  This paper introduces a new systematic algorithm for constructing periodic
Euclidean weaving diagrams with combinatorial arguments. It is shown that such
a weaving diagram can be considered as a specific type of four-regular periodic
planar tiling with over or under information at each vertex. Therefore, a
weaving diagram can be constructed using two sets of cycles, one to build a
tiling, and a second to define the crossing information. However, this
construction method does not guarantee the uniqueness of the diagram, so we
define the notion of equivalence classes of weaving diagrams using the concept
of crossing-matrices. Finally, we present a classification of our periodic
structures according to the minimum number of crossings on a unit cell.
