Once again I come back to the shifting of the center. This time I want to show the result of rotating the seed pattern of the Näpfchenstein labyrinth.
Ill. 1 shows this labyrinth and its seed pattern for the Ariadne’s Thread.
Rotating this seed pattern results in two figures. In fact this is only one figure, the labyrinth itself, either in clockwise or anti-clockwise rotation. This pair of figures is then repeated five times.
A closer look at the seed pattern shows the reasons. The seed pattern is not only composed of 2 similar halves, but made up of six identic sixth parts. Moreover, each of these sixths is symmetric in itself (Ill. 3). This explains why the two figures differ only in their rotational direction (clock- or anticlockwise).
A seed pattern that is composed of multiple similar elements, after a certain number of rotation steps will look exactly the same as in its original position. The Näpfchenstein seed pattern is very well suited to illustrate this.
In ill. 4 we first fix the original position of the seed pattern (grey) in fig. a. Then, we place a copy (black) on top of it (fig. b). In the original position, this copy exactly covers the original. Therefore only the black copy can be seen. Third, let’s rotate the (black) copy by one step and connect the next end of it with the center (fig. c). Now the original is uncovered and becomes visible. In fig. d we rotate the copy one step further, and it completely covers the original again.
After only two rotation steps the seed pattern is self-covering. This, of course, generates the same figure as in the original position.
The seed pattern of Rockcliffe Marsh is self-covering after six rotation steps. The seed pattern of my demonstration labyrinth needs a full circle rotation.
The number of rotation steps needed for a seed pattern to be self-covering corresponds with the number of figures that can be generated with it. With the seed pattern of my demonstration labyrinth, 12 figures, with the one of Rockcliffe Marsh 6 figures, and with the Näpfchenstein seed pattern, 2 figures were generated. It has to be kept in mind that the same figure in clockwise and anti-clockwise rotation is counted as two figures. In some seed patterns with an inherent symmetry in each element, the number of different figures may be further reduced.