  We show several properties related to the structure of the family of classes
of two-dimensional periodic continued fractions. This approach to the study of
the family of classes of nonequivalent two dimexsional periodic continued
fractions leads to the visualization of special subfamilies of continued
fractions with torus triangulations (i.e. combinatorics of their fundamental
domains) that possess explicit regularities.Several cases of such subfamilies
are studied in detail; the method to construct other similar subfamilies is
given.
