Where you see fancy fish, engineers see mathematics Alan Turing | Technology news

Two engineers at the University of Colorado at Boulder are particularly intrigued by the spots, stripes and hexagons found on an Australian species of the self-made fish family: the ornate boxfish. They found that the species’ skin patterns could be described and reproduced using decades-old mathematics once explored by Alan Turing, often called the father of modern computing.

Engineers Siamak Mirfendereski and Ankur Gupta recently presented a mathematical model capable of accurately recreating the image of a decorated artificial fish, even accounting for the graininess and other imperfections seen in nature.

According to Gupta, the model described in the study, which was published last week in the journal Matter, brings scientists one step closer to understanding the mechanisms by which such patterns form in nature in fish and other organisms.

“It helps bridge the gap between mathematical models and the messy beauty of biological reality,” Gupta wrote in an email.

Someday, he added, the knowledge could lead to biologically inspired camouflage fabrics, or advances in soft robotics that move away from typical hard hardware to build machines out of softer materials like silicone.

Gupta’s work is an extension of a theoretical model published by Turing in 1952. Turing’s model relied on the interplay between diffusion—the process by which particles spread to regions where they are less populated—and the chemical reactions those particles encounter.

Diffusion usually results in something uniform: drop a little food coloring into a glass of warm water, for example, and the liquid will end up being one shade of color. But under certain conditions, Turing argued, a combination of diffusion and chemical reaction can spontaneously cause particles to organize into stripes, spots, and other patterns. These became known as Turing patterns.

The mathematics behind Turing’s patterns helped explain how nature creates leopard spots, swirls on seashells, and other patterns found in biological systems. It has also been used to describe the formation of human fingerprints, the formation of sand ripples, and the distribution of matter across galaxies.

Computer programs that simulate diffusion and reaction processes have been able to replicate some biological patterns. But, Gupta says, they often produce overly idealized results that fail to capture natural imperfections, including changes in size or thickness, line breaks and graininess. The simulations developed by Gupta’s group, which mimicked the behavior of pigment cells in the skin of decorated fish, also produced designs that appeared blurred at the edges rather than sharp like in real life.

“A diffusive system is, by definition, diffusive,” Gupta said. “So how do you get clear patterns?”

In 2023, a student in Gupta’s group solved this problem by adding a different kind of cell movement to the simulation. Gupta explained that cells in a fluid can stick together and move together, attracted by the motion of other diffusing particles. This process, known as diffusion phoresis, is also how soap pulls dirt out of clothes during washing.

As a result, the modeled drawings of the boxes looked sharper and more expressive. To introduce flaws into these patterns, Mirfendereschi further tweaked the simulations by taking into account individual cells bumping into each other.

As patterns emerged, so did flaws. The ship’s simulated strips were thin in some parts but thicker in others, and also broken off in some places. The sides of some hexagons never formed, while others appeared split or bulbous. The spots inside the hexagons stretched or flowed into each other.

According to Gupta, these imperfections can be corrected. But a simulation is still a simplified version of reality. It does not take into account the more complex interactions between cells. And, like Turing’s original mathematical model, it lacks features of pigment production and other biological mechanisms.

However, Turing’s model laid the foundation for scientists to control pattern formation in real-world programs, biological and otherwise. Researchers have used it to develop patterns in growing colonies of E. coli and changing stripes in zebrafish. Others have used it to develop more efficient filters for salt water and to understand trends in urban development.

“We’re going to learn how biology does it so we can replicate it,” Gupta said, though he admitted he was doing the work primarily out of curiosity. He’s eager, he added, to learn how nature creates the “imperfect but distinctive patterns that have fascinated biologists for decades.”