It may be hard to imagine, but a newly defined geometric shape exists. Based on recent calculations, mathematicians have described a new classification they call “soft cells.” In their most basic form, soft cells take the form of geometric building blocks with rounded corners that can be connected with cusp-like corners to fill two- or three-dimensional space. If you find this concept surprisingly elementary, you’re not alone.
“Simply put, no one has ever done this before,” Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics who was not connected to the study, said of the classification. Nature September 20. “It’s amazing how many basic things there are to consider.”
[Related: How to prove the Earth is round.]
Experts have known for thousands of years that certain polygons, like triangles, squares, and hexagons, can completely cover a two-dimensional plane. But in the 1980s, researchers discovered the following structure: Penrose Tile Building on these and other geometric advances, a team led by Gábor Domokos of the Budapest University of Technology and Economics recently began to explore these concepts in more detail, which included revisiting “periodic polygon tilings” and the notion of what happens if you round some of the corners.
The results were published in the September issue. PNAS Nexusrevealed what Domokos and colleagues call “soft cells” — rounded shapes that can fill space perfectly thanks to certain corners that are deformed into “cusp shapes.” These cusps have zero interior angles and edges that meet tangentially to fit other rounded corners. Using a new algorithmic model, the mathematicians explored what could be done with shapes that obey these new rules. While tiles need to have at least two cusp corners in two-dimensional space, when extended to 3D they can fill voluminous space even without such corners. In particular, they calculated quantitative means to measure the “softness” of 3D tiles and found that the “softest” iterations contain winged edges.
Examples of 2D soft tissues in nature include the cross-section of an onion, biological tissue cells, and islands formed by river erosion. In 3D, the shape can be seen in fragments of a nautilus shell. Observing these mollusks was a “turning point,” Domokos said. NatureThat’s because cross sections of the compartment looked like 2D soft cells with a pair of horns. Despite this work, co-author Christina Regos theorized that the shell chamber itself doesn’t have horns.
“It was an incredible story, but it turned out she was right,” Domokos said.
But why have geometers failed to define soft cells concretely for hundreds of years? The answer, Domokos argues, lies in the relative simplicity of soft cells.
“The world of polygon and polyhedral tilings is so fascinating and rich that mathematicians have never needed to expand their playground,” he said, adding that many contemporary researchers mistakenly assume that discoveries require sophisticated mathematical equations or algorithmic programs.
Even though it has never been explicitly explained, humans seem to have intuitively understood soft cell design for many years. Heydar Aliyev Center The Sydney Opera House relies on that fundamental principle to achieve its iconic rounded features.
