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The original and the dual labyrinth have the same pattern, although rotated by 180° (see related posts below). The pattern indicates the course of the pathway. On its course, the pathway covers the circuits in a certain sequence. This level sequence indicates which circuit is covered by the pathway as the first, which circuit follows as the second, third and so forth until the last circuit.

But how do we have to label the circuits? Normally they are numbered in ascending order from the outermost to the innermost circuit. This enumeration corresponds with the direction into the labyrinth. However, each path can be followed in two directions – i.e from the outside into the labyrinth and from the inside out.

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Figure 1. Numbering the Circuits

If we follow the direction into the labyrinth (fig. 1, left image), we first meet the outermost and last the innermost circuit. On the way out of the labyrinth (fig. 1, right image), however, we first meet the innermost and last the outermost circuit. The direction in which the path is followed, thus, determines how we have to number the circuits.

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Figure 2. Coloring the Circuits

Next, let us determine the level sequence as follows (fig. 2): We color the circuits in the sequence they are covered by the pathway with the colors yellow, green, blue, magenta, and red. Thus, the circuit covered by the path as the first will be colored yellow, the second green, and so forth til the last circuit in red color. In the order of these colors we then can read the numbers of the circuits.

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Figure 3. Level Sequences of Original Labyrinth

Fig. 3 shows on the left image the level sequence into and on the right image the level sequence out of the original labyrinth. On its course into the labyrinth, the pathway first covers circuit 3, then circuit 4, circuit 5, circuit 2, and finally circuit 1. The level sequence into the labyrinth therefore is: 3-4-5-2-1. However, the level sequence out of the original labyrinth is: 5-4-1-2-3.

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Figure 4. Level Sequences of Dual Labyrinth

Fig. 4 shows the same for the dual labyrinth. The level sequence into the dual labyrinth is: 5-4-1-2-3. The level sequence out of the dual labyrinth is 3-4-5-2-1. As can be seen: the level sequence into the original labyrinth is the same as out of the dual labyrinth: 3-4-5-2-1. Likewise, the level sequence into the dual labyrinth is the same as out of the original labyrinth: 5-4-1-2-3.

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Figure 5. Level Sequences in the Pattern

A closer look at the pattern can explain this (fig. 5). The patten can be followed either from top left to bottom right or in the opposite direction. The direction from top left to bottom right corresponds with the pathway into the original and out of the dual labyrinth. The directions from bottom right to top left on the other hand corresponds with the way into the dual and out of the original labyrinth. As we walk into the original labyrinth we walk out of the dual labyrinth and vice versa.

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Let us consider the duality once more based on the pattern. The two labyrinths that are dual to each other have the same pattern.

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Figure 1. The Pattern and the two Dual Labyrinths

The pattern is no labyrinth. It has no closed form and is indifferent with respect to the outside and inside. It can be transformed into a labyrinth in two directions. In the last post (see related posts below: the Dual Labyrinth) we have unrolled the Ariadne’s Thread of the original labyrinth from below and obtained the pattern using method 2 (see below: From the Ariadne’s Thread to the Pattern – Method 2). Then, we have re-curled in the pattern to the other side, i.e. upwards, and thus obtained the Ariadne’s Thread of the dual labyrinth. This, however, lay with the entrance on top. In order to compare the original and the dual labyrinths, we have rotated the dual so that its entrance was from below.

In rotating a labyrinth we rotate the pattern too. By the way, this can be already seen from my earlier post (see below: What’s the Use of the Pattern?). In this post, fig. 5 showed the Ariadne’s Thread of the Chartres type labyrinth in a representation by Niels Mejlhede Jensen with the entrance on the right side. This is one quarter of a circle anticlockwise against our usual orientation with the entrance from below. Consequently in this figure the pattern was also rotated by one quarter of a circle and standing on its left outer side.

In our case here we have the dual labyrinth rotated by half a circle lying on its head. Here I want to show how by rotating the labyrinth, the pattern is rotated too.

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Figure 2. Isolating the Dual Labyrinth

In fig. 2 we first isolate the dual labyrinth and also carry-over the pattern lying on it.

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Figure 3. The Pattern of the Original and Dual Labyrinths

Then we rotate the isolated labyrinth with the pattern on it (fig. 3) and place it next to the original labyrinth. Both labyrinths now lie with their entrances from below and the pattern placed on top of the figure. The pattern of the dual labyrinth is the same as the pattern of the original labyrinth, however, rotated by half a circle.

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Figure 4. From the Original to the Dual Labyrinth

An important consequence arises from this. As shown in fig. 4 it is also possible to proceed as follows in order to transform the original into the dual labyrinth: In a first step we generate the pattern from the original labyrinth. Then we rotate the pattern by half a circle. Finally we can curl it in again downwards and by this generate the dual labyrinth.

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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“.

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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.

 

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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.

 

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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).

 

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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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For every labyrinth exists a second or dual one. And in special cases the dual one looks like the original one. Then this is a self-dual labyrinth.

These connections should be explained here.

Andreas Frei has done this on his website under the topic “Grundlagen” (basics), to this day only in German. I expressly recommend to take a look at it, there are some meaningful drawings also.

Here again we will see it from the practical side. Hence, it is a continuation of the post from the 1st of September, 2013 about the circular 7 circuit labyrinths. Through the dual labyrinths here we will get six more to add to the seven there. So we will have 13 new labyrinths in all.

How we will reach for that, should be shown step by step. Maybe a little bit awkwardly, but I hope, understandably.

We number all labyrinths from the outside inwards in black. “0” stands for the  outside and “8” for the center. The path sequence, that is the order in which we walk through the circuits to arrive at the center, is noticed on the bottom left in black.
Then we number all circuits once again from the inside outwardly in green. “0” is now the center and “8” is now the outside. We write down the circuits in the order in which we walk them while going backwards from the middle. This path sequence is noticed on the bottom right in green.

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As already mentioned, there is to every (original) labyrinth a second (dual) one. And this arises when we interchange inside and outside, when we turn inside out. The path sequence which we will get, is normally different from the one of the original labyrinth.

If it is the same, we speak of a self-dual labyrinth. Then an internal symmetry is given. Or differently expressed: The rhythm and the motion sequence is the same when stepping inside or outside.  In our examples this applies to the first (well-known Cretan) labyrinth, and to the last, a new labyrinth.


The remaining six have another path sequence and, hence, are to be taken for new, different labyrinths.
Here the six new types (click to enlarge, print or save):

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These examples shows that always at first the middle is circled around. After that one moves inside the the labyrinth and finally one enters the center from the 3rd or the 5th circuit.

In the case of the types introduced in the last article the entry into the center was always from the outermost, the first circuit. Here we have the circling around the middle immediately after stepping into the labyrinth.

The motion sequences are completely different.
It would be of interest exploring that by a temporary or even a permanent labyrinth. Worldwide there are still no labyrinths of this kind.
The shape must not necessarily be perfectly circular. It is important only to adhere to the path sequence.
For the rest, they can be as simply build in sand like the types introduced in the below mentioned post.

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