# The patterns on the scales of lizards vary according to mathematical laws

The patterns on the scales of lizards vary according to mathematical laws

An international group of scientists proved that the appearance of patterns on the skin of lizards is subject to the action of two mathematical models — the equation of a Turing machine and von Neumann.

The study involved the winner of the award of the fields, the head of the laboratory. Chebyshev St. Petersburg State University Stanislav Smirnov. The result of the study published in the journal Nature.

The coloration of young specimens of the South-Western European lizards is a ring of three colors, which are located across scales. Like most reptiles, the color of these rings has nothing to do with the structure of the scales.

But when the lizard Matures, the scales are entirely painted in two colors — green or black and form a winding pattern, like a maze

The scales change their color.

Watching older lizards, head of laboratory of artificial and live the evolution of the University of Geneva, Professor Michel Milinkovich noted that the behavior of the skin of reptiles like the evolution of a cellular automaton. A cellular automaton system, proposed by one of the fathers of the modern computer, mathematician John von Neumann. The system consists of a set of rules by which cells are recoloured depending on the colors of its neighbors. However, even simple rules can lead to very complex behaviors, orderly or chaotic. An example is the “game of life” on graph paper, which is often mentioned in the popular literature, such as books by Martin Gardner.

Michelle Milinkovic and his colleagues for four years watching the lizards, the color of which changes with age.

It turned out that the skin of the lizards is really similar to a cellular automaton — the cells in disguise depending on how many neighbors of each color.

However, the most common reason for the appearance of patterns on the skin of animals has nothing to do with cellular automata. It was proposed by the famous mathematician Alan Turing in one of his recent articles. This interaction of the pigments described by equations of reaction-diffusion.

If the pigment just diffuserwith, then, for example, black spot spreads, and the whole skin becomes dull. But if diffusibility between different pigments is non-trivial reactions, the behavior becomes more complicated. Turing noticed that with very simple coefficients, the solutions can behave differently — be, or spots, or stripes, or spirals. Sometimes the patterns move or pulsate.

The conceptualization of the hypotheses Turing took half a century, and now it has become apparent that the coloring of many animals comes from his equations

Illustrative example is found in tropical seas, Zebra fish, complex patterns which result from simple equations.

The question who formulated Michel Milinkovich was: how can you type equations, reaction-diffusion derive a cellular automaton? Together with Stanislav Smirnov Milinkovich suggested that the change in the thickness of the skin between the scales of lizards should reduce in these places the diffusion coefficients in the equation of Turing.

“I was able to not only confirm our General assumption, — said Stanislav Smirnov, but to show how in this case the equations of Turing is reduced to a discrete form at the grid scales (where it is considered that each scale is painted in one color), then the cellular automaton. Because instead of a whole region of points it all comes down to the study of flowers of several thousand scales, the task is greatly simplified. In turn, this new model was able to associate with the cellular automaton”.

A mathematical proof of this new observation is obtained largely through research in the field of chemistry and biology. It is interesting that the equations Turing were associated with the cellular automata of von Neumann, and this relationship is manifested in nature as a result of evolution of Darwin.

The mechanism of occurrence of patterns described by scientists in Geneva and in St. Petersburg, is very different from those that were considered their colleagues. According to Stanislav Smirnov, the results obtained in this study can be developed and applied in various fields of science, including biology and physics in the study of spontaneously formed patterns.