Mathematicians Discover a New Class of Shape: the ‘Soft Cell’

When humans cover a space using tiles of the same shape without overlap or gaps—a geometrical process called tiling—we will usually opt for a shape with sharp corners and straight lines, like squares or triangles. Think of bathroom floors, with regular polygons repeating in a pattern wall-to-wall.

Nature, meanwhile, rarely resorts to straight edges and pointed angles. In the words of IFL Science’s Katie Spalding, “as smart as we as a species are, Mother Nature almost always seems to have us beat.”

Now, an international team of mathematicians has discovered a new class of shapes capable of tiling with a minimal amount of sharp corners, which they define as “soft cells.” Their findings were published in the journal PNAS Nexus on September 10.

It’s not so much a discovery, however, as it is a revelation—nature has been using soft cells in organisms for far longer than humans have been aware of them.

“These shapes emerge in art, but also in biology,” lead author Gábor Domokos, a researcher of geometric modeling at the Budapest University of Technology and Economics, told New Scientist’s Alex Wilkins in February, when the paper was shared as a preprint. “If you look at sections of muscle tissue, you’ll see the cells having just two sharp corners, which is one less than the triangle—it is a very special kind of tiling.”

In two dimensions, soft cells have curved boundaries with two pinched corners called cusps. Shapes of this type are observable in nautilus shells, zebra stripes, river islands and even in the cross-section of an onion, per a statement from the University of Oxford. All 2D soft cells must have at least two of these teardrop-like corners, which come to such a narrow point that they have an internal angle of zero degrees.

But in three dimensions, things get more interesting—because 3D soft cells, called z-cells, have no corners at all. For example, a two-dimensional cross-section of a nautilus shell shows shapes with curved boundaries and two cusps. But when the researchers created a CT scan of the structure, it revealed the inner chamber is actually composed of 3D soft cells without corners.

Study co-author Krisztina Regős had suspected this before conducting the CT scans of the shell, but it still “sounded unbelievable,” Domokos tells Nature News’ Philip Ball. “But later we found that she was right.” They also realized that architects like Zaha Hadid had already used soft cell shapes to avoid straight lines and corners in their projects.

The team created an algorithm to soften the edges and corners of regular geometric tiles in two and three dimensions, shifting them into soft cells. They measured the degree of softness necessary for these cells to successfully tile a 3D space, and they found the softest ones take on a saddle shape, with curved wing-like flaps extending outward. The team suspects that any tiling pattern of standard, straight-edged polyhedrons can be converted into its own unique tiling with the “greatest possible softness,” reports Nature News, though they haven’t yet proven this idea.

The edge bending algorithm

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The fact that soft cells seem to be the “geometric building blocks of biological tissue,” per the statement, holds implications for both geometry and biology. For example, soft cells could shed light on a type of cell expansion called tip growth, which is widely used in nature, such as by algae and fungi. The conditions that lead to soft tiling could also explain nature’s preference for certain patterns.

“They’ve come up with a language for describing cellular materials that might be more physically realistic than the strict polyhedral model that mathematicians have been playing with for millennia,” Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics who wasn’t involved in the study, told New Scientist. “The way that geometry influences the mechanical properties of tissue is really very poorly understood.”

Nature’s aversion to corners and straight edges might serve to save energy, the team writes in the paper, since forces like surface tension and elasticity end up smoothing out corners naturally. For this reason, they weren’t really surprised to find that soft cells are common in the world.

“Nature not only abhors a vacuum,” co-author Alain Goriely, a mathematician at the University of Oxford, says in the statement, “she also seems to abhor sharp corners.”

Margherita Bassi | READ MORE

Margherita Bassi is a freelance journalist and trilingual storyteller. Her work has appeared in publications including BBC Travel, Discover magazine, Live Science, Atlas Obscura and Hidden Compass.