  We introduce and study a three-folded linear operator depending on three
parameters that has associated a triangular number tilling of the plane. As a
result the set of all triples of integers is decomposed in classes of
equivalence organized in four towers of two-dimensional triangulations. We
provide the full characterization of the represented integers belonging to each
network as families of certain quadratic forms. We note that one of the towers
is generated by a germ that produces a covering of the plane with {L\"oschian}
numbers.
