Boxfish have enchanted scientists and marine enthusiasts for decades.
Despite their boxy shape, they are agile swimmers, inspiring research into how they get around. They are also cute, with pouting mouths and cubic bodies that come in a variety of bright, striking patterns: yellow with black polka dots, gray with blue stripes and more.
Two engineers at the University of Colorado Boulder are particularly intrigued with the spots, stripes and hexagons found on an Australian species in the boxfish family: the ornate boxfish. The designs on that species’ skin, they discovered, can be described and reproduced using decades-old mathematics once explored by Alan Turing, who is often referred to as the father of modern computing.
The engineers, Siamak Mirfendereski and Ankur Gupta, recently unveiled a mathematical model able to accurately re-create the ornate boxfish’s patterns, even accounting for the graininess and other imperfect features that are observed in nature.
According to Gupta, the model, described in a 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 on fish and other organisms.
“This helps bridge the gap between mathematical models and the messy beauty of biological reality,” Gupta wrote in an email.
Someday, he added, that knowledge may lead to bio-inspired fabrics for camouflage or advances in soft robotics, which stray from typical rigid hardware to build machines with softer materials, such as silicone.
Gupta’s work is an extension of a theoretical model that Turing published in 1952. Turing’s model relied on the interplay between diffusion — a process by which particles spread into regions where they are less populated — and the chemical reactions experienced by those particles.
Diffusion generally leads to something uniform: Drop a bit of food coloring in a glass of warm water, for example, and the liquid will eventually become one consistent hue. But under certain conditions, Turing argued, the combination of diffusion and chemical reaction could spontaneously cause particles to organize into stripes, spots and other designs. These came to be known as Turing patterns.
The mathematics underlying Turing patterns has helped explain how nature creates the spots on a leopard, the swirls on seashells and other designs found in biological systems. It has also been used to describe the formation of human fingerprints, the shaping of sand ripples and the spread of matter across galaxies.
Computer programs that simulate diffusion and reaction processes have been able to replicate some biological patterns. But according to Gupta, they often produce results that are too idealized, failing to capture natural imperfections, including variations in size or thickness, line breaks and graininess. Simulations developed by Gupta’s group, which mimicked the behavior of pigment cells on the skin of ornate boxfish, also created designs that appeared blurry around the edges, rather than sharp as in real life.
“A diffusive system is, by definition, diffuse,” Gupta said. “So how can you get sharp patterns?”
In 2023, a student in Gupta’s group solved this problem by adding a different kind of cell movement to the simulation. Cells in a fluid can clump and move together, pulled by the motion of other diffusing particles, Gupta explained. This process, known as diffusiophoresis, is also how soap pulls dirt out of clothes during washing.
As a result, the simulated boxfish patterns appeared sharper and more defined. To introduce imperfections into those patterns, Mirfendereski tweaked the simulation further, accounting for individual cells bumping into each other.
As the patterns emerged, so, too, did flaws. The simulated boxfish stripes were thin in some parts but thicker in others, as well as broken off in some places. The sides of some hexagons never formed, while others appeared splintered or bulbous. Spots inside the hexagons stretched or bled into one another.
According to Gupta, these imperfections can be tuned. But the simulation is still a simplified version of reality. It does not account for more complicated interactions between cells. And like Turing’s original mathematical model, it lacks specifics about pigment production and other biological mechanisms.
Still, Turing’s model laid a foundation for scientists to control pattern formation in real-world applications, biological and beyond. Researchers have used it to engineer patterns in growing E. coli colonies and rearrange the stripes on zebrafish. Others have used it to develop more efficient saltwater filters and understand trends in human settlement.
“We learn how biology does it so that we can replicate it,” Gupta said, though he admitted that he primarily did the work for curiosity’s sake. He is keen, he added, to learn how nature creates “the imperfect but distinctive patterns that have fascinated biologists for decades.”
Closeup shot of yellow Cichlidae cichlid at home in the aquarium. (Image: Freepik) 
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