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Posts Tagged ‘Demonstration Labyrinth’

The pattern is a transformation of the Ariadnes’s Thread from the closed form into the rectangular form. This is not a labyrinth any more. The exterior of the labyrinth is represented by the area above the pattern, the interior by the area below the pattern. The circuits have transformed to horizontal lines. The arms are all oriented vertically.

AF Demo

Figure 1. The Ariadne’s Thread of my Demonstration Labyrinth

Fig. 1 shows the Ariadne’s Thread of my demonstration labyrinth. In accordance with the direction into the labyrinth, I number the circuits from the outside in. Thus 1 is the outermost, 5 the innermost circuit.

M Demo

Figur 2. The Pattern of my Demonstration Labyrinth

Fig 2 shows how this is represented in the pattern. There, 1 is the uppermost, 5 the lowest horizontal line. In the transformation, the axis is split in a left and right half. These halves come to lie on the left and right outer verticals of the pattern. As long as the pathway follows a circuit, in the pattern this is represented by a horizontal course. When it changes to another circuit, it moves axially and by this forms the axis. In the pattern this is represented vertically. However, this can only be seen clearly in labyrinths with multiple arms.

AF Comp

Figure 3. The Ariadne’s Thread of the Compiègne-Type Labyrinth

Fig. 3 shows the Ariadne’s Thread of the Compiègne labyrinth. This labyrinth has also 5 circuits, but 4 arms. Labyrinths with multiple arms generally are composed of a main axis. This is where the pathway enters the labyrinth and from which it also reaches the center. In addition these labyrinths have one or more side-arms.

First, it has to be made clear, how we want to refer to the arms of the labyrinth. I give the number that corresponds with the number of arms of a labyrinth to its main axis. So, in four-arm labyrinths, I number the main axis with 4. I start the enumeration of the arms with the first side-arm next to the main axis in clockwise direction. The reason for this is, that I always start with the labyrinth orientated such that the entrance is at the bottom and the labyrinth in clockwise rotation when I transform it into the rectangular form.

M Comp

Figure 4. Pattern of the Compiègne-Type Labyrinth

Fig. 4 shows how this affects the pattern. The main axis is split. This is the same as with the only axis in one-arm labyrinths. Both halves of the main axis come to lie at the left and right outer verticals. The side-arms are not split for the transformation. In a four-arm labyrinth, therefore, we can find five vertical lines in the pattern. Two for both halves of the main axis and one for each side-arm. (By the way: I have shifted the labellings of the arms. In the pattern these should lie on top according to the positioning of the labels on the outside of the Ariadne’s Thread, however it reads better this way.)

So, the pattern can be thought as lying on a grid of horizontal and vertical lines. The horizontal lines indicate the circuits, and the path in the pattern follows on these horizontal lines. The vertical lines indicate the arms of the labyrinth. These lie between or aside the turns of the pathway.

What these considerations also show is that we here have read the pattern in one direction. Keep this in mind, it is important.

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In several previous posts I have shown, that different variants can exist for a certain labyrinth or seed pattern.

KSAF_var

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.

KS Umf1

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.

KS Umf2

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.

KS Muster

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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