3.2.5  Lunda designs

If we enumerate by 1, 2, 3, ¼ the small squares through which the single mirror curve passes until the closed curve is completely enumerated, and then reduce all the numbers modulo 2 (replacing every number by its remainder, when dividing it by 2), the result will be a 0-1 (or "black"-"white") mosaic: a Lunda design (Gerdes, 1997, 1999). Lunda designs possess the local equilibrium property: the sum of the integers in every two border unit squares with the joint vertex is the same, and the sum of the integers in the four unit squares between two arbitrary neighboring grid points is always twice the preceding sum. This gives the global equilibrium property: the sums in all rows are equal, and the same holds for the columns. Local and the resulting global equilibrium property hold even if we enumerate the curve and reduce all the numbers modulo 4. 

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 vertex is orderly surrounded by numbers 0,1,2,3 and the disposition of the sequences around the points is alternately clockwise and anti-clockwise. 

The correspondence between monolinear mirror-curves (i.e., the corresponding arrangements of mirrors) and Lunda designs is many-to-one, so the same Lunda design could originate from several classes, consisting of different mirror arrangements. Open question is defining classes of mirror arrangements. 

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