Hard as it may be to imagine, there’s a newly defined geometric shape on the books. Based on recent calculations, mathematicians have described a new classification they now call a “soft cell.” In its most basic form, soft cells take the form of geometric building blocks with rounded corners capable of interlocking at cusp-like corners to fill a two- or three-dimensional space. And if you think this concept is surprisingly rudimentary, you aren’t alone.
“Simply, no one has done this before,” Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics not affiliated with the work, said of the classification to Nature on September 20. “It’s really amazing how many basic things there are 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 arrange to cover a 2D plane without any gaps. In the 1980’s, however, researchers discovered structures such as Penrose tilings capable of filling a space without regularly repeating arrangements. Building on these and other geometry advances, a team led by Gábor Domokos at the Budapest University of Technology and Economics recently began exploring these concepts in more detail. This included a reexamination of “periodic polygonal tilings,” and the concept of what might happen if some corners are rounded.
The results, published in the September issue of PNAS Nexus, reveal what Domokos and colleagues describe as soft cells—rounded forms capable of filling a space entirely thanks to specific corners deformed into “cusp shapes.” These cusps feature an internal angle of zero with edges meeting tangentially to fit into other rounded corners. Using a new algorithmic model, the mathematicians examined what one can do using shapes that follow these new rules. Tiles require at least two cusp corners in two-dimensional space, but when expanded into 3D, volumetric spaces can fill without even needing such corners. In particular, they calculated a quantitative means for measuring “softness” of 3D tiles, and discovered the “softest” iterations include winged edges.
Examples of 2D soft cells in nature include an onion’s cross-section, biological tissue cells, and islands formed by erosion in rivers. In 3D, the shapes can be found in nautilus shell segments. Observing these mollusks was a “turning point,” Domokos told Nature, because their compartment cross-sections looked like 2D soft cells with a pair of corners. Despite this study co-author Krisztina Regős theorized the shell chamber itself possessed no corners.
“That sounded unbelievable, but later we found that she was right,” Domokos said.
But how could geometers not concretely define soft cells for hundreds of years? The answer, Domokos argues, is in their relative simplicity.
“The universe of polygonal and polyhedral tilings is so fascinating and rich that mathematicians did not need to expand their playground,” he said, adding that many modern researchers incorrectly assume discoveries require advanced mathematical equations and algorithmic programs.
Even if not explicitly explained, it appears that humans have intuitively understood soft cell designs for years—architectural designs such as the Heydar Aliyev Center and the Sydney Opera House rely on their underlying principles to attain their iconic rounded features.
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