can be modelled by nonlinear differential equations. Turing discovered that patterns could be created if the chemical reaction not only produced catalyst A, but also produced an inhibitor B that slowed down the production of A. If A and B then diffused through the container at different rates, then you could have some regions where A dominated and some where B did. In order to calculate the extent of this, Turing would have needed a powerful computer, but these were not so freely available in 1951, so he had to use linear approximations in order to solve the equations by hand. Fortunately these calculations gave the right qualitative results, and produced, for example, a uniform mixture that oddly enough had regularly spaced fixed red spots. The Russian biochemist Boris Belousov had performed experiments with similar results, but could not get his papers published because of the contemporary prejudice that any such thing violated the second law of thermodynamics. For a modern view of living organisms and the second law see The Second Law of Thermodynamics Section#7. Unfortunately Belousov was not aware of Turing's paper in the Philosophical Transactions of the Royal Society.[109]

Although published before the structure and role of DNA was understood, Turing's work on morphogenesis remains relevant today, and is considered a seminal piece of work in mathematical biology.[110] One of the early applications of Turing's paper was the work by James Murray explaining spots and stripes on the fur of cats, large and small.[111][112][113] Further research in the area suggests that Turing's work can partially explain the growth of "feathers, hair follicles, the branching pattern of lungs, and even the left-right asymmetry that puts the heart on the left side of the chest."[114] In 2012, Sheth, et al. found that in mice, removal of Hox genes causes an increase in the number of digits without an increase in the overall size of the limb, suggesting that Hox genes control digit formation by tuning the wavelength of a Turing-type mechanism.[115] Later papers, though promised, were not available until Collected Works of A. M. Turing was published in 1992.[116]