G4G's Virtual Celebration of Mind | June 2023

Frieze patterns and Farey graphs

In the 1970s, Coxeter studied arrays of integers that form friezes in the plane. He and Conway discovered an elegant way of classifying these friezes using triangulated polygons. Recently, research in friezes has revived, in part of because of connections to continued fractions, cluster algebras, hyperbolic geometry, and other topics. Here we will take a tour of the basic theory of friezes using the geometry and arithmetic of an infinite graph called the Farey graph. If you’re lucky, we might catch a glimpse of a wild frieze…

Ian Short is the Director of Research in Mathematics and Statistics at the Open University, a distance-learning university in the UK. His research interests are geometry and dynamics. He lives in Woburn Sands, nearby to Woburn Abbey, which is the family seat of the Duke of Bedford and was the workplace of the geologist John Farey (1728–1798), whose letter to the Philosophical Magazine in 1816 sparked key ideas used in this talk.