In the thermodynamic Bethe ansatz approach to 2D integrable, ADE-relat
ed quantum field theories, one derives a set of algebraic functional e
quations (a Y system) which play a prominent role. This set of equatio
ns is mapped onto the problem of finding finite triangulations of cert
ain 3D manifolds. This mapping allows us to find a general explanation
of the periodicity of the Y system. For the A(N) related theories, an
d more generally for the various restrictions of the fractionally supe
rsymmetric sine-Gordon models, we find an explicit, surprisingly simpl
e solution of such functional equations in terms of a single unknown f
unction of the rapidity. The recently found dilogarithm functional equ
ations associated to the Y system simply express the invariance of the
volume of a manifold for deformations of its triangulations.