Hailed as one of the most significant achievements in the field of science, the Mendeleev Periodic Table of Chemical Elements is celebrating its 150th anniversary in 2019. Not only has it revolutionised the way that scientists understand the basis of chemistry, but it has also inspired other disciplines to create a similar common language. One such project is the work of a British research team, who aim to identify and chronicle the fundamental building blocks of complex 4D shapes, achieving for geometry what the Periodic Table has achieved for science. With European Research Council (ERC) support, this ambitious project is a unique collaboration of fundamental theory, geometrical mathematics, physics and art.

“Shapes can be described using equations,” explains Tom Coates, Professor of Mathematics at Imperial College London, UK. “By manipulating these equations, we can break shapes up into simpler pieces, in the same way that molecules are made up from atoms.”

To put it another way, Prof. Coates is interested in identifying and classifying the fundamental building blocks of 4D shapes, named ‘Fano manifolds’ after Gino Fano – the Italian mathematician who first described their existence in geometry. One example of a manifold is the shape of the universe itself. To better understand their impact within algebra, geometry and theoretical physics, Prof. Coates is aiming to classify these shapes and create a directory, much like the Periodic Table.

An unusual element of the project has been the involvement of visual artists like Gemma Anderson, who has created sculptures and etchings based on the Fano manifolds in partnership with Prof. Coates and his team. These have been exhibited in gallery spaces and science museums.

“We were extremely lucky,” he says. “Anderson was at Imperial College and happened to read about our aim to find the atomic pieces of shapes.” Her involvement has provided the project with another dimension. “Artists think in a different way,” continues Prof. Coates. “Vision and form, and not logic and thought, are their primary concerns. I know shapes from the equation side of things, but working with Anderson to visualise them has enabled me to know them in a way I didn’t before.”

Other artists have been involved in designing Christmas tree decorations based on the shapes, which were displayed at Imperial College’s Christmas Symposium. “People really got into it,” says Prof. Coates. “These decorations helped people to understand what we are doing, and they also got a cool shape to hang on their Christmas tree!”

Prof. Coates accepts that it is very hard to visualise space-time in four dimensions, because our everyday life is three-dimensional. “Imagine yourself standing on a sunny hillside, watching a plane fly overhead,” he says. “The shadow that falls on the valley below will not follow a straight path, even though the plane is flying straight; it will curve.” Similarly, space-time is curved by gravitation, as Einstein’s Theory of General Relativity demonstrated. Prof. Coates hopes that the project will make this visualisation easier.

“A big chunk of the methodology behind this project actually comes from string theory and theoretical physics, and this interplay between physics and maths has been essential,” he says. “Usually we apply fundamental maths to physics, but here, we’ve used physics as a lens to view mathematics.” Software tools to carry out computational algebra at scale have also been developed, in order to find these atomic geometric pieces. These tools will be a key legacy of the project, enabling researchers in the future to carry out mathematical experiments at scale.

“All this would not have been possible without ERC support,” says Prof. Coates. “The ERC is not prescriptive, and it trusts researchers to get on with their work. I’ve been able to spend two years on this and not have to worry about delivering immediate results. This long-term perspective has been vital to our work.” Once the basic pieces of 4D shapes have been identified, grouped and classified, Prof. Coates and his team will turn their focus to an even bigger challenge – 5D shapes.