From the Ariadne’s Thread to the Pattern

Method 1

In my last post I have shown how the seed pattern can be transformed into the pattern. The same result can be obtained by transforming the Ariadne’s Thread into the rectangular form.

Figure 1. Ariadne’s Thread and Situation of the Seed Pattern

Fig. 1 shows the Ariadne’s Thread of my demonstration labyrinth with the seed pattern highlighted. In addition the situation of the entrance (arrow) and of the center (bullet point) is indicated.

Figure 2. Rotating the Right Half of the Axis…

In fig. 2 we now fix the left half of the axis and rotate the right half anticlockwise a full turn along the circuits. By this, the circuits are continually shortened. Immediately before the right half reaches the left half of the seed pattern on its opposite side, the circuits have reduced to very short lines. But, as can be seen, it is really the circuits of the labyrinth, that connect the ends of both halves of the seed pattern.

Figure 3. … Until it Meets the Left from the Other Side

At the point where both halves meet each other, these remaining pieces of the circuits disappear. In lieu of them the straight of the meander appears. This is composed of the outer vertical lines of the original auxiliary figure of the seed pattern.

Therefore, it is absolutely justified to straighten-out the meander at the point where it intersects with the vertical straight. The lines that connect the ends of the seed pattern really represent the circuits of the labyrinth.

In fig. 3 we have now generated the meander starting from the Ariadne’s Thread, fixing one half of the seed pattern and rotating the other by a full turn. I refer to this way of generating the pattern as method 1. I had fixed the left half and rotated the right half.

Figure 4. Rotating the Left Half of the Axis by a Full Turn

Fig. 4 shows that we could also fix the right half and rotate the left. In the result, this makes no difference.

Figure 5. Result: Pattern with Entrance on Below Right and Center on Top Left

The result of this method is in both cases the same meander that is straightened-out to the pattern as described previously.

Important: Please notice, that after this transformation, in the pattern the entrance lies on the bottom right and the center on top left. This result is against our spontaneous intuition and also contradicts with how we are used to read. It is a result of the applied method 1.

Related posts:

• Seed Pattern and Pattern
• Meanders in Labyrinths – Why Straighten-out?