DESIGNS

If we enumerate the small squares through which the singular mirror
curve passes by 1,2,3,... until the closed curve is complete,
and then reduce all the numbers *modulo* 2
(replacing every number by its reminder,
when dividing it by 2), the result will be a 0-1
(or "black"-"white") mosaic: a * Lunda design*
Lunda designs possess the local equilibrium property:
the sum of the integers in every two border unit squares
with the joint reference point is the same, and the sum
of the integers in the four unit squares between
two arbitrary neighbouring grid points is always
tvice the preceding sum. From this, results
the global equilibrium property: the sum
in each row is equal, and the same holds for the
coloumns. This local, and global equilibrium
property resulting from it, holds as well if we enumerate
the curve and reduce all the numbers

In particular, enumerating a regular curve (with the mirrors
incident to the grid edges) and reducing all the numbers
*modulo* 4, we obtain four-colored Lunda designs,
where every reference point is orderly surrounded by numbers 0,1,2,3
and the disposition of that sequences around the points is alternately
clockwise and anti-clockwise.

The correspondence between monolinear mirror-curves and Lunda-designs is
*many-to-one*, so the same Lunda-design could originate from
some class, consisting of different mirror-curves. Try to find such
classes of mirror-curves.