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Figure 29.

As Lunda-designs may be considered as matrices, it is quite natural to define addition of Lunda-designs in terms of matrix addition: the sum of two (or more) matrices (of the same dimensions) is the matrix in which the elements are obtained by adding corresponding elements (see the example in Figure 29).

The sum of k mxn Lunda-designs may be called a mxn Lunda-k-design. The Lunda-k-designs inherit the following symmetry properties:

  • (i) The sum of the elements in any row is equal to km;
  • (ii) The sum of the elements in any column is equal to kn;
  • (iii) The sum of the integers in two border unit squares of any grid point in the first or last rows or columns is equal to k;
  • (iv) The sum of the integers in the four unit squares between two arbitrary (vertical or horizontal) neighbour grid points is always 2k.
    Once again, properties (i) and (ii) guarantee a global equilibrium for each row and column. Properties (iii) and (iv) guarantee more local equilibria.

    The characteristics (i), (ii), (iii), and (iv) may be used to define Lunda-k-designs of dimensions mxn. The characteristics (iii) and (iv) are sufficient for this definition, as they imply (i) and (ii).

    57Kb 43Kb

    Figure 30.


    Figure 31.


    Figure 32.

    Figure 30 displays the forty 3x3 Lunda-3-designs of the type d4', with white ( = 0) being the colour of the first unit square (with vertices (0,0), (1,0), (1,1), and (0,1)). Figure 31 shows an example of a 8x8 Lunda-4-design. Figure 32 displays examples of a 4x4 Lunda-5-design, an 8x4 Lunda-5-design, and a 5x3 Lunda-4-design. This time the colour chosen for the first unit square is different from white.

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