  We defined exponential maps with one parameter, associated with geodesics on
the parameter surface. By group theory we proposed a formula of the critical
points, which is a direct sum of the Lie subalgebras at the critical
temperature. We consider the self similar transformations as symmetric
operations. In the opinion of symmetry we analyzed the hexagon lattice system,
and got its three cluster spin states: single, double, and threefold, then its
critical point is calculated. There are two cases for lattice-Ising model in
thermodynamic equilibrium. In one case the periodic boundary conditions are
present without the infinite self similar transformations; in another the
system is in the possibility of the infinite self similar transformations
without the conditions. We think the real exact critical points close to our
results.
