A team of mathematicians from the Budapest University of Technology and Economics, the HUN-REN network, and the University of Oxford have discovered a novel, universal class of shapes that tile space seamlessly and without sharp corners. Aptly named 'soft cells,' these forms are prevalent in nature, appearing in structures ranging from seashells to muscle cells.
The findings offer profound insights into the geometry of biological tissues and hold potential for innovative architectural designs devoid of sharp corners.
Mathematicians have long been fascinated by tiling—how shapes can fit together to cover surfaces or fill spaces without gaps or overlaps—ever since Plato’s theory identified the Elements with regular polyhedra. Traditionally, this field has focused on shapes with sharp corners and flat faces, such as triangles, quadrilaterals, cubes, and other polyhedra. However, such forms are seldom observed in the natural world. This discovery marks a significant departure from conventional approaches, aligning more closely with the organic, curved geometries found in nature.
Nature has favored intricate shapes throughout its evolutionary journey. Living organisms employ a diverse array of space-filling patterns for growth and development, evident in biological tissues and chambered shells. These natural patterns are notable for their highly curved edges, non-flat faces, and few, if any, sharp corners.
Until recently, the mathematical explanation for how nature achieves such geometric complexity with these 'soft shapes' has remained elusive.
The solution to this long-standing geometric puzzle, discovered by Gábor Domokos, Krisztina Regős, and Ákos G. Horváth (HUN-REN-BME Morphodynamics Research Group and Budapest University of Technology and Economics), along with Alain Goriely (University of Oxford), is presented in their recently published article in PNAS Nexus. They define a new class of mathematical shapes called soft cells, which are characterized by having a minimal number of sharp corners and the ability to cover space without gaps and overlaps.
In two dimensions (2D), soft cells feature curved boundaries with only two corners. These tiling patterns are observed in a variety of natural and man-made structures, including muscle cells, zebra stripes, river island shapes, onion bulb layers, and even in architectural designs.
In three dimensions (3D), these soft cells reveal even greater complexity and intrigue. The research team first established that soft cells have no corners at all. They demonstrated that starting with known hard tilings, such as the cubic grid, these structures could be transformed into soft cells by suitably deforming the edges and eliminating all sharp corners by applying this process, The procedure led them to discover entirely new classes of soft cells with diverse tiling properties.
Their research further revealed that many of these shapes occur naturally in various contexts. The most remarkable discovery was that the inner chambers of the iconic Nautilus shell exemplify soft cells. The cross-section of these chambers shows a planar soft cell with two corners. Krisztina Regős noted, "Despite having two corners in the plane, it appeared to me that there would be no corners in three dimensions." Detailed CT images confirmed that the chambers of the Nautilus and many Ammonites are indeed soft cells, filling the shell without any sharp corners.
Gábor Domokos remarked, “We found that architects, including Zaha Hadid, have intuitively constructed these kinds of shapes whenever they wanted to avoid corners. In fact, a team of young architects constructed one of our three dimensional soft cells using the geometry of the Gömböc shape as an inspiration.”
Soft cells appear to be geometric building blocks of biological tissues, opening up a myriad of questions in both geometry and biology. Understanding the necessary conditions for generating soft tilings could provide new insights into why certain patterns are favored by nature. The concept of soft cells could explain not only the static geometry of tissues but also processes like tip growth, which is one of the most ubiquitous mechanisms in biological shape evolution.
Geometric model of the Nautilus shell. Right panel: comparison of the Nautilus chamber and the soft cell in the geometric model. Credit: Lajos Czeglédi, Krisztina Regős
Soft version of the truncated octahedron.
Credit: Krisztina Regős, Gábor Domokos