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Among the one-arm labyrinths we have not found any pairs of uninteresting labyrinths complementary to each other (see related posts, below). In labyrinths with multiple arms, however, such pairs do exist, at least if we consider labyrinths as uninteresting in which the path enters on the outermost circuit or reaches the center from the innermost circuit. This is shown in the following example.

Figure 1. Complementary, Uninteresting Labyrinths

Labyrinth a has 2 arms and 3 circuits. The pathway enters on the outermost circuit. Therefore it is an uninteresting labyrinth. The path also reaches the center from the outermost circuit.

The complementary of it, labyrinth b, is also an uninteresting labyrinth. In this, the path enters the labyrinth on the innermost circuit and also reaches the center from the innermost cirucit.

So far, this is nothing special. But in this labyrinth we can observe another special feature. This can be seen, if we also view the two duals of these labyrinths. This is shown in the already familiar manner in figure 2.

Figure 2. The Dual and the Complementary Labyrinths are the Same

The dual (b) to the original labyrinth (a) ist the same as the complementary (c). The dual (d) to the complementary (c) is the same as the original (a). The two labyrinths that are dual-complementary to each other are the same.

Now this is not valid for all pairs of complementary uninteresting labyrinths. However, other labyrinths exist, in which this is also the case. In figure 3 I show two such examples of labyrinths and their patterns (only originals). In these labyrinths also, the complementary and the duals are the same.

Figure 3. Other Labyrinths with this Property

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Among all one-arm labyrinths with up to and including 7 circuits, there are no two uninteresting labyrinths complementary to each other. The reason for this is that in such labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit (see related posts, below). However, there exist uninteresting labyrinths with more than 7 circuits in which this is not the case.

In order to show this, I begin with the example of the 11-circuit Cakra-Vyuh labyrinth (see related posts). Figure 1 shows this labyrinth and the pattern of it.

Figure 1. The 11-circuit Cakra Vyuh Labyrinth

 

As can be seen, the pathway enters the labyrinth on the first circuit and reaches the center from the innermost circuit. So, the outer- and innermost circuits can simply be cut-off (grey lines in the right image). This then results in a labyrinth with 9 circuits, in which the pathway does not enter on the outermost circuit and doesn’t either reach the center from the innermost circuit. The pattern of this labyrinth is shown in figure 2.

Figure 2.The Pattern of the Uninteresting Labyrinth with 9 Circuits

Because we removed the grey circuits, the course of the pathway in the remaining pattern is from top right to bottom left. If we want to show the pattern in the usual form, we have to mirror it horizontally. This does not affect the pattern itself nor the labyrinth related to it, except for the labyrinth changing its rotational direction (see related posts).

Even though the pathway of this labyrinth enters on the 3rd circuit and reaches the center from the 7th circuit, this is an uninteresting labyrinth. This, because it is made up of two elements of the type Knossos on circuits 1 – 3 and 7 – 9 (indicated with brackets in the right image) and three internal trivial cirucits 4, 5, 6 between them (indicated with dashes). Although this labyrinth is uninteresting, it is self-dual.

Parenthesis: This labyrinth has similarities with the well known basic type (former: Cretan type) labyrinth. However, the basic type is a very interesting (that is interesting and self-dual) labyrinth.

Figure 3. The Pattern of the Basic Type Labyrinth

As shown in figure 3, this is also made-up of two elements of the type Knossos. However, between these there is only one circuit. And this is by no means trivial as it is needed to connect the two elements. But adding further circuits in the shape of a serpentine will result in an uninteresting labyrinth.

Let us get back to the uninteresting labyrinth with 9 circuits. How does the complementary labyrinth look like? Is it may be also an uninteresting labyrinth?

Figure 4. The Two Complementary Labyrinths with 9 Circuits

In order to generate the complementary, we mirror the original labyrinth vertically and let the connections with the environment and the center uninterrupted. Then the pathway enters on the 7th circuit and reaches the center from the 3rd circuit. The three trivial internal circuits are still recognizable. However, they are enclosed by the axial pieces of the pathway that lead into the labyrinth and to the center. So they are nested one level deeper. Therefore, this is no more an uninteresting, but an interesting, and, as it is self-dual, a very interesting labyirnth.

Thus it seems, that also in larger one-arm labyrinths there are no pairs of uninteresting labyirnths that are complementary to each other.

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There are 42 different one-arm alternating labyrinths with 7 circuits. Among these there is one pair of complementary interesting labyrinths. Now how does it look in pairs of complementary unintersting labyrinths? This question has already been indirectly answered in my last post (see related posts below): There is none! This sounds surprising. Therefore I address it further here. The 42 labyrinths form 21 complementary pairs. One of it is composed of 2 interesting labyrinths. We also know there are 22 interesting labyrinths. So the other 20 pairs are made up of an interesting and an uninteresting labyrinth each. Therefore no possibility remains for a pair with two complementary uninteresting labyrinths. What is the reason for that?

As we have seen, only in alternating labyrinths with an odd number of circuits it is possible to derive a complementary (see related posts). In such labyrinths the pathway always enters on an odd-numbered ciruit and also reaches the center from an odd-numbered circuit. Further, in one-arm labyrinths the pathway cannot enter the labyrinth on the same circuit from which it reaches the center.

In uninteresting labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The complementary is derived by mirroring. By this, the outermost is transformed to the innermost circuit and vice versa. If in an original labyrinth the pathway enters on the first circuit, it is an uninteresting labyrinth. In the complementary the path will enter on the innermost circuit. Thus the complementary is not an uninteresting labyrinth, unless the path would reach the center from the innermost circuit. This, however is not possible, as it already enters the labyrinth on this circuit. The original is an unintersting, but the complementary an interesting labyrinth. The other alternative would be that the path in the original labyrinth reached the center from the innermost circuit. But then in the complementary it would reach the center from the outermost circuit what is not an unintersting labyrinth. Therefore the complementary could only be an unintersting labyrinth, if the path would enter it on the outermost circuit. This, however is impossible, as the path reaches the center from this circuit.

These results are only valid for one-arm labyrinths with up to 7 circuits. In labyrinths with mulitiple arms, the pathway may reach the center from the same circuit on which it enters the labyrinth. Thus, for example it could enter the original labyrinth on the first circuit and also reach the center from the first circuit. This would consitute an uninteresting labyrinth. In the complementary, the pathway would then enter the labyrinth on the innermost circuit and also reach the center from the innermost circuit, what again would qualify for an uninteresting labyrinth. In one-arm labyriths with more thean 7 circuits the definition of what constitutes an uninteresting labyrinth can be extended. In these cases trivial circuits can be added not only at the outside or inside of smaller interesting labyrinths (what generates uninteresting labyrints) but also on central circuits between other interesting elements at the inside and outside of the labyrinth, what also may generate uninteresting labyrinths.

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I have already elaborated on uninteresting and interesting labyrinths (see related posts, below). Unintersting labyrinths can be generated by simply attaching additional trivial circuits to the outside or inside of smaller labyrinths. Interesting labyrinths cannot be obtained this way. This particularly implies, that in interesting labyrinths the pathway may not enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The dual of an interesting labyrinth is another interesting labyrinth too.

This is different if we derive the complementary labyrinth of an interesting labyrinth. The resultung labyrinth may very well be an uninteresting labyrinth. Complementary labyrinths exist only for alternating labyrinths with an odd number of circuits. To obtain the complementary, the pattern of the original labyrinth is vertically mirrored without interrupting the connections between the outside or center of the labyrinth with their corresponding circuits. Labyrinths with an odd number of circuits always have a central circuit. When the pattern is mirrored, this circuit remains in position, whilst the other circuits change their positions symmetrically around it.

Figure 1. Mirroring

In a labyrinth with seven circuits, e.g., the central circuit is the one with number 4. After the mirroring, this remains in its position as number 4. The outermost circuit, number 1, transforms to the innermost circuit and obtains number 7, circuit 2 changes to circuit 6, circuit 3 to circuit 5, and vice versa.

Now, if in an interesting labyrinth the pathway first leads to the innermost circuit or reaches the center from the outermost circuit, then the complementary to this labyrinth will be an uninteresting labyrinth. This, because in the complementary, the path will enter the labyrinth on the outermost or reach the center from the innermost circuit. Thus, there exist pairs of complementary labyrinths, both of which are interesting and others in which one of the labyrinths is interesting and the other uninteresting.

Now I want to find out which are the pairs of interesting complementary labyrinths. The website of Tony Phillips provides best material for such a purpose. On one page, HERE, are included the seed patterns (left figures) and the patterns (right figures) of the interesting alternating labyrinths with up to 7 circuits. I therefore reproduce the page in fig. 2 and in the following add some comments to the items indicated with red letters:

Figure 2. Interesting Labyrinths

  • a) In addition to the circuits, Tony also counts the exterior (= 0) and the center (= one greater than the number of circuits) of a labyrinth. He refers to this as the depth of the labyrinth. A labyrinth with depth 4, thus, has three circuits, one with depth 6 has 5 circuits and so on.
  • b) Below the two figures (seed pattern und pattern), in each case the sequence of circuits is listed. This also contains the zero for the exterior and the number for the center, here indicated with red boxes. The true sequence of circuits is the sequence of numbers between these boxes.
  • c) If the labyrinth is self-dual, this is indicated as „s.d.“ after the sequence of circuits.
  • d) If this is not the case, anyway only one of each dual example is shown in the figures. However, the sequence of circuits of the dual not shown is listed in parentheses below the sequence of circuits of the labyrinth shown.
  • e) The patterns are drawn in such a manner that the course of the pathway leads from top right to bottom left. This is different from how I do it. I draw the pattern from top left to bottom right. As a consequence, the labyrinth that corresponds with the pattern by Tony rotates anti-clockwise, whereas in my case it rotates clockwise.
  • f) Now, lets consider all interesting (including very interesting) labyrinths with 7 circuits. Of these, there are 22 (6 of them very interesting) interesting labyrinths. In fig. 2 the seed patterns and patterns of only 14 labyrinths are depicted. The missing 8, however, are duals, represented by the sequences of circuits in parentheses.

Among the interesting labyrinths with 7 circuits, only 2 exist, in which the pathway does not enter the labyrinth on the innermost circuit nor reach the center from the outermost circuit. And these two form the only pair of interesting labyrinths complementary to each other. We already know this pair from the first post of this series. It is the basic type labyrinth (g) and the labyrinth with the S-shaped course of the pathway (h).

Figure 3. Complementary and Interesting Labyrinths

These are self-dual and thus very interesting labyrinths. In the other 20 interesting labyrinths, the complementary in each case is an uninteresting labyrinth.

Thus, there are 42 different alternating labyrinths with one arm and 7 circuits. Among these, there are 8 pairs of interesting dual labyrinths, 6 self-dual very interesting labyrinths, but only 1 pair of interesting complementary labyrinths. In addition, there is no pair of interesting complementary labyrinths with less than 7 circuits.

Pairs of complementary interesting labyrinths seem to be relatively rare and thus something special.

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