Mathematicians discover new universal class of shapes to explain complex biological forms

Mathematicians have long studied how shapes can fit together to cover surfaces without gaps. However, their typical approach – using shapes with sharp corners and flat faces – is rarely seen in the natural world.

Nature not only abhors a vacuum, she also seems to abhor sharp corners.

Professor Alain Goriely (Mathematical Institute, University of Oxford)

Instead, living organisms use a dazzling array of patterns to form and grow, for instance in muscle tissues. Most strikingly these patterns are characterised by shapes with curved edges, non-flat faces, and few, if any, sharp corners.

Up to now, how nature achieves geometrical complexity using these ’soft shapes’ has eluded mathematical explanation.

The answer, discovered by mathematicians Professor Alain Goriely (Mathematical Institute, University of Oxford) and Professor Gábor Domokos, Krisztina Regős and Professor Ákos G. Horváth (Budapest University of Technology and Economics) is a new class of mathematical shapes called soft cells. These shapes have a minimal number of sharp corners, and cover space without gaps.

In 2D, these soft cells have curved boundaries with only two corners. Such tiling patterns are found, among others, in muscle cells, zebra stripes, the shapes of river islands, in the layers of onion bulbs, and even in architectural design.

In 3D, these soft cells become more complex and interesting. The team first established that, in 3D, soft cells have no corners at all. Then, starting with conventional 3D tiling systems such as the cubic grid, the team showed that they can be softened by allowing the edges to bend whilst minimising the number of sharp corners in this process. Through doing this, they found entire new classes of soft cells with different tiling properties.

Soft version of the truncated octahedron. Model: Krisztina Regős. Photo: Gábor Domokos.

 A central part of the study, relying on CT images, demonstrates how the inner chambers of the iconic nautilus are natural examples of 3D soft cells without corners. Surprisingly, the planar section of the chambers are 2D soft cells.

‘Nature not only abhors a vacuum, she also seems to abhor sharp corners’ explained Professor Alain Goriely.

Professor Domokos added: ‘Soft cells help explain why, when you look at a cross section of a chambered shell, it shows corners but the 3D geometry of the chambers doesn't.’

Soft cells appear to be geometric building blocks of biological tissue and their existence opens up an array of questions in geometry and biology. Necessary conditions for generating soft tilings could shed new light on why certain patterns are preferred by nature. For instance, the concept of soft cells could help to explain not only the static geometry of tissues but also tip growth, one of the most ubiquitous biological shape evolution processes.

The study ‘Soft cells and the geometry of seashells’ has been published by PNAS Nexus.

Soft cells help explain why, when you look at a cross section of a chambered shell, it shows corners but the 3D geometry of the chambers does not. Image credit: Krisztina Regõs and Lajos Czegledi.