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Stellenbosch mathematician offers historical view of connection between unique mathematical forms
Author: Wiida Fourie-Basson (Media: Faculty of Science)
Published: 29/04/2024

​​Stellenbosch University mathematician Bruce Bartlett's article on computing the roots of a quintic polynomial was chosen as the feature article in the April edition of the Notices of the American Mathematical Society (AMS), regarded as the world's most widely read magazine aimed at professional mathematicians.

The article, titled “The Quintic, the Icosahedron, and Elliptic Curves" tells the story of the remarkable correspondence between these unique mathematical forms while at the same time framing this correspondence in a new and systematic way. In the process, he also uses this correspondence to come up with an efficient numerical method to calculate the roots of the quintic.

Moreover, the article is written in such a way that it takes the reader on a “carpet ride through the mathematics of the last four centuries". This includes Kepler's Platonic model of the solar system (1597), Gauss' diary entry from 1799 and experiencing “the atmosphere in Trinity College Hall during the wonderful moment Ramanujan burst on to the scene in 1913", he writes in the article. Srinivasa Ramanujan (1887-1920) is regarded as one of India's greatest mathematical geniuses, and the focus of the film “The man who knew infinity", released in 2016.

Bartlett says the article has been more than ten years in the making. As a topologist, his interest was first sparked when listening to a maths talk by the French mathematician Etienne Ghys at the International Congress of Mathematics in Madrid in 2006: “In this incredible talk, Ghys mixed three-dimensional topology, which is the field I specialise in, with elliptic curves, a field in number theory that I was not familiar with."

Bruce Bartlett.jpg

In 2014 he wrote a blog post on this connection between seemingly unrelated objects for The n-Category Café, a group blog on math, physics and philosophy.

However, in 2020, while on a visit at the Max Planck Institute for Mathematics in Germany, he attended two lectures by the renowned number theorist Don Zagier on the icosahedron, the Roger-Ramanujan identities and beyond: “I simply had to attend these lecturers. I sat in the front row and took diligent notes!" he recounts.

As it turned out, even a world leader in number theory like Zagier still found the relationship between the quintic and the icosahedron a bit mysterious: “I realised then that there is scope for an article, providing a coherent picture of how to compute the roots of a quintic polynomial via the icosahedron and elliptic curves. I realised that there were very few people, if any, who had this global picture of the way it worked," he explains.

But Barlett is also a firm believer that, in the words of his intellectual mentor John Baez, “(In) mathematics, every sufficiently beautiful object is connected to all others".  

In this regard, it is worth remembering that the icosahedron is one of the five Platonic solids. Bartlett explains: “The ancient Greeks discovered that there are only five ways that you can build a closed convex shape in three dimensions by gluing regular polygons, such as equilateral triangles, or squares, or pentagons etc., together. These are the five Platonic solids."

The first four correspond with the elements of nature: the tetrahedron with fire, the octahedron with air, the cube with earth, and the dodecahedron with water.

“The icosahedron, on the other hand, played a mysterious role. According to Plato the gods used it for arranging the constellations of the whole heaven. It certainly has an air of mystery around it!" he adds.

Back in the 21st century, Bartlett says the whole theme of his research in quantum topology is to try and connect objects from diverse mathematical areas together: “This is what excites me about mathematics, relating A to B where A and B live in very different areas. That is also why I need to have a broad background."

Taking deep dives into mathematical history is another topic that excites him. While writing the article for AMS, he had to obtain copyright to include a picture of equation 5 on page 9 of Ramanujan's original handwritten letter to Godfrey Hardy at Trinity College in 1913.

After a long search, and with the help of the makers of the film, The man who knew infinity, it was traced to Cambridge University Library. An editor at AMS, Masahiro Yamada, managed the communication with Cambridge and after paying a fee, Bartlett is now the proud owner of a reproduction right to page 9 of Ramanujan's letter to Hardy.

“So, now I also own a little piece of mathematical history!"  

On the photo above, SU mathematician Bruce Bartlett with a copy of equation 5 on page 9 of Ramanujan's original handwritten letter to Godfrey Hardy at Trinity College in 1913. Photo: Wiida Fourie-Basson