Pattern Formations in Nature

That geometric patterns exist in nature was recognized by Leonardo da Vinci and described half a millennium later by the Scottish mathematical biologist D’Arcy Wentworth Thompson. Thompson analyzed the structure of organisms in physical and mathematical terms, and showed numerous correlations among living beings in his 1917 classic book On Growth and Form.

Angelfish and zebras are striped, while jaguars and ladybugs are spotted. According to Alan Turing, these pattern formations are attributable to activator and inhibitor morphogens.

English mathematician Alan Turing examined pattern formations in nature from a highly theoretical perspective. Turing was no ordinary mathematician. During World War II, he was a leading figure at Bletchley Park, Britain’s code-breaking center. His Turing machine cracked the German Enigma Machine-generated coded messages, decoding that was successfully used by the Allies in the naval Battle of the Atlantic. After the war, he was instrumental in conceiving the first computing machine and artificial intelligence. In 1952, until his death by suicide two years later, Turing turned his attention to mathematical biology and published his only biological paper, The Chemical Basis of Morphogenesis. (Morphogenesis refers to the “beginning of shape” and generally refers to the development of form and structure in an organism growing from embryo to adult.) In this paper, Turing proposed a mathematical model, explaining the formation of natural patterns based on physical laws that govern how certain chemicals react and spread through the skin. He then formulated a set of “reaction-diffusion” equations to produce patterns that simulated actual animal patterns.

These equations would provide the basis for explaining the formation of the diverse patterns that exist in plants and animals: the patterns on a sunflower and daisy, the stripes on a tiger and zebra fish, the spots on a jaguar, and the spacing of hair follicles on a mouse’s paw. He theorized that patterns are formed by the interaction of two chemicals, called morphogens, which diffuse at different rates. One of these morphogens is an activator, which expresses the characteristic pattern (stripes, spots), while the other is an inhibitor, which shuts off the activator, leaving a blank space. The Turing pattern mechanism remained highly theoretical for six decades until 2012, when two chemicals were identified that behave as activator and inhibitor morphogens.