In several previous posts I have shown, that different variants can exist for a certain labyrinth or seed pattern.
Illustration 1. Variants of the Same Seed Pattern
In Ill. 1 I again show some variants of the seed pattern for the Ariadne’s Thread of my demonstration labyrinth. This same seed pattern can be drawn e.g. with a circular, elliptic, petal-shaped or rectangular outline. The outline figure is only an auxiliary figure. The seed pattern itself is formed by the system of lines within this outline figure. Depending on the shape of the outline figure, also the orientation and rounding of the seeds may somewhat differ. However, they are always ordered the same way. On top left one (not-nested) turn, on bottom left two nested and on the right three nested turns. Which variant of the seed pattern is best suited depends on the purpose for which it is used.
In this post I want to show the relationship between the seed pattern and the pattern. For this purpose, the rectangular variant is best suited. The seed pattern can be transformed to the pattern in a few steps.
Illustration 2. From Seed Pattern to Meander
The left figure of ill. 2 shows the rectangular variant of the seed pattern. This is also shown as baseline in grey in the right figure. As a first step, the right half of the seed pattern is shifted against the left (shown in red), until it comes to lie on the other side of the left half.
Illustration 3. From Meander to Pattern
The result of this shift is a meander. It is one of Arnol’d’s figures. This meander is in a next step straightened-out, as has already been shown here. For this, the right half of the seed pattern is shifted somewhat further to the left. The ends opposite each other are then connected with lines.
Illustration 4. Pattern
The result of this process is shown in ill. 4. Apparently, in transforming the meander to the pattern, the first and most important step is the horizontal straightening-out. By this the situation of the circuits in the pattern are made apparent. Next, one can easily straighten-out the axial segments and finalize the pattern.
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