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).