GENERAL
SYMMETRY
PROPERTIES
Searching for the common characteristics of
Lundadesigns (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);
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).
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 blackandwhite design that
satisfies the properties (i), (ii), (iii), and (iv)
is a Lundadesign.
In other words, for any rectangular blackandwhite
design that satisfies the properties (i),
(ii), (iii), and (iv) there exists a (rectanglefilling)
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 Lundadesigns of dimensions mxn
(abreviately: mxn Lundadesigns).
In fact, it may be proven that the characteristics (iii) and
(iv) are sufficient for this definition, as they imply (i) and (ii).
