Frieze patterns over algebraic numbers

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 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 Q(\sqrt{d}) where d