Since time immemorial, science has learned how to effectively cover space without leaving gaps. It turns out that, the usual way of using sharp shapes to tile a seamless floor, or to decorate the walls using carved tessellations, is not the first natural choice.
A team of mathematicians from the University of Oxford and the Budapest University of Technology and Economics discovered a new universal type of structures called ‘soft cells’, which seem to control the structure of nature: they are found in everything from nautilus shells, zebra stripes, how seeds. pack in the vegetables, to the layers of the onion. These cells have curved edges with cusp-like corners, their ends pinched together into a narrow point at an internal angle of zero degrees.
Examples of soft cell types in nature and their use in Zaha Hadid’s architecture. (Domokos et al./PNAS Nexus 2024/CC BY-NC 4.0)
“These trends come from art, but also from biology,” lead author Prof Gábor Domokos he told Young Scientist. “If you look at parts of muscle tissue, you’ll see cells that have just two sharp corners, which is one part less than a triangle—they’re a very special type of tiling.”
Tiling is a subfield of mathematics, in that it explores how polygons can cover a 2D plane without leaving gaps or overlapping. Its concepts are inspiring puzzle packaging designjigsaws, centuries of Middle Eastern weaving patterns, and futuristic architecture.
It’s in the 3D plane, however, that soft cells get really interesting. “The team first discovered that, in 3D, soft cells have no corners at all,” explained the Oxford statement. “Then, by starting with standard 3D tiling systems like the cubic grid, the team showed that it could be softened by allowing the edges to curve while reducing the number of sharp corners. in this process. By doing this, they discovered new classes of soft cells with different tile structures.”
Dongdaemun Plaza, an area of Seoul designed using the ethos of this new phenomenon by Zaha Hadid.
When examining the geometry of complex life, researchers discovered that nature seems to “abhor sharp corners”. They also noticed that architects such as Zaha Hadid were ‘precisely made’ new soft cell-like shapes when they wanted to avoid corners in buildings. Hadid’s topsy-turvy designs can be seen at Seoul’s Dongdaemun Plaza, the London Aquatics Center and the Heydar Aliyev Center in Baku.
Key to the study was the reliance on CT images, which show that the inner chambers of the nautilus, a marine mollusc, are a natural example of 3D smooth cells without corners. The findings seem to be hidden in plain sightsince group analysis is used in ancient geometry that has been known for centuries.
The cells are soft which is why the cross section of the chambered shell shows the corners, but the 3D view of the chambers does not. (Photo credit: Krisztina Regõs and Lajos Czegledi)
“The universe of polygonal and polyhedral tiles is so interesting and rich that mathematicians didn’t need to expand their playing field,” Domokos said. The environmentit adds that it disproves the idea that advanced mathematics or computing is necessary to make significant discoveries.
Hobbyist mathematician David Smith proved this in 2023 by revealing the ‘Einstein tile’ – a 13-sided shape that can fill an infinitely large plane without repeating its pattern. Smith, 64, was a retired typist who stumbled upon it while ‘looking around’. Also known as ‘hat’the tile showed aperiodic (irregular) tiling, as seen below.
Einstein’s tiles show a non-repeating pattern; and (right) a scutoid example.
Other features found in recent memory are a ‘reinvented wheel’ that looks like a ‘multi-dimensional guitar pick’, and a scutoid. Spanish biologists had defined the latter as the 3D shape adopted by the epithelial cells of our body in certain situations. Like a twisted prism, it allowed the body’s muscles to bend, allowing our organs to form properly during growth. Several scutoids can pack together to fill the space between two similar surfaces.
Such discovery allows creators to look at new frontiers in art and mathematics – two fields that seem far away, but which are always considered to be united by polymaths like Leonardo Da Vinci and Luca Pacioli. This integrated approach is also successful in everyday innovation, as puzzle designer Manish Rathod discovered. You can read more about him it’s a classic Rubik’s Cube twist here.
For more brain teasers and new ways of thinking, we invite you to check out @iepuzzles on Instagram.