As hard as it is to imagine, there is a newly defined geometric shape in the books. Based on recent calculations, mathematicians have described a new classification that they now call a ‘soft cell’. In its most basic form, soft cells take the form of geometric building blocks with rounded corners that can interlock at cusp-like angles to fill a two- or three-dimensional space. And if you think this concept is surprisingly rudimentary, you’re not alone.
“Just, no one has done this before,” Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics who is not affiliated with the work, said of the classification Nature on September 20. “It’s really amazing how many basic things you have to consider.”
[Related: How to prove the Earth is round.]
Experts have understood for thousands of years that specific polygonal shapes such as triangles, squares and hexagons can be arranged to cover a 2D plane without gaps. However, in the 1980s, researchers discovered structures such as Penrose tilings able to fill a space without regularly repeating arrangements. Building on these and other developments in geometry, a team led by Gábor Domokos of the Budapest University of Technology recently began investigating these concepts in more detail. This included a re-examination of ‘periodic polygonal tilings’ and the concept of what might happen if some corners were rounded.
The results, published in the September issue of PNAS nexusreveal what Domokos and colleagues describe as soft cells: rounded shapes that can completely fill a space thanks to specific angles that are deformed into ‘cuboid shapes’. These cusps have an internal angle of zero, with the edges meeting tangentially to fit other rounded corners. Using a new algorithmic model, the mathematicians investigated what you can do with shapes that follow these new rules. Tiles require at least two vertices in two-dimensional space, but when extended to 3D, volumetric spaces can be filled without such vertices even being necessary. Specifically, they calculated a quantitative way to measure the “softness” of 3D tiles, and found that the “softest” iterations included winged edges.
Examples of 2D soft cells in nature include the cross-section of an onion, biological tissue cells and islands formed by erosion in rivers. In 3D, the shapes can be found in nautilus shell segments. Observing these molluscs was a “turning point,” Domokos said Naturebecause the cross-sections of their compartments resembled 2D soft cells with a few corners. Despite this study, co-author Krisztina Regős theorized that the shell chamber itself had no corners.
“That sounded incredible, but later we found out she was right,” Domokos said.
But how could geometers not concretely define soft cells for hundreds of years? The answer, Domokos argues, lies in their relative simplicity.
“The universe of polygonal and polyhedral tilings is so fascinating and rich that mathematicians did not need to expand their playing field,” he said, adding that many modern researchers wrongly assume that discoveries require sophisticated mathematical equations and algorithmic programs.
Even if not explicitly explained, it seems that people have been intuitively understanding soft cell designs for years – architectural designs like the Heydar Aliyev center and the Sydney Opera House rely on their underlying principles to achieve their iconic circular features.
