Frieze patterns over algebraic numbers

Abstract

AbstractConway 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 Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field where . We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases all non‐zero frieze patterns over the rings of integers for have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns.