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Inverting a labyrinth brings us to the dual labyrinth of it. If we now invert the Cretan-type labyrinth, we will obtain another Cretan-type labyrinth, although with the entrance on top.

O-D Kretisch

Figure 1. Inverting the Cretan-type Labyrinth

Fig. 1 shows the process and result of this inversion.

In order to compare the original and dual labyrinths, as previously shown, we isolate the dual labyrinth and carry-over the pattern on it. Then we rotate the dual labyrinth with the pattern lying on it, so that it is directed with the entrance from below, and place it next to the original labyrinth.

SD Kret

Figure 2. The Original and Dual Labyrinths are the Same: Self-dual

As shown in Fig. 2, the original and dual labyrinths are the same. The two labyrinths that are dual to each other have the same pattern, although the pattern is rotated by 180°. This is the case here too. So the right image really shows the pattern rotated by 180°. However, this pattern, after it has been rotated, is self-covering. This is not the case in “normal” dual labyrinths.

Now let us also have a look at the level sequences. As the Cretan-type labyrinth has 7 circuits, we need to use 7 colors.

UF 7 Farben

Figure 3. The Colors of the Circuits

Fig. 3 shows the sequence of the colors. In addition to the first five colors from our last post, we use the color Bordeaux for the circuit that is covered as the 6th and orange for the circuit that is covered as the last by the pathway.

UF Muster Kret

Figure 4. Level Sequences in the Pattern

Fig. 4 shows the level sequence directly on the pattern. As usual, the left image indicates the level sequence into the original and out of the dual labyrinth, whereas the right image indicates the level sequence into the dual and out of the original labyrinth. Both level sequences are identical.

Labyrinths in which the original and dual labyrinths are the same, are referred to as self-dual. These are particular labyrinths and they have a higher inner order than “normal” dual labyrinths.

Muster d sd

Figure 5. Pattern of a Dual (left) and Self-dual (right) Labyrinth

This can also be seen in comparing the patterns of dual and self-dual labyrinths (fig. 5). In dual labyrinths (left image), the courses of the first (grey) and the second (black) halves of the pathways are different, whereas they are congruent in self-dual labyrinths (right image).

Some of the most excellent labyrinths are self-dual, such as the Otfrid, Chartres, Reims, Auxerre, Saffron Walden and some others.

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Labyrinths have the following well known properties:

  • Closed form
  • one entrance and a center
  • a pathway that leads from the entrance to the center and is the only way back
  • the path is free of crossings or junctions and has no dead ends

However, labyrinths have another property too, that is less well known

  • they can be turned their inside out

Inverting a labyrinth in such a way results in the dual labyrinth of it. I refer to the baseline labyrinth as the „original“ and to the labyrinth resulting after the inversion as the “dual” labyrinth. The meaning of „original“ here is only in relation with the transformation process to the dual labyrinth. Every labyrinth can be used as baseline and in this respect can be „original“.

OL M OL

Figure 1

We have already shown how the pattern can be obtained from the Ariadne’s Thread (fig. 1 left image). Bending the pattern back downwards (fig. 1, right image) reverses this process and brings us back to the original Ariadne’s Thread.

 

M_US

Figure 2

However, generating the pattern from the original labyrinth is also the first half in the inversion process (fig. 2). So let us continue with it.

 

DL

Figure 3

For this purpose we bend over the pattern and re-curl it in to the other side from where it was uncurled, that is upwards. This results in another labyrinth, which is the dual and lies with the entrance on top (fig. 3).

 

OL-DL

Figure 4

In order to compare both labyrinths, we rotate the dual labyrinth and place it next to the original labyrinth (fig. 4). As can be seen, the two dual labyrints are different, but they have a resemblance. The dual labyrinth has the same pattern. This pattern, however, is followed in the opposite direction. The way out of the original labyrinth corresponds with the way into the dual and vice versa.

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