Have you ever wondered why zebras have curved stripes instead of perfect squares? Or why seashells don’t have sharp corners? It turns out that nature has been solving complex math problems all this time, and scientists have just discovered its secret: soft cells.
For centuries, mathematicians have been obsessed with a seemingly simple question: how to completely fill space with shapes? Human response has always been to turn to shapes with sharp corners and flat surfaces, such as triangles, squares, and hexagons in two dimensions, and cubes and other polyhedra in three dimensions. We can imagine the tiles in the bathroom: they are probably perfect squares or hexagons.
But nature, the master of innovation, has another idea. Opt for soft curves and wavy edges instead of sharp corners. And until now, no one knew exactly how he did it.
“soft cells”
Now a new study is published in the journal PNAS Nexus, and led by Alain Goriely from the University of Oxford and a team from the Budapest University of Technology and Economics finally solved this puzzle. They discovered a new class of mathematical shapes called “soft cells”. These ingenious shapes have the minimum number of sharp corners necessary to fill the space without leaving gaps and are more reminiscent of natural structures than classic geometric patterns.
In two dimensions, these soft cells have curved edges with only two vertices, and the most surprising thing is that these shapes are not just a mathematical oddity, but commonly found in nature. A clear example of this is an onion: if we cut it in half, the inner layers show perfectly connected shapes without sharp corners. Similarly, smooth muscle cells, when viewed in cross-section, show mosaics of smooth, elongated shapes that fill space without leaving gaps.
Other examples of soft cells cited by researchers include zebra stripes, river island formations, and even some innovative architectural designs.
But beyond two dimensions, soft cells become even more complex and interesting as we move into three dimensions. The team discovered that in 3D these shapes have no corners at all.
Starting with conventional three-dimensional mosaics such as a cubic grid, they showed that it is possible to soften these shapes by curving the edges and minimizing sharp corners. In this way, they discovered completely new classes of soft cells with distinctive properties.
“We found that architects, including Zaha Hadid, built these types of shapes intuitively when they wanted to avoid corners,” commented Professor Gábor Domokos. “A team of young architects actually built one of our three-dimensional soft cells inspired by the geometry of the Gömböc shape,” he added.
Why is it important for nature?
Soft cells not only explain why living organisms prefer these shapes, but could also shed light on fundamental biological processes such as cell growth or the formation of natural patterns. For example, the inner chambers of the nautilus, that spiral-shaped marine mollusk, are a perfect example of a three-dimensional soft cell without horns and with a geometry adapted to optimal use of space.
“Nature hates not only a vacuum, but also sharp angles,” said Professor Alain Goriely of Oxford. And it makes sense. Maintaining sharp corners in physical cells is difficult and energetically expensive. “Surface tension and elasticity naturally tend to smooth corners. It is therefore not surprising that many smooth mosaics are found in nature,” the researchers noted.
This discovery not only opens new doors in geometry and biology, but may also have practical applications in fields as diverse as architecture, industrial design and even medicine, where a better understanding of natural forms could inspire new solutions to complex problems.
Edited by Felipe Espinosa Wang with input from PNAS Nexus, University of Oxford and IFL Science.
