Readers who have been visiting this blog for some time, will know the following figure.
Figure 1: Snail Shell Labyrinth
Erwin calls it the Snail Shell Labyrinth. He had generated this labyrinth by completing the seed pattern for the Ariadne’s Thread of the Cretan-type labyrinth. But instead of the usual end he completed it around an other one of its ends. So he did, what I refer to as shifting the center. Somebody else also had the same idea, as can be seen in this source on page 14 (scroll down on the link page). Both shifted the center to a neighbouring end in the same quadrant, but did not continue with this process.
What might be unknown to the most is, that the Snail Shell Labyrinth is the only other figure that can be generated by rotating the seed pattern of the Cretan. With the statements from my previous posts about the shifting of the center (see related posts below), we are able to demonstrate this.
Figure 2: Seed Pattern
The seed pattern for the Ariadne’s Thread of the Cretan has 16 ends (fig. 2). So there are 16 possibilities to place the center. However, these reduce to four, as the seed pattern is made up of four similar quadrants. After four rotational steps, the seed pattern is self-covering.
Figure 3: Symmetry
In addition, each of these quadrants is symmetric in itself (fig. 3). This reduces the range of figures to two pairs. Each pair is made up of the same figure rotating either clockwise or anti-clockwise.
Figure 4: The ends and the pairs of figures
This is shown in fig. 4. Connecting the first end with the center generates the first figure in clockwise rotation. I therefore label this end as 1 →. If we connect the second end with the center, this generates figure 2, in clockwise rotation (2 →). The third end connected to the center generates also figure 2, although in anticlockwise rotation (← 2). And finally, the fourth end generates figure 1 rotating anticlockwise (← 1). Figure 1 is the Cretan, figure 2 the Snail Shell Labyrinth. We now have completed the first quadrant. The fifth end of the seed pattern is the first of its second quadrant and with this, the whole process begins anew.
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This is shown in fig. 4. Connecting the first end with the center generates the first figure in clockwise rotation. I therefore label this end as 1 →. If we connect the second end with the center, this generates figure 2, in clockwise rotation (2 →). The third end connected to the center generates also figure 2, although in anticlockwise rotation (← 2). And finally, the fourth end generates figure 1 rotating anticlockwise (← 1). Figure 1 is the Cretan, figure 2 the Snail Shell Labyrinth. We now have completed the first quadrant. The fifth end of the seed pattern is the first of its second quadrant and with this, the whole process begins anew.
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