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## The Complementary Labyrinth

If we turn the inside out of a labyrinth, we obtain the dual labyrinth of it. The dual labyrinth has the same pattern as the original labyrinth, however, the pattern is rotated by a half-circle, and the entrance and the center are exchanged. This has already been extensively described on this blog (see related posts, below).

Now, there is another possibility for a relationship between two labyrinths with the same pattern. In this kind of relationship, the pattern is not rotated, but mirrored vertically. Also – other than in the relationship of the duality – the entrance and the center are not exchanged. At this stage, I term this relation between two labyrinths the complementarity in order to distinguish it from the relationship of the duality.

Here I will show what is meant with the example of the most famous labyrinth.

This labyrinth is the „Cretan“, „Classical“, „Archetype“ or how soever called alternating, one-arm labyrinth with 7 circuits and the sequence of circuits 3 2 1 4 7 6 5, that I will term the „basic type“ from now on.

Figure 1.The Original Labyrinth

Figure 1 shows this type in the concentric style.

The images (1 – 6) of the following gallery (figure 2) show how the pattern of the complementary type can be obtained starting from the pattern of the original type.

Image 1 shows the pattern of the basic type in the conventional form. In image 2 this is drawn slightly different. By this, the connection from the outside (marked with an arrow downwards) into the labyrinth and the access to the center (marked with a bullet point) are somewhat enhanced. This in order to show, that when mirroring the pattern, the entrance and the center will not be exchanged. They remain connected with the same circuits of the pattern. In images 3 til 5 the vertical mirroring is shown, divided up in three intermediate steps. Vertical mirroring means mirroring along a horizontal line. Or else, flipping the figure around a horizontal axis – here indicated with a dashed line. One can imagine, a wire model of the pattern (without entrance, center and the grey axial connection lines) being rotated around this axis until the upper edge lies on bottom and, correspondingly, the lower edge on top. In the original labyrinth, the path leads from the entrance to the third circuit (image 3). With this circuit it remains connected during the next steps of the mirroring (shown grey in images 4, 5 and 6). After completion of the mirroring, however, this circuit has become the fifth circuit.The path thus first leads to the fifth circuit (image 6) of the complementary labyrinth. A similar process occurs on the other side of the pattern. In the original labyrinth, the path reaches the center from the fifth circuit. This circuit remains connected with the center, but transforms to the third circuit after mirroring.

Figure 3: The Complementary Labyrinth

In the pattern of the complementary labyrinth we can find a type of labyrinth that has already been described on this blog (see related posts). It is one of the six very interesting (alternating) labyrinths with 1 arm and 7 circuits. That is to say the one with the S-shaped course of the pathway.

So, what is the difference between the dual and the complementary labyrinth?

Let us remember that the basic type is self-dual. The dual of the basic type thus is a basic type again.

The complementary to the basic type is the type with the S-shaped course of the pathway.

By the way: In this case, the dual to the complementary is the same complementary again, as also the complementary of the basic type is self-dual (otherwise it would not be a very interesting labyrinth).

This opens up very interesting perspectives.

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## The Labyrinth by Al Qazvini

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).

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

## An Eleven-Circuit Cakra Vyuh Labyrinth

A very beautiful labyrinth example (fig. 1) named Cakra-vyuh can be found in Kern’s Book° (fig. 631, p. 294).

Figure 1: Cakra-Vyuh Labyrinth from an Indian Book of Rituals

The figure originates from a contemporary Indian book of rituals. In this, a custom of unknown age, still in practice today, was described, in which the idea of a labyrinth is used to magically facilitate birth-giving. To Kern this is a modified Cretan type labyrinth. I attribute it to a type of it’s own and name it after Kern’s denomination type Cakra-Vyuh (see Related Posts: Type or Style / 14).

The seed pattern is clearly recognizable. One can well figure out that this labyrinth was constructed based on the seed pattern. Despite this, I hesitate to attribute it to the Classical style. For this, the calligraphic looking design deviates too much from the traditional Classical style. The walls delimiting the pathway all lie to a mayor extent, i.e. with about 3/4 of their circumference on a grid of concentric circles. Therefore it has also elements of the concentric style. The labyrinth even somewhat reminds me of the Knidos style with its seamlessly fitting segments of arcs where the walls delimiting the path deviate from the circles and connect to the seed pattern.

Therefore I have not attributed this labyrinth to any one of the known styles, but grouped it to other labyrinths (Type or Style /9). However, I had also drawn this labyrinth type in the Man-in-the-Maze style already (How to Draw a Man-in-the-Maze Labyrinth / 5).

Figure 2: Composition of the Seed Pattern

Fig. 2 shows how the seed pattern is made-up. We begin with a central cross. Tho the arms of this cross are then attached half circles (2nd image). Next, four similar half circles are fitted into the remaining spaces in between. Thus the seed pattern includes now 8 half circles (3rd image). Finally, a bullet point is placed into the center of each half circle. We now have a seed pattern with 24 ends, that all lie on a circle.

In the pattern it can be clearly seen, that the labyrinth has an own course of the pathway. Therefore, to me it is a type of it’s own.

Figure 3: Pattern

Furthermore it is a self-dual, even though, according to Tony Phillips, uninteresting labyrinth (Un- / interesting Labyrinths). This because it is made-up of a very interesting labyrinth with 9 circuits with one additional, trivial circuit on both, the inside and the outside.

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## Type or Style / 6

### What is a Type

What is interesting in labyrinths is the manner in which the pathway takes it’s course through the labyrinth. Therefore I use exclusively the course of the pathway for a typology. This is the approach that is already recognizable in Kern, although it has not been elaborated to a full typology yet.

The course of the pathway can be represented in different ways, for instance by using the level sequence or with the pattern. To me the pattern is the easier way. Therefore my rule is: labyrinths with the same pattern are of the same type. Labyrinths with different patterns belong to different types.

All labyrinths with this pattern are of the Cretan Type

All labyrinths with this pattern are of the Reims Type

All labyrinths with this pattern are of the Chartres Type

asf.

I have already described here at length how the pattern can be obtained (see realted posts below).

Erwin sometimes uses the level sequence to describa a type of labyrinth. This has the great advantage, that a type is directly given a name (even though a somewhat abstract one). So, for instance, type 3 2 1 4 7 6 5. However, there are two reasons, why I do not use the level sequence:

• Only in alternating one-arm labyrinths, there exists exactly one type of labyrinth for each level sequence. If we also consider non-alternating labyrinths where the pathway traverses the axis, there can exist multiple types of labyrinths for the same level sequence.
• In many labyrinths with multiple arms the level sequence is much longer and more complex and therefore less understandable.

This is why I use the pattern for the classification of labyrinths. This approach has advantages and disadvantages.

Types of labyrinths can be clearly defined. Each labyrinth example can be unequivocally assigned to a type. The typology is given in the form of a rule. One has to know and apply this rule. Therefore it is not necessary to provide all possible types in advance. It is sufficient to keep a listing of the types that have already been realized. If a new type is discovered or designed, this can be easily added to the existing ones.

A countless number of types are thinkable. However, in practice, it can hardly be expected that ten thousands or even thousands of types of labyrinths will persist. Rather I expect it to be some hundreds. There exist about 100 different historical types of labyrinths. A comprehensive list of all contemporary labyrinths is missing. Also, many designs and sketches may fall into oblivion. However, theoretically there is a vast number of possible types of labyrinths. This can be already seen from Tony Phillips’ work, that is limited to alternating one-arm labyrinths.

It is therefore necessary to aggregate the types of labyrinths on higher levels. For instance they could be aggregated to sub-groups, groups and families or else. This, however, is also a necessity in other typologies. We have seen this already in the typologies by TLS and BL.

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## Self-dual Labyrinths

Inverting a labyrinth brings us to the dual labyrinth of it. If we now invert the Cretan-type labyrinth, we will obtain another Cretan-type labyrinth, although with the entrance on top.

Figure 1. Inverting the Cretan-type Labyrinth

Fig. 1 shows the process and result of this inversion.

In order to compare the original and dual labyrinths, as previously shown, we isolate the dual labyrinth and carry-over the pattern on it. Then we rotate the dual labyrinth with the pattern lying on it, so that it is directed with the entrance from below, and place it next to the original labyrinth.

Figure 2. The Original and Dual Labyrinths are the Same: Self-dual

As shown in Fig. 2, the original and dual labyrinths are the same. The two labyrinths that are dual to each other have the same pattern, although the pattern is rotated by 180°. This is the case here too. So the right image really shows the pattern rotated by 180°. However, this pattern, after it has been rotated, is self-covering. This is not the case in “normal” dual labyrinths.

Now let us also have a look at the level sequences. As the Cretan-type labyrinth has 7 circuits, we need to use 7 colors.

Figure 3. The Colors of the Circuits

Fig. 3 shows the sequence of the colors. In addition to the first five colors from our last post, we use the color Bordeaux for the circuit that is covered as the 6th and orange for the circuit that is covered as the last by the pathway.

Figure 4. Level Sequences in the Pattern

Fig. 4 shows the level sequence directly on the pattern. As usual, the left image indicates the level sequence into the original and out of the dual labyrinth, whereas the right image indicates the level sequence into the dual and out of the original labyrinth. Both level sequences are identical.

Labyrinths in which the original and dual labyrinths are the same, are referred to as self-dual. These are particular labyrinths and they have a higher inner order than “normal” dual labyrinths.

Figure 5. Pattern of a Dual (left) and Self-dual (right) Labyrinth

This can also be seen in comparing the patterns of dual and self-dual labyrinths (fig. 5). In dual labyrinths (left image), the courses of the first (grey) and the second (black) halves of the pathways are different, whereas they are congruent in self-dual labyrinths (right image).

Some of the most excellent labyrinths are self-dual, such as the Otfrid, Chartres, Reims, Auxerre, Saffron Walden and some others.

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## The Pattern and the Dual Labyrinths

Let us consider the duality once more based on the pattern. The two labyrinths that are dual to each other have the same pattern.

Figure 1. The Pattern and the two Dual Labyrinths

The pattern is no labyrinth. It has no closed form and is indifferent with respect to the outside and inside. It can be transformed into a labyrinth in two directions. In the last post (see related posts below: the Dual Labyrinth) we have unrolled the Ariadne’s Thread of the original labyrinth from below and obtained the pattern using method 2 (see below: From the Ariadne’s Thread to the Pattern – Method 2). Then, we have re-curled in the pattern to the other side, i.e. upwards, and thus obtained the Ariadne’s Thread of the dual labyrinth. This, however, lay with the entrance on top. In order to compare the original and the dual labyrinths, we have rotated the dual so that its entrance was from below.

In rotating a labyrinth we rotate the pattern too. By the way, this can be already seen from my earlier post (see below: What’s the Use of the Pattern?). In this post, fig. 5 showed the Ariadne’s Thread of the Chartres type labyrinth in a representation by Niels Mejlhede Jensen with the entrance on the right side. This is one quarter of a circle anticlockwise against our usual orientation with the entrance from below. Consequently in this figure the pattern was also rotated by one quarter of a circle and standing on its left outer side.

In our case here we have the dual labyrinth rotated by half a circle lying on its head. Here I want to show how by rotating the labyrinth, the pattern is rotated too.

Figure 2. Isolating the Dual Labyrinth

In fig. 2 we first isolate the dual labyrinth and also carry-over the pattern lying on it.

Figure 3. The Pattern of the Original and Dual Labyrinths

Then we rotate the isolated labyrinth with the pattern on it (fig. 3) and place it next to the original labyrinth. Both labyrinths now lie with their entrances from below and the pattern placed on top of the figure. The pattern of the dual labyrinth is the same as the pattern of the original labyrinth, however, rotated by half a circle.

Figure 4. From the Original to the Dual Labyrinth

An important consequence arises from this. As shown in fig. 4 it is also possible to proceed as follows in order to transform the original into the dual labyrinth: In a first step we generate the pattern from the original labyrinth. Then we rotate the pattern by half a circle. Finally we can curl it in again downwards and by this generate the dual labyrinth.

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## The Dual Labyrinth

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“.

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.

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.

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).

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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