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Labyrinths have the following well known properties:

  • Closed form
  • one entrance and a center
  • a pathway that leads from the entrance to the center and is the only way back
  • the path is free of crossings or junctions and has no dead ends

However, labyrinths have another property too, that is less well known

  • they can be turned their inside out

Inverting a labyrinth in such a way results in the dual labyrinth of it. I refer to the baseline labyrinth as the „original“ and to the labyrinth resulting after the inversion as the “dual” labyrinth. The meaning of „original“ here is only in relation with the transformation process to the dual labyrinth. Every labyrinth can be used as baseline and in this respect can be „original“.

OL M OL

Figure 1

We have already shown how the pattern can be obtained from the Ariadne’s Thread (fig. 1 left image). Bending the pattern back downwards (fig. 1, right image) reverses this process and brings us back to the original Ariadne’s Thread.

 

M_US

Figure 2

However, generating the pattern from the original labyrinth is also the first half in the inversion process (fig. 2). So let us continue with it.

 

DL

Figure 3

For this purpose we bend over the pattern and re-curl it in to the other side from where it was uncurled, that is upwards. This results in another labyrinth, which is the dual and lies with the entrance on top (fig. 3).

 

OL-DL

Figure 4

In order to compare both labyrinths, we rotate the dual labyrinth and place it next to the original labyrinth (fig. 4). As can be seen, the two dual labyrints are different, but they have a resemblance. The dual labyrinth has the same pattern. This pattern, however, is followed in the opposite direction. The way out of the original labyrinth corresponds with the way into the dual and vice versa.

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Many of those who are involved with labyrinths use the rectangular form, e.g. Jo Edkins, Niels Mejlhede Jensen, some authors in Caerdroia and many others. Erwin and myself have used it in many posts on this blog. Not all use the same rectangular form and not all use it the same way. However they all intend to achieve a better understanding of the labyrinth. In the following I show some examples with the rectangular form of the Chartres labyrinth.

RFChartresSteafel

Figure 1

Thorn Steafel (fig. 1) uses the rectangular form for the walls obtained with method 1 in order to compare the patterns of the Chartres and the Bayeux labyrinths (Steafel T. Reappraising the Bayeux Labyrinth. Caerdroia 2014; 43: 40-45).

RFChartresEdkins

Figure 2

Jo Edkins shows on his website the rectangular form  for the Ariadne’s Thread using method 1 and analyzes the course of the pathway (fig. 2).

diagram_chartres_rechteck_dual

Figure 3

The same rectangular form (for the Ariadne’s Thread, applying method 1) is used by Erwin in this post (fig. 3). He analyzes the course of the pathway and the duality. This latter can be seen in the different numberings of the circuits on the left and right outer sides. In all three rectangular forms obtained with method 1, the entrance is at bottom right and the center on top left.

RF Chartres

Figure 4

Since it can be read from top left to bottom right, I always use the rectangular form for the Ariadne’s Thread obtained with method 2 (fig. 4). This is the form I refer to as the pattern.

RFChartresJensen

Figure 5

Actually, Niels Mejlhede Jensen uses also the rectangular form for the Ariadne’s Thread obtained with method 2 (fig. 5). However, he starts from a labyrinth the main axis of which is oriented to the right. Therefore, his version of the rectangular form stands on one of the outer sides, the arms are represented in horizontal, the circuits in vertical order. The rectangular form or pattern shows the essential of a labyrinth without confusions that may be caused by circular or polygonal layouts, varying lengths of circuits, decorative artwork etc. This is useful for

  • the analysis of the course of the pathway. This may serve for further purposes such as
  • comparing labyrinths in order to identify communities or differences between labyrinths
  • presenting the inner structure and particularities of specific labyrinths
  • the research of relationships between different labyrinths
  • the demonstration of an important general property of labyrinths: the duality

The pattern provides an unambigous criterion for grouping similar and distinguishing between different labyrinths.

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The pattern is a transformation of the Ariadnes’s Thread from the closed form into the rectangular form. This is not a labyrinth any more. The exterior of the labyrinth is represented by the area above the pattern, the interior by the area below the pattern. The circuits have transformed to horizontal lines. The arms are all oriented vertically.

AF Demo

Figure 1. The Ariadne’s Thread of my Demonstration Labyrinth

Fig. 1 shows the Ariadne’s Thread of my demonstration labyrinth. In accordance with the direction into the labyrinth, I number the circuits from the outside in. Thus 1 is the outermost, 5 the innermost circuit.

M Demo

Figur 2. The Pattern of my Demonstration Labyrinth

Fig 2 shows how this is represented in the pattern. There, 1 is the uppermost, 5 the lowest horizontal line. In the transformation, the axis is split in a left and right half. These halves come to lie on the left and right outer verticals of the pattern. As long as the pathway follows a circuit, in the pattern this is represented by a horizontal course. When it changes to another circuit, it moves axially and by this forms the axis. In the pattern this is represented vertically. However, this can only be seen clearly in labyrinths with multiple arms.

AF Comp

Figure 3. The Ariadne’s Thread of the Compiègne-Type Labyrinth

Fig. 3 shows the Ariadne’s Thread of the Compiègne labyrinth. This labyrinth has also 5 circuits, but 4 arms. Labyrinths with multiple arms generally are composed of a main axis. This is where the pathway enters the labyrinth and from which it also reaches the center. In addition these labyrinths have one or more side-arms.

First, it has to be made clear, how we want to refer to the arms of the labyrinth. I give the number that corresponds with the number of arms of a labyrinth to its main axis. So, in four-arm labyrinths, I number the main axis with 4. I start the enumeration of the arms with the first side-arm next to the main axis in clockwise direction. The reason for this is, that I always start with the labyrinth orientated such that the entrance is at the bottom and the labyrinth in clockwise rotation when I transform it into the rectangular form.

M Comp

Figure 4. Pattern of the Compiègne-Type Labyrinth

Fig. 4 shows how this affects the pattern. The main axis is split. This is the same as with the only axis in one-arm labyrinths. Both halves of the main axis come to lie at the left and right outer verticals. The side-arms are not split for the transformation. In a four-arm labyrinth, therefore, we can find five vertical lines in the pattern. Two for both halves of the main axis and one for each side-arm. (By the way: I have shifted the labellings of the arms. In the pattern these should lie on top according to the positioning of the labels on the outside of the Ariadne’s Thread, however it reads better this way.)

So, the pattern can be thought as lying on a grid of horizontal and vertical lines. The horizontal lines indicate the circuits, and the path in the pattern follows on these horizontal lines. The vertical lines indicate the arms of the labyrinth. These lie between or aside the turns of the pathway.

What these considerations also show is that we here have read the pattern in one direction. Keep this in mind, it is important.

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Recently I became aware of this illustration:

The Rauner Library from Hanover (USA) acquired recently a specially remarkable manuscript. The manuscript is a copy of the Taj Torah produced in Yemen c. 1400-1450. This is one of only three known Hebrew manuscripts with illustrated carpet pages.

The labyrinth in the Taj Torah

The labyrinth in the Taj Torah (Illustration with kind permission of © Rauner Library)

It is a 6 circuit Jericho labyrinth. The walls are drawn with a thick red line and with a black double line. The way into the middle (Ariadne’s thread) is indicated through a banner. With this the structure of the labyrinth is revealed very well. The entrance is on top.

In a great number of manuscripts the city of Jericho is shown as a labyrinth or in the center of a labyrinth. This tradition is proved in different cultural spheres from the 9th up to the 19th century. The labyrinth type used for the Jericho Labyrinth is from the Classical 7 circuit (Cretan) on to the Chartres labyrinth.

Under them are also some 6 circuit labyrinths with 7 walls which are probably based on the Jewish tradition of the seven walls around the city of Jericho. They show an advancement of the labyrinth form.

The oldest known Jericho labyrinth with the same alignment as in the Taj Torah can be seen on a page of a Hebrew Old Testament which was completed by Josef von Xanten in 1294.

That is the reason why Andreas Frei name this type as “von Xanten”.

The Jericho Labyrinth (type von Xanten) in a Hebrew Bible

The Jericho Labyrinth (type von Xanten) in a Hebrew Bible from 1294 / Source: Hermann Kern, Labyrinths, fig. 225

In the book of Hermann Kern one can find other examples of this type. One is found in the Farhi Bible from the 14th century.

The Jericho Labyrinth (type von Xanten) in the Farhi Bible

The Jericho Labyrinth (type von Xanten) in the Farhi Bible from 1366-1383 / Source: Hermann Kern, Labyrinths, fig. 227

On a Hebrew scroll from the 17th century is this drawing. Here, as well as in the upper labyrinth, the entrance is at the bottom. The first turn goes to the right.

The Jericho Labyrinth (type von Xanten) on a Hebrew scroll

The Jericho Labyrinth (type von Xanten) on a Hebrew scroll from the 17th century / Source: Hermann Kern, Labyrinths, fig. 229

The essential of a labyrinth figure can be shown through a geometrical construction in  a drawing.

The Jericho Labyrinth type von Xanten

The Jericho Labyrinth type von Xanten

On the following drawings the labyrinth is mirrored, the path first turns to the left. The path sequence, numbered from the outside inwards (in green), is 0-3-4-5-2-1-6-7. If one numbers the paths from the inside outwardly (in blue) the path sequence will be 0-1-6-5-2-3-4-7.

The original Jericho Labyrinth type von Xanten

The original Jericho Labyrinth type von Xanten

If one draws a labyrinth according to this order, one receives the dual labyrinth. This looks here differently than the original one. This shows that this type of labyrinth is not selfdual as for example the Cretan 7 circuit labyrinth. One obtains a new labyrinth with another structure.

For more information about that context I recommend the website of my co-author Andreas Frei about the pattern (in German). Link >

The dual Jericho Labyrinth type von Xanten

The dual Jericho Labyrinth type von Xanten

Here is the alignment of the original Labyrinth depicted as a rectangular diagram:

The pattern of the Jericho Labyrinth (type von Xanten) as diagram

The pattern of the Jericho Labyrinth (type von Xanten) as diagram

In this diagram with the entrance below on the right, and the center on the top on the right too, the path structure can also be presented very nicely. It is simply Ariadne’s thread as an uninterrupted line in angular form. Andreas calls this the pattern, and shows it a little bit differently. However, the essential things are identical.

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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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I have shown with the example of my demonstration labyrinth how the pattern of a labyrinth can be obtained. This was a one-arm labyrinth. Of course, it is also possible to transform labyrinths with multiple arms into the rectangular form.

Compiegne

Figure 1. Compiègne

I will show this here with the Compiègne labyrinth as an example (fig. 1). This is a labyrinth with four arms. It is presented with the walls shown. Erwin has already used this type of labyrinth in this blog (see related posts). In order to obtain the rectangular form, I will use the Ariadne’s Thread and apply method 2, as usual.

Lage Achsen

Figure 2. The Ariadne’s Thread, situation of the arms

Fig. 2 shows the baseline situation. The labyrinth is represented by the Ariadne’s Thread with the entrance at the base and in clockwise rotation. The main axis and the side-arms are highlighted.

Muster Merhachs Meth2

Figure 3. Flipping the Arms

As can be seen in fig. 3, both side-arms on the left and right side are flipped upwards by about 1/4 of the arc of a circle. There they meet the upper side arm which remains unchanged. Only the main axis is split into two halves, and these are flipped upwards each by ca. Half the arc of a circle, where they attach to the side-arms.

Muster MA Erg2

Figure 4. Finalization of the Pattern

Fig. 4 shows the straightening-out process and, as a result of it, the pattern of the Compiègne labyrinth.

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Method 2

In the last post I have shown a first method of how to transform the Ariadne’s Thread into the rectangular form. For this, one of the halves of the axis was fixed and the other rotated by a full turn along the circuits. This resulted in the pattern with the entrance on bottom right and the center on top left. Here I will show a second method.

Lage KS

Figure 1. Ariadne’s Thread and Situation of the Seed Pattern

We start from the same baseline situation as in method 1. The labyrinth is presented with it’s Ariadne’s Thread with the entrance at the bottom and in clockwise rotation (fig. 1).

Muster Meth2

Figure 2. Rotating Both Halves of the Axis Upwards by Half a Circle

In method 2, however, each half of the axis is rotated by half a turn along the circuits (fig. 2).

Both halves then meet on top of the circuits. Perhaps, this figure shows even better, how by flipping up both ends of the axis the circuits are shortened from full circles to short lines.

Muster Erg2

Figure 3. Result: Pattern with Entrance on Top Left and Center on Bottom Right

After straightening-out the result shows the same pattern as in method 1. However it now lies with the entrance on top left and the access to the center on bottom right.

In both methods we started from the same labyrinth in the same basic situation. Both methods lead to the same pattern. However, in method 1, the pattern lies with the entrance on bottom right and the center on top left. In method 2 this is rotated by 180 degrees so that the entrance lies on top left and the center on bottom right. This orientation of the pattern corresponds better with the way we are used to read. For that reason, I prefer method 2.

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