The Farey framework for SL₂-tilings

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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.

July 2026 highlights

• New preprint: All Y-friezes come from SL₂-friezes.
• 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 A_n, 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

An SL₂-tiling with the greatest possible wild density and unbounded rank

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 (I.S., M.v.S.), EP/W524098/1 (A.Z.), and EP/T518165/1 (S.B.).