  It is known that any infinite frieze comes from a triangulation of an annulus
by Baur, Parsons and Tschabold. In this paper we show that each periodic
infinite frieze determines a triangulation of an annulus in essentially a
unique way. Since each triangulation of an annulus determines a pair of
friezes, we study such pairs and show how they determine each other. We study
associated module categories and determine the growth coefficient of the pair
of friezes in terms of modules as well as their quiddity sequences.
