In an earlier post I have rotated the seed pattern for the Ariadne’s Thread of my demonstration labyrinth and generated 12 different figures. Six of these figures rotate clockwise, the others anticlockwise.
Each labyrinth with five circuits has a seed pattern with 12 ends. Thus, the framework presented in my earlier post can also be used to rotate the seed pattern of other labyrinths with five circuits. I have done this with the core-labyrinth of Rockcliffe Marsh (Arnol’d’s figure 8).
Ill. 1 shows the Rockcliffe Marsh labyrinth on the left with its core-labyrinth marked. On the right, the script version of the core-labyrinth is shown.
Ill. 2 compares the seed pattern of my demonstration labyrinth (left figure) with the one of Rockcliffe Marsh (right figure). The seed pattern of Rockcliffe Marsh is made up of 2 similar halves. This is a characteristic of self-dual labyrinths. In my demonstration labyrinth, the figure that results when connecting the end 7 of the seed-pattern with the center (figure 7) is the dual of figure 1. Self-dual means, that the two duals are identic. Therefore, in Rockcliffe Marsh, figure 7 is identic with figure 1. The same holds for figure 2 and 8 and so forth. It is therefore sufficient to only connect the first six ends of the Rockcliffe Marsh seed pattern with the center, as the ends 7 to 12 will simply reproduce the figures 1 – 6. In the seed pattern of Rockcliffe Marsh the ends 7 – 12 therefore were not numbered.
Ill. 3 shows the result. The numbers of the figures indicate which end of the seed pattern was connected with the center to generate the figure.
- First: the number of different figures reduces to six. Three of them rotate clockwise, three anti-clockwise.
- Second: A closer look reveals that there are only three different figures, each in clockwise and anti-clockwise rotation. These pairs of figures have been arranged on the same line in the illustration (figure 1 and 6, fig. 2 and 5, and fig. 3 and 4).
The reason for this is that the seed pattern of Rockcliffe Marsh not only is made up of 2 similar halves. In addition, each of these halves is symmetric around the dashed line (illustration 4).
Self-duality reduces the number of different figures from 12 to 6, the symmetry of the seeds in both halves reduces it further to only three different figures.