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Posts Tagged ‘dual’

In the last post I have presented the complementary labyrinth. I did this with the example of the basic type labyrinth. This is a self-dual labyrinth. The complementary is different from the dual labyrinth. This can be better shown using non-self-dual labyrinths. I want tho show this here and for this choose an alternating labyrinth with 1 arm and 5 circuits. As already shown in this blog, there exist 8 such labyrinths (see related post below: Considerung Meanders and Labyrinths). Of these, 4 are self-dual (labyrinths 1, 3, 6, and 8) and 4 are not self-dual (labyrinths 2, 4, 5, and 7).

I thus choose one of the non-self-dual labyrinths, nr. 2, and use the pattern of it. With the pattern, two activities can be performed:

  • Rotate

  • Mirror

Figure 1 shows the result of performing these actions with pattern 2.

Figure 1. Rotating and Mirroring of the Pattern

Rotation leads to the pattern of labyrinth 4
Mirroring leads to pattern 7

So we have already three labyrinths. Now it is possible to go even further. Rotating the dual again brings it back to the original labyrinth. However, the dual can also be mirrored. This results then in the complementary of the dual. And similarly, the complementary can be rotated, which results in the dual to the complementary.

Mirroring of the dual (pattern 4) leads to the complementary pattern of labyrinth 5
Rotation of the complementary (pattern 7) leads to the dual of it – which is also pattern 5.

Figure 2. Relationships

Figure 2 shows the labyrinths corresponding to the patterns. The labyrinths are presented in basic form (i.e shown with their walls delimiting the pathway) in the concentric style. All four non-self-dual alternating labyrinths with 1 arm and 5 circuits are in a relation of either dualtiy or complementarity to each other.

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If we turn the inside out of a labyrinth, we obtain the dual labyrinth of it. The dual labyrinth has the same pattern as the original labyrinth, however, the pattern is rotated by a half-circle, and the entrance and the center are exchanged. This has already been extensively described on this blog (see related posts, below).

Now, there is another possibility for a relationship between two labyrinths with the same pattern. In this kind of relationship, the pattern is not rotated, but mirrored vertically. Also – other than in the relationship of the duality – the entrance and the center are not exchanged. At this stage, I term this relation between two labyrinths the complementarity in order to distinguish it from the relationship of the duality.

Here I will show what is meant with the example of the most famous labyrinth.

This labyrinth is the „Cretan“, „Classical“, „Archetype“ or how soever called alternating, one-arm labyrinth with 7 circuits and the sequence of circuits 3 2 1 4 7 6 5, that I will term the „basic type“ from now on.

Figure 1.The Original Labyrinth

Figure 1 shows this type in the concentric style.

The images (1 – 6) of the following gallery (figure 2) show how the pattern of the complementary type can be obtained starting from the pattern of the original type.

Image 1 shows the pattern of the basic type in the conventional form. In image 2 this is drawn slightly different. By this, the connection from the outside (marked with an arrow downwards) into the labyrinth and the access to the center (marked with a bullet point) are somewhat enhanced. This in order to show, that when mirroring the pattern, the entrance and the center will not be exchanged. They remain connected with the same circuits of the pattern. In images 3 til 5 the vertical mirroring is shown, divided up in three intermediate steps. Vertical mirroring means mirroring along a horizontal line. Or else, flipping the figure around a horizontal axis – here indicated with a dashed line. One can imagine, a wire model of the pattern (without entrance, center and the grey axial connection lines) being rotated around this axis until the upper edge lies on bottom and, correspondingly, the lower edge on top. In the original labyrinth, the path leads from the entrance to the third circuit (image 3). With this circuit it remains connected during the next steps of the mirroring (shown grey in images 4, 5 and 6). After completion of the mirroring, however, this circuit has become the fifth circuit.The path thus first leads to the fifth circuit (image 6) of the complementary labyrinth. A similar process occurs on the other side of the pattern. In the original labyrinth, the path reaches the center from the fifth circuit. This circuit remains connected with the center, but transforms to the third circuit after mirroring.

Figure 3: The Complementary Labyrinth

In the pattern of the complementary labyrinth we can find a type of labyrinth that has already been described on this blog (see related posts). It is one of the six very interesting (alternating) labyrinths with 1 arm and 7 circuits. That is to say the one with the S-shaped course of the pathway.

So, what is the difference between the dual and the complementary labyrinth?

Let us remember that the basic type is self-dual. The dual of the basic type thus is a basic type again.

The complementary to the basic type is the type with the S-shaped course of the pathway.

By the way: In this case, the dual to the complementary is the same complementary again, as also the complementary of the basic type is self-dual (otherwise it would not be a very interesting labyrinth).

This opens up very interesting perspectives.

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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.

M 2L

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.

DL isolieren

Figure 2. Isolating the Dual Labyrinth

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

Muster o-d

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.

OL-DLnu

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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