  The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were
introduced by Tateo in the 90's in the study of the integrable deformation of
conformal field theory by the thermodynamic Bethe ansatz method. The
periodicity property and the dilogarithm identities concerning these Y-systems
were conjectured by Tateo, and only a part of them have been proved so far. In
this paper we formulate these Y-systems by the polygon realization of cluster
algebras of types A and D, and prove the conjectured periodicity and
dilogarithm identities in full generality. As it turns out, there is a
wonderful interplay among continued fractions, triangulations of polygons,
cluster algebras, and Y-systems.
