  The structures of the configuration space of the six-vertex models with
various boundaries and boundary conditions are investigated, and it is derived
that the free energies depend on the boundary conditions, and that they are
classified by the fractal structures. The "n-equivalences" of the boundary
conditions are defined with a property that the models with n-equivalent
boundary conditions result in the identical free energy. The configurations
which satisfy the six-vertex restriction are classified, through the
n-equivalences, into sets of configurations called islands. It is derived that
each island shows a fractal structure when it is mapped to the real axis. Each
free energy is expressed by the weighted fractal dimension (multi-fractal
dimension) of the island. The fractal dimension of the island has a strict
relation with the maximum eigenvalue of the corresponding block element of the
transfer matrix. It is also found that the fractal dimensions of islands take
the absolute maximum when the boundary is "alternative"; in this case, the free
energy equals to the solution obtained by Lieb and Sutherland.
