Where there are no Complementary Labyrinths

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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Sequence of Segments in One-arm Labyrinths

With the coordinates for segments from the last post (see related posts below) we have now found an understandable notation for the sequence of segments of labyrinths. Here it seems important to me to add that such coordinates can also be used for one-arm labyrinths. I will show this with the examples for which I had already shown the sequences of circuits (see related posts). For this, each circuit has to be divided into two segments.

Partitioning of Circuits in Segments

Next we write the sequences of segments for the three examples and also compare them straightaway with their sequences of circuits.

 

 

A unique notation for one-arm labyrinths can also be achieved, if we can write two different numbers on the same circuit, one for each side of the axis. For this, the circuits have to be partitioned into two segments. This allows us to write unique sequences of segments for alternating and non-alternating labyrinths. Also it is possible to use the same form of notation in one-arm and multiple-arm labyrinths. However, this notation will always need 14 coordinates for each one-arm labyrinth with 7 circuits. This is clearly more digits than are needed for the sequences of cirucits with separators.

 

 

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The Level Sequence in One Arm Labyrinths

In an earlier post „Type or Style / 6“ (see related posts, below) I had already mentioned the level sequence. And I had stated two reasons for why I do not use it for naming types of labyrinths.

  • Among the one-arm labyrinths only in alternating labyrinths there exists exactly one type of labyrinth for each level sequence. If we also consider non-alternating labyrinths, in which the pathway traverses the axis, there can exist multiple courses of the pathway for the same level sequence.
  • In labyrinths with multiple arms the level sequence may rapidly increase to a length and complexity that is difficult to memorize.

Here I want to address the first issue further. I do this because there is a very simple solution for it. In one-arm labyrinths every circuit is represented by one number. In real practice only few of the larger labyrinths will have more than 15 – 17 circuits. Most one-arm labyrinths have a markedly smaller size. Therefore these labyrinths could be quite simply be named with their level sequence. But there remains the problem with the ambiguity. Erwin had elaborated on it in his post “The Classical 7 Circuit Labyrinth with Crossed Axis“ (see related posts, below). I will illustrate it here and use some figures of Erwin’s post.

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Figure 1. Level Sequence 3 2 1 4 7 6 5

In Figure 1 three labyrinths with the level sequence 3 2 1 4 7 6 5 are shown. The first image shows the alternating Cretan type, the second and third images show non-alternating labyrinths with the same level sequence. In the second image, the pathway traverses the axis when changing from the 1st to the 4th circuit. In the third image it traverses the axis from the 4th to the 7th circuit. (There is an other labyrinth with the pathway traversing the axis twice, first from the 1st to the 4th and second from the 4th to the 7th circuit). We thus are here presented with the only one alternating and several non-alternating types of labyrinths with the same level sequence.

Now there is a simple solution, to take account of this in the level sequence. For this it has to be considered, that the single numbers (not numerals) of the level sequence are separated. This separation can be obtained in different ways, using blanks, commas, semicolons etc. These separators, however, can also be used to indicate how the path will continue on the next level. Therefore we could e.g. define: if the path changes direction from the former to the next circuit, we will separate the numbers with a vertical slash. If, on the other hand, the path continues in the same direction and thus traverses the axis, we separate with a hyphen. This enables us to specify the level sequence so that it is unique also in non-alternating labyrinths. I show this in figure 2 using the images from figure 1.

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Figure 2. Level Sequence with Separators


Here we see for each labyrinth the unique level sequence with separators. The sequence of numbers is the same 3 2 1 4 7 6 5 in all three labyrinths. However, whereas in the alternating Cretan type all numbers are separated by slashes (as the path always changes direction when progressing from one circuit to an other), the level sequence in the second labyrinth is written with a hyphen between 1 and 4, and the level sequence in the third image with a hyphen between 4 and 7.

Indeed, the notation can be even simplified by separating with blanks and using hyphens only to indicate where the pathway traverses the axis. The level sequences would then be written as follows:

for the  1st image: 3 2 1 4 7 6 5
for the  2nd. image: 3 2 1-4 7 6 5
for the  3rd image: 3 2 1 4-7 6 5

What matters is that in the level sequence it is indicated where the path traverses the axis. With this specification it is now possible to give a unique level sequence to each course of the pathway and thus a unique name to each alternating and non-alternating type of labyrinth.

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How to Draw a Man-in-the-Maze Labyrinth / 6

Non-alternating Labyrinths

In all previous posts of this series with the exception of the second part (see related posts below) I have shown alternating labyrinths. In alternating labyrinths the pathway does not traverse the axis. However, there exist also labyrinths in which the path traverses the axis (in multiple-arm labyrinths: the main axis). These are termed non-alternating. A beautiful example of such a labyrinth is depicted in a manuscript from the 10./11. century of the Stiftsbibliothek St. Gallen. Erwin has already presented it on this blog, and I have published on it in Caerdroia 38 (2008).

Illustration 1. St. Gallen Labyrinth

Illustration 1. St. Gallen Labyrinth

From part / 2 of this series, we know that in principle also non-alternating labyrinths can be drawn in the MiM-style, as the Snail Shell labyrinth is non-alternating. The pathway of this labyrinth traverses the axis twice. Once when it skips from the first to the second circuit and second when skipping from the second inner to the innermost circuit.

Illustration 2. The Ariadne's Thread

Illustration 2. The Ariadne’s Thread

The pathway of the St.Gallen labyrinth (ill. 2), however, comes in clockwise from the outer circuit, turns to the right and moves axially to the innermost circuit, where it turns to the left and continues without changing direction (clockwise). How does this affect the seed pattern and its variation into the MiM-style of this labyrinth?

Illustration 3. Seed Patterns Compared

Illustration 3. Seed Patterns Compared

Ill. 3 shows the seed pattern of my demonstration labyrinth from part / 5 of this series (figures a and b) and compares it with the seed pattern of the St. Gallen labyrinth (figures c and d). The seed pattern of the demonstration labyrinth has one central vertical line. This represents the central axial wall to which are aligned the turns of the pathway (fig. a). This is the case with all alternating labyrinths. Variation of seed patterns of alternating labyrinths into the MiM-style leaves the central line and the innermost ring untouched (fig. b). The auxiliary figures of alternating labyrinths all have two vertical spokes and an intact innermost ring.

This is different with the labyrinth of St. Gallen. The seed pattern of this labyrinth has two equivalent vertical lines (fig. c). Between these lines the pathway continues along the central axis. If we vary this seed pattern into the MiM-style, we find no central wall and the innermost ring interrupted (fig. d). The auxiliary figure of the St. Gallen labyrinth therefore has no vertical spoke.

Illustration 4. Labyrinth of St. Gallen in the MiM-style

Illustration 4. Labyrinth of St. Gallen in the MiM-style

Non-alternating labyrinths can be drawn in the MiM-style in the same way as alternating labyrinths. The seed pattern of the St. Gallen labyrinth has two elements with single and two elements with two nested turns, and in addition the segment of the path that traverses the axis. In the MiM-auxiliary figure this seed pattern covers two circuits. This corresponds with the elements that are made-up of two nested turns. The pathway segment traversing the axis needs no additional circuit, as for this the innermost ring is interrupted to let the path continue through the middle of the seed pattern.

Illustration 5. My Logo in the MiM-style

Illustration 5. My Logo in the MiM-style

And, finally, here is my logo in the MiM-style (ill. 5).

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