The Farey framework for SL2-tilings
This website is devoted to the project on using a combinatorial approach to explore the connections between number theory and geometry that arise around Conway–Coxeter friezes and SL2-tilings over integer numbers and other rings, carried out at the Open University, UK.
April 2025 highlights
- Our paper Frieze patterns and Farey complexes has been published in Advances in Mathematics. The arXiv preprint has been updated to the accepted version.
- Meet us at the conference Frieze patterns in algebra, combinatorics and geometry at the CIRM, Marseille, 12–16 May 2025.
- Triangulations and friezes, an interactive demonstration of the connections between triangulated polygons, paths in the Farey graph, friezes, and cluster variables in the cluster algebra of type An, now can also be found at the Visual Cluster Algebras portal, together with other CA-related apps (although the updates appear here first, for now).
Gallery
Two paths in the Farey graph in the Poincaré disk, corresponding to a certain SL₂-tiling
Portions of the Farey graph (grey), Ford circles (blue), and the truncated Farey graph (black) in the upper half-plane
Farey surface of a certain wild SL₂-tiling
The Farey complexes over integers modulo 5, 6, 7: icosahedron, hexagonal torus, Klein quartic
The project is funded by EPSRC grants EP/W002817/1, EP/W524098/1, and EP/T518165/1 (S.B.'s internship).
SL2-tilings