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GENERAL
SYMMETRY
PROPERTIES



Searching for the common characteristics of Lunda-designs (of dimensions mxn), the following symmetry properties were observed and proven:

  • (i) In each row there are as many black as white unit squares;
  • (ii) In each column there are as many black as white unit squares;
  • (iii) Of the two border unit squares of any grid point in the first or last row, or in the first or last column, one is always white and the other black (see Figure 14);


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    Figure 14: Possible border situations.

  • (iv) Of the four unit squares between two arbitrary (vertical or horizontal) neighbouring grid points, two are always black and two are white (see Figure 15).


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    Figure 15: Possible situations between vertical or horizontal neighbouring grid points.

    Properties (i) and (ii) guarantee a global equilibrium between black and white unit squares for each row and column. Properties (iii) and (iv) guarantee more local equilibria. From (i) it follows that the number of black unit squares of any row is equal to m, and from (ii) that the number of black unit squares of any column is equal to n.

    Inversely, the following theorem can be proven:

  • any rectangular black-and-white design that satisfies the properties (i), (ii), (iii), and (iv) is a Lunda-design.

    In other words, for any rectangular black-and-white design that satisfies the properties (i), (ii), (iii), and (iv) there exists a (rectangle-filling) mirror curve that produces it in the discussed sense (Figure 7). Moreover, in each case, such a mirror curve may be constructed. The characteristics (i), (ii), (iii), and (iv) may be used to define Lunda-designs of dimensions mxn (abreviately: mxn Lunda-designs). In fact, it may be proven that the characteristics (iii) and (iv) are sufficient for this definition, as they imply (i) and (ii).


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