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Posts Tagged ‘complementary’

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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It is not possible to design a complementary counterpart for each labyrinth. The complementary is obtained by horizontally mirroring of the pattern whilst the connections between the entrance, the center and their corresponding circuits in the labyrinth are left uninterrupted. If the entrance and the access to the center are situated on the same side of the axis, this does not work.

Figure 1. Alternating Labyrinth with an even Number of Circuits

Fig. 1 shows this with the example of the alternating, one-arm labyrinth with 6 circuits and the sequence of circuits 3 2 1 6 5 4. As can be seen from the pattern (figure in the middle), the entrance and the access to the center are situated on the same side of the axis. The pathway first leads to the 3rd cricuit and finally reaches the center from the 4th circuit. If we want to mirror this pattern and let the connections with the entrance and the center unbroken, then the lines intersect at the position marked with a black circle. Such a figure is not free of crossroads any more and thus no labyrinth. In alternating labyrinths with an even number of cirucits, therefore, there exist no complementary labyrinths.

Now there are also non-alternating labyrinths with an even number of circuits in which the entrance to the labyrinth and the access to the center lie on the opposite sides of the axis. The labyrinth shown in fig. 2 is such an example and has already been repeatedly discussed in this blog.

Figure 2. Non-alternating Labyrinth with an even Number of Circuits

This non-alternating, one-arm labyrinth with 6 circuits has the sequence of circuits 3 2 1-6 5 4. That is the same sequence of cirucits as in labyrinth shown in fig. 1 with the difference, that the pathway traverses the axis between circuit 1 and 6. So we are here presented a labyrinth with an even number of circuits, but with the entrance and access to the center on the opposite sides of the axis. Despite this, it is not possible to form a complementary labyrinth to this. If we mirror the pattern vertically without interrupting the connections with the entrance and the center, this results in two crossroads (highlighted with black circles).

Thus, complementary counterparts can only be found in alternating labyrinths with an odd number of circuits.

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There are sets of four labyrinths each, from which the labyrinths are in a complementary or dual relationship with each other. This is also expressed in their sequences of circuits. If we write down the sequences of circuits of complementary labyrinths below each other, they add up at each position to One greater than the number of circuits. In fig. 1 I show what this means.

Figure 1. Sequences of Circuits in Complementary Labyrinths

First we write down the sequence of circuits for each of the four patterns. The patterns in the same column are complementary. Next we extract the sequences of circuits of dual labyrinths 2 and 4 and in the line below write the sequences of circuits of dual labyrinths 7 and 5. Now we can add the numbers below each other and will find that at each position they sum up to 6. This is 1 greater than the number of 5 circuits.

Now there is another relationship between the sequences of circuits. This is illustrated in figure 2.

Figure 2. Sequences of Circuits in Dual-Complementary Labyrinths

The sequences of circuits of the dual-complementary labyrinths are mirror-symmetric. Thus, in this case, the labyrinths that are in a diagonal relationship to each other are considered. Labyrinth 5 is the complementary of the dual (4) and the dual of the complementary (7), respectively, i.e. the dual-complementary to labyrinth 2. This connection is highlighted by a black line with square line ends. The sequences of circuits of these labyrinths are also written in black color. If we write the sequence of circuits of labyrinth 2 in reverse order this results in the sequence of circuits of labyrinth 5 and vice versa (black sequences of circuits).
Labyrinth 7 is the complementary of the dual (2) and the dual of the complementary (5), i.e. the dual-complementary to labyrinth 4. This is highlighted by a grey line with bullet line ends. The sequences of circuits of these labyrinths are also written in grey. Also in this case it is true: the sequence of circuits of labyrinth 4 written in reverse order corresponds with the sequence of circuits of labyrinth 7 and vice versa.

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It is known that there are 8 alternating labyrinths with 1 arm and 5 circuits (see “Considering Meanders and Labyrinths”, related posts, below). Of these, four are not self-dual. These four all are in a relationship to each other via the duality and complementarity (see “The Complementary versus the Dual Labyrinth”, related posts, below). The other four labyrinths are self-dual.

I had already pointed to the relationship between complementary and self-dual labyrinths (see “The Complementary Labyrinth”, related posts, below). Here I want to elaborate on it further. For this purpose I use the same form of diagram I had already used in my previous post (see “The Complementary versus the Dual Labyrinth”). I also use the same numbers of the labyrinths according to the numbering of Arnol’d’s meanders (see “Considering Meanders and Labyrinths”), that underlie them.

Figure 1. Labyrinths 1 and 6

The first of the Arnol’d’s labyrinths, number 1, is self-dual. In the diagram, the dual is situated in the same row, the complementary in the same column with the original labyrinth. The dual of number 1 is again number 1 (what actually is the meaning of selfdual). The complementary of number 1 is number 6. And – of course – is the dual to the complementary again number 6. So in the case of self-dual labyrinths, we only captured two different labyrinths, whereas it were four in the case of not self-dual labyrinths.

Thus, two more labyrinths are still missing. We need another diagram to capture labyrinths number 3 and number 8 (fig. 2).

Figure 2. Labyrinths 3 and 8

And, indeed, these two are complementary to each other. So in self-dual labyrinths, only two different labyrinths are in a relationship to each other.

Here the question arises: Do there also exist self-complementary labyrinths? Up to now we have not yet found such a labyrinth. So let us remember, what self-dual imples. The patterns of the original and self-dual labyrinths are self-covering. In fig. 3 I show what that means. The two patterns in the same row are dual. If we shift them together, we can easily see, what I mean.

Figure 3. Self-dual patterns are self-covering

Thus, self-complementary would imply that the original and complementary pattern would also be self-covering.

Figure 4. Complementary patterns are not self-covering

Fig. 4 shows, that even though there is a certain similarity between these two patterns, they are not self-covering. In my opinion there are no self-complementary labyrinths. This is because vertical mirroring with uninterrupted connections to the entrance and center modifies the sequence of circuits. This, however, woult have to remain unaltered.

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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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