Frieze patterns over algebraic numbers

Michael Cuntz, Thorsten Holm

Published 2023-06-21Version 1

Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jorgensen and the authors. In this note we determine rings of algebraic numbers such that there are finitely many non-zero frieze patterns for any given height. We show that this is only the case for subrings of quadratic number fields $\mathbb{Q}(\sqrt{d})$ where $d<0$. We then concentrate on the imaginary quadratic case and show as a second main result that apart from the cases $d\in \{-1,-2,-3,-7,-11\}$ all non-zero frieze patterns over the rings of integers $\mathcal{O}_d$ for $d<0$ have only integral entries and hence are known as (twisted) Conway-Coxeter frieze patterns.