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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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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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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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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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The Cretan is the most frequently encountered type of labyrinth, and so for this type we can find a broad range of individual variants. Here I want to show some examples that are of particular interest for various reasons. Unless stated otherwise, all figures are sourced from the book Through the Labyrinth by Hermann Kern. The details can be found here.

Figure 1. Pylos

Figure 1. Pylos

This graffito on a clay tablet from Pylos dates from 1200 BCE at the latest and is the oldest securely dated labyrinth. It shows the Cretan-type on a rectangular layout.

Figure 2. Silver  coin, Knossos

Figure 2. Silver coin, Knossos

This figure shows the labyrinth with a concentric layout on a silver coin from Knossos, ca. 190 100 BCE. The center of the labyrinth covers with the middle of its circuits. The axis, however, is somewhat eccentric, as the pathway reaching the center is aligned centrally.

Figure 3. Walahfrid

Figure 3. Walahfrid

On this drawing from a parchment manuscript by Walahfrid Strabo (808-849), the labyrinth is shown in full concentric form. The axial wall that connects the innermost with the outermost wall of the labyrinth is aligned centrally with the center.

The following examples show, that variants of the layout are not limited to standard forms, such as circles or rectangles.

Figure 4. Heart labyrinth

Figure 4. Heart labyrinth

This heart-labyrith by Mario Höhn is of the Cretan-type, although with an additional closed circuit at the inside. Not all circuits are in parallel course (as with a supposed 7-lane roundabout). Circuits 7 and 6 are limited to the right heart chamber. Circuit 5 leads to the left chamber, where it is connected with the closed 8th circuit.

Figure 5. Double labyrinth

Figure 5. Double labyrinth

An other method to generate a heart labyrinth was used by Marty Kermeen and Jeff Saward. They apply a double labyrinth (DL). This is made up of two identic labyrinths (L) that are mirrored horizonally and connected to each other. So the actual labyrinth is one of these two part-labyrinths. This is a Cretan-type projected on a half-hearted layout.

Figure 6. Abhuyumani Tantra

Figure 6. Abhuyumani Tantra

This tantric drawing from Rajasthan, India, 19th century, shows the labyrinth arranged on three quarters of a circle – most of it actually is unrolled to a semi circle. Only the turn from the first to the fourth circuit covers the whole third quadrant. The fourth quadrant is not covered by the figure.

Figure 7. Nîmes

Figure 7. Nîmes

This roman mosaic labyrinth from Nîmes, France, 1st century, has an inconspicious rectangular outline. But, like no other, it shows that the layout of a labyrinth is not only limited to its outline (circle, rectangle, heart, etc.). It is also important to consider how the course of the pathway is organized within this outline form. And this is really tricky. Just try to identify the seed pattern of this labyrinth. The course of the pathway is special in at least three points.

  • All circuits do not rotate by a full (360°) but only a 3/4 (270°) circle. This is the same as with the Indian labyrinth described above. It is a sort of a 3/4 labyrinth. However, the layout covers all four quadrants.
  • The inner circuits are completely embedded in quadrants 1 and 2. Normally all circuits cover all quadrants.
  • Only the outer 4 circuits cover all quadrants.

These shiftings and transformations vary the layout of the labyrinth so that it is barely recognizable.

But what makes me classify all these different examples as Cretan-type labyrinths? What do all these have in common? What defines a Cretan-type labyrinth has been repeatedly described on this blog and elsewhere:

  • One-arm labyrinth
  • alternating, i.e., the pathway does not traverse the axis
  • 7 circuits
  • level sequence: 3-2-1-4-7-6-5.

It is important to keep in mind that we are dealing with alternating labyrinths. There exist also non-alternating labyrinths. Only among the alternating labyrinths there is exactly one type of labyrinth for each level sequence. The other way round, this allows us to unequivocally describe each type of an alternating labyrinth by its level sequence.

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With Arnol’d’s figures we already know all 8 alternating labyrinths with one arm and five circuits. Alternating means that the path does not traverse the axis. Whenever the pathway changes to another circuit it also changes direction from clockwise to anticlockwise (or vice-versa). Among these 8 labyrinths with 5 circuits there are 4 uninteresting, 2 interesting and 2 very interesting examples.

With an increasing number of circuits the number of different labyrinths increases rapidly. So there are 42 labyrinths with 7 circuits: 20 uninteresting, 16 interesting and 6 very interesting examples. The seed patterns for the walls and the patterns of the interesting and very interesting labyrinths are accessible on Tony Phillips’ website. These patterns generate six beautiful very interesting labyrinths. I therefore have reproduced the patterns and added the labyrinths in script form (i.e. on circular layout, with the entrance at the base of the design and in clockwise rotation). Here are the results:

Figure 1

Figure 1

Fig. 1: This is the well-known, most widespread labyrinth – the Cretan.

Figure 2

Figure 2

Fig. 2: A principle that appears also among Arnol’d’s figures: serpentine from the inside out. This can also be conceived as serpentine enclosed in a single double-spiral like meander (Erwin’s type 4 meander).

Figure 3

Figure 3

Fig. 3: A beautiful pattern with an S-shaped course of the pathway.

Figure 4

Figure 4

Fig. 4: Also a beautiful pattern – sort of a Yin/Yang movement.

Figure 5

Figure 5

Fig. 5: A serpentine enclosed by a two-fold double-spiral like meander (Erwin’s type 6 meander).

Figure 6

Figure 6

Fig. 6: This principle is also well known from Arnol’d’s figures: double-spiral type meander here in its three-fold manifestation (Erwin’s type 8 meander).

The Cretan type labyrinth therefore belongs to a group of six matching self-dual interesting alternating  one-arm labyrinths with 7 circuits.

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