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Posts Tagged ‘rectangular form’

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).

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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).

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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.

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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.

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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.

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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.

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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.

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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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Summary

As is the case with the labyrinth itself and the seed pattern, there are also two representations of the rectangular form: this can be represented either with the walls or with the Ariadne’s Thread. In addition, there are two methods to obtain the rectangular form and therefore two versions of it. Ill. 1 summarizes this with the example of my demonstration labyrinth.

L:SP:P Rep

Illustration 1. Overview

This illustration shows on the first line the labyrinth (figures 1), on the second line the seed pattern (figures 2), on the third line the rectangular form obtained with method 1 (figures 3) and on the bottom line the rectangular form obtained with method 2 (figures 4). Each of these are shown in the representation with the walls (left figures a) and with the Ariadne’s Thread (right figures b).

  • When we speak of a „labyrinth“ we usually mean the labyrinth in its representation with the walls. This is shown in fig. 1 a. But also the representation with the Ariadne’s Thread is in widespread use and generally well known (fig. 1 b). This is usually simply referred to as the Ariadne’s Thread.
  • Fig. 2 a shows the seed pattern for the walls, fig. 2 b the seed pattern for the Ariadne’s Thread. As Erwin and I have written so much about this in recent posts, I don’t elaborate more on it here.
  • If we start from the labyrinth (fig. 1 a) or from the Ariadne’s Thread (fig. 1b) and apply method 1, we will as a result obtain the rectangular forms shown in line 3. Thus, there exists a rectangular form for the walls (fig. 3a) as well as for the Ariadne’s Thread (fig. 3b).
  • If we apply method 2 this results in the rectangular forms of line 4. These are the same as the figures on line 3, although rotated by half the arc of a circle.

For what I termed “rectangular form” here, in the literature we can find also the terms „compression diagram“ or „line diagram“ or else. And, most often, we will encounter rectangular forms for the walls obtained with method 1, i.e. figures like fig. 3 a.

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Illustration 2. Figure 3a

I, however, always use the rectangular form for the Ariadne’s Thread. This is the simpler graphical representation. Furthermore, I use the version obtained with method 2, as the result can be read from top left to bottom right, what corresponds better with the way we are used to read. This figure (e.g. fig. 4 b), the rectangular form for the Ariadne’s Thread obtained with method 2, is what I refer to as the pattern.

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Illustration 3. Figure 4b

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If we dissect a labyrinth along its axis and uncurl it symmetrically on both sides, we can transform it into a rectangular form. The Ariadne’s Thread in the rectangular form is what I refer to as the pattern of a labyrinth. In this pattern, the entrance is at top left and the way into the center at bottom right.

Figure 1. From the Ariadne's Thread to the Pattern

Figure 1. From the Ariadne’s Thread to the Pattern

Figure 1 shows this process in abbreviated form for the Snail Shell labyrinth. The labyrinth, represented by its Ariadne’s Thread is dissected along the axis (2 vertical black lines). Both halves of the axis are flipped upwards by half the arc of a circle around the center. By this, the Ariadne’s Thread is transformed from a circular closed form into a rectangular form.

The path of the Snail Shell labyrinth traverses the axis twice. This is indicated with the black circles. When transforming the Ariadnes Thread into the rectangular form, the segments of the pathway that lie on the axis are dissected too and come to lie on both sides of the rectangular form. These segments are drawn as dashed lines in the pattern and also indicated by circles.

Figure 2. Labyrinths Contained in the Snail Shell Labyrinth

Figure 2. Labyrinths Contained in the Snail Shell Labyrinth

In the Snail Shell labyrinth, a Knossos-type labyrinth (single double-spiral like meander) is included. To this are attached at the inside and outside one circuit with the pathway changing direction. Therefore, in the Snail Shell labyrinth, the labyrinth that corresponds with Arnol’d’s figure 3 is also included. To this are attached at the in- and outside one circuit without the pathway changing direction. This is where the path traverses the axis. Thus, the Snail Shell labyrinth, in the terminology of Tony Phillips, is a non-alternating uninteresting labyrinth.

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