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Posts Tagged ‘non-alternating’

An interesting labyrinth is reproduced in the book of Kern (fig. 200, p. 119)°. A drawing by Arabian geographer Al Qazvini in his cosmography completed in 1276 is meant to show the ground plan of the residence of the ruler of Byzantium, before the large city of Constantinople was built up.

This non-alternating labyrinth has 10 circuits and a unique course of the pathway. I will show this using the Ariadne’s Thread and the pattern. In my post “From the Ariadne’s Thread to the Pattern – Method 2” (see related posts, below), I have already described how the pattern can be obtained. When deriving the pattern I always start with a labyrinth that rotates clockwise and lies with the entrance from below. The labyrinth by Qazvini rotates in clockwise direction, however it lies with the entrance from above. Therefore I rotate the following images of the labyrinth by a semicircle so that the entrance comes to lie from below. So it is possible to follow the course of the pathway with the Ariadne’s Thread and in parallel see how this is represented in the pattern.

Four steps can be distinguished in the course of the pathway.

Phase 1

The path first leads to the 3rd circuit. The entrance is marked with an arrow pointing inwards. In the pattern, axial sections of the path are represented by vertical, circuits by horizontal lines. The way from the outside in is represented from above to below.

Phase 2

In a second step, the path now winds itself inwards in the shape of a serpentine until it reaches the 10th and innermost circuit. Up to this point the course is alternating.

Phase 3

Next follows the section where the pathway leads from the innermost to the outermost circuit whilst it traverses the axis. In order to derive the pattern, the labyrinth is split along the axis and then uncurled on both sides. As the pathway traverses the axis, the piece of it along the axis has to be split in two halves (see related posts below: “The Pattern in Non-alternating Labyrinths”). This is indicated with the dashed lines. These show one and the same piece of the pathway. In the pattern, as all other axial pieces, this is represented vertically, however with lines showing up on both sides of the rectangular form and a course similarly on both sides from bottom to top.

Phase 4

Finally the pathway continues on the outermost circuit in the same direction it had previously taken on the innermost circuit (anti clockwise), then turns to the second circuit, from where it reaches the center (highlighted with a bullet point).

Related Posts:

°Kern, Hermann. Through the Labyrinth – Designs and Meanings over 5000 Years. Munich: Prestel, 2000.

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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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Up to now we have always considered the patterns of alternating labyrinths. Most of all well-known labyrinths are alternating. In these labyrinths, the pathway does not traverse the (main) axis. Every time when it skips to another circuit it also changes direction (clockwise or anticlockwise). If we transform such labyrinths into the rectangular form, we split the main axis along the central axial wall to which are aligned the turns of the pathway. Both halves are then flipped upwards. By this, the turns of the pathway come to lie on the left and right outer sides of the rectangular form. The pathway, however, is not interrupted. The entrance to the labyrinth and the way to the center lie on the outermost left and right verticals of the rectangular form.

However, there exist also labyrinths in which the pathway traverses the main axis. Two examples of such labyrinths have been repeatedly shown on this blog: the Snail Shell labyrinth by Erwin and the labyrinth of St. Gallen (see related posts). If we want to transform such labyrinths into the rectangular form, the pathway has to be interrupted in the positions where it traverses the (main) axis. This can best be demonstrated with the labyrinth of St.Gallen.

Figure 1. Labyrinth of St. Gallen

Figure 1. Labyrinth of St. Gallen

Fig. 1 shows the labyrinth with the Ariadne’s Thread inscribed and the Ariadne’s Thread isolated. Even it the Ariadne’s Thread, due to the construction of the labyrinth, appears slightly skewed, it is immediatly evident that the pathway in this labyrinth traverses the axis. On its way across the axis it follows the axis in full length from the outside in. Contrastingly the labyrinth has no central axial wall that would connect the innermost with the outermost wall.

If we want to transform this Ariadne’s Thread into the rectangular form, the axial piece of the pathway has to be split in 2 halves.

Figure 2. Transformed into the Rectangular Form

Figure 2. Transformed into the Rectangular Form

Fig. 2  from top to bottom shows the process and its result. As can be seen, the axial segment of the pathway is split on its full length (in two dashed lines), and these are flipped upwards on each side.

Figure 3. The Pattern

Figure 3. The Pattern

Fig. 3 shows the pattern once again. In the rectangular form, the Ariadne’s Thread cannot be drawn in one single line. Multiple lines, this case two interweaving lines, are needed for this. Beginning at the entrance on top left, the first line ends at the outer right side (dashed line). This is the right half of the pathway segment that traverses the axis and therefore was split. Its course is in direction from top to bottom. The second line begins as the dashed line on the outer left side, which is the corresponding left half of the pathway segment that traverses the axis. This line must be drawn in the same direction (from top to bottom) as the right half. Both halves of the same segment of pathway, of course, follow the same direction. These two halves now mark the outermost vertical lines of the rectangular form. The pathway segments for the entrance to the labyrinth and the access to the center lie further inside.

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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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