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Posts Tagged ‘complementary’

Quite simply: By leaving off the barriers in the minor axes. I have already tried this with the Chartres labyrinth (see related posts below). But is that also possible with every other Medieval labyrinth?

As an example I have chosen the type Auxerre that Andreas showed here recently. This labyrinth is self dual as are Chartres and Reims, therefore of special quality. And they all have a complementary version.

The Auxerre labyrinth

The Auxerre labyrinth

Here the original with all the lines and the path in the labyrinth, Ariadne’s thread. The barriers in the minor axes are identical with those of the Chartres type. There is only another arrangement of the turning points (the lanes 4, 5, 7, 8) in the middle of the main axis.

The original Auxerre labyrinth without the barriers

The original Auxerre labyrinth without the barriers

The barriers are omitted. When drawing Ariadne’s thread, I found that four tracks could not be inserted. Hence, I have anew numbered the circuits and there remain now 7 circuits instead of the original 11. However, this also means that by changing this Medieval labyrinth into a concentric Classical labyrinth through this method no 11 circuit labyrinth is generated, but a 7 circuit.

The 7 circuit circular Cretan labyrinth

The 7 circuit circular Cretan labyrinth

If one looks more exactly at it, one recognises the well-known path sequence: 3-2-1-4-7-6-5-8. We got a Cretan labyrinth in concentric style.


Now we turn to the complementary labyrinth:

The complementary Auxerre labyrinth

The complementary Auxerre labyrinth

The complementary labyrinth is generated by mirroring the original one. The upper barriers remain, right and left they run differently and in the main axis, the turning points shift. The entrance into the labyrinth changes to the middle (lane 9) and the entrance into the center is from further out (lane 3).

The complementary Auxerre labyrinth without the barriers

The complementary Auxerre labyrinth without the barriers

As with the original, four lanes can not be inserted (4, 5, 7, 8). Therefore, the result is again a 7 circuit labyrinth. I renumbered the lanes and have redrawn the labyrinth.

This is how it now looks like:

The complementary 7 circuit circular Cretan labyrinth

The complementary 7 circuit circular Cretan labyrinth

The labyrinth is entered on the 5th lane, the center is reached from the 3rd lane. The path sequence is: 5-6-7-4-1-2-3-8. This labyrinth is not one of the historically known labyrinths. But it showed up in this blog several times (see related posts below). Because it belongs to the interesting labyrinths among the mathematically possible 7 circuit labyrinths.

The surprising fact is that no 11 circuit Classical labyrinth could be generated through the transformation. But for that  the 7 circuit Cretan labyrinth. Therefore we can say that the heart of the Medieval Auxerre labyrinth is the Cretan (Minoan) labyrinth as it is in the Chartres labyrinth.

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In addition to the universally known labyrinth of Chartres and the less popular labyrinth of Reims a third, much less known, very interesting (interesting and self-dual) medieval labyrinth with 4 arms and 11 circuits has been preserved. This is sourced from a manuscript that is stored in the municipal library of Auxerre. Therefore I have named it as Type Auxerre.

At the end of this series I want to show these three labyrinths and their complementaries.

In the three following figures I start with the original labyrinth (image on top left).

From this I obtain the pattern by unrolling the Ariadne’s Thread of it (image on top right).

Then I mirror the pattern vertically without interrupting the connections to the exterior and to the center. This results in the pattern of the complementary labyrinth (image on bottom right).

Then I curl in this pattern to obtain the complementary labyrinth (image on bottom left).

Fig. 1 shows this procedure with the example of the labyrinth of Auxerre. This labyrinth is not recorded in Kern [1]. The image of the original labyrinth was taken from Saward [2] who sourced it from Wright [3].

Figure 1. Labyrinth of Auxerre and Complementary

Fig. 2 shows the labyrinth of Reims and the complementary of it. The image of the original labyrinth was sourced from Kern [1].

Figure 2. Labyrinth of Reims and Complementary

Finally, the labyrinth of Chartres and it’s complementary are presented in fig. 3. The image of the original labyrinth was sourced from Kern [1].

Figure 3. Labyrinth of Chartres and Complementary

With these considerations I wanted to point out that three historical labyrinths exist with a similar degree of perfection as Chartres. Together with their complementaries we now have present six very interesting labyrinths with 4 arms, 11 circuits and a similar degree of perfection.

[1] Kern, H. Through the Labyrinth. Prestel, Munich 2000.
[2] Saward J. Labyrinths and Mazes. Gaia, London 2003.
[3] Wright C. The Maze and the Warrior. Harvard University Press, Cambridge (Massachusetts) 2001.

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Just like the labyrinth from Ravenna, the Wayland’s House labyrinth is also a historical type of labyrinth with 4 arms and 7 circuits. There exist even two different Wayland’s House labyrinths (figure 1).

Figure 1. The Two Wayland’s House Labyrinths

 

I have named them as Wayland’s House 1 and Wayland’s House 2. Wayland’s House 1 first appeared in a manuscript of the 14th century, Wayland’s House 2 in a manuscript of the 15th century, both from Iceland. This can be easily looked up in Kern. In the following I refer to Wayland’s House 1.

In this type of labyrinth, the pathway does not enter on the first circuit and does not reach the center from the last circuit either. Therefore it is an interesting labyrinth. And also the complementary of it is an interesting labyrinth. This, however, is not the most important reason for why I present this type of labyrinth and its’ relatives here. Whereas no existing examples of any relative of the Ravenna labyrinth are known, there exists a contemporary type of labyrinth for each, the dual, complementary and complementary-dual of the Wayland’s House labyrinth.

Figure 2 shows the patterns of the original Wayland’s House labyrinth (a), the dual (b), complementary (c) and complementary-dual (d) labyrinths.

Figure 2. The Relatives of the Wayland’s House Type – Patterns

The original (a) and dual (b) both are interesting labyrinths. The complementaries of them, (c) and (d), are likewise interesting labyrinths.

Figure 3 shows the labyrinths corresponding to the patterns in their basic form with the walls delimiting the pathway shown on concentric layout and in clockwise rotation.

Figure 3. The Relatives of the Wayland’s House Type – Basic Forms

The relatives of the Wayland’s House type (a) are three of the so-called neo-medieval labyirnth types (there are other neo-medieval types of labyrinths too). These relatives are: dual (b) = „Petit Chartres“, complementary (c) = „Santa Rosa“, and complementary-dual (d) = „World Peace“ labyrinth.

So these contemporary types of labyrinths can be easily generated simply by rotating or mirroring of the pattern of Wayland’s House. This having stated I do not mean to pretend, these types of labyrinths have intentionally or knowingly been derived in such a way from the Wayland’s House type. Rather, available information suggests that they were created in a naive way, i.e without their designers having known about this relationship with the Wayland’s House type labyrinth. Nevertheless, actually, these modern neo-medieval labyrinths are the relatives of Wayland’s House.

The Wayland’s House labyrinth at first glance has some similarities with the Chartres type labyrinth. However it is not self dual and its course of the pathway is guided by an other principle yet. And this applies to its relatives too. Therefore the choice of the name „Petit Chartres“ to me seems inconvenient. It seems like this name was chosen because this type of labyrinth originally was designed in the Chartres-style. So this type seems to have been named after its style.

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Among labyrinths with mulitple arms it is also common that one labyrinth is interesting and the complementary to it is uninteresting. An example for this is the labyrinth of the type Ravenna (figure 1).

Figure 1. The Labyrinth of Ravenna

This labyrinth has 4 arms and 7 circuits. The pathway enters it on the innermost circuit and reaches the center from the fifth circuit. It is, thus, an interesting labyrinth. This type of labyrinth has been named after the example laid in church San Vitale from Ravenna. What is really special in this example is the graphical design of the pathway. This is designed by a sequence of triangles pointing outwards. The effect is, that the direction from the inside out is strongly highlighted. This stands in contrast to the common way we use to approach a labyrinth and seems just an invitation to look up the dual of this labyrinth. Because the course of the pathway from the inside out of an original labyrinth is the same as the course from the outside into the dual labyrinth.

I term as relatives of an original labyrinth the dual, complementary, and complementary-dual labyrinths of it. In fig. 2 the patterns of the Ravenna-type labyrinth (a, original), the dual (b), the complementary (c), and the complementary-dual (d) of it are presented.

Figure 2. The Relatives of the Ravenna-type Labyrinth – Patterns

The original (a) and the dual (b) are interesting labyrinths. The complementaries of them are uninteresting labyrinths, because in these the pathway enters the labyrinth on the outermost circuit (c) or reaches the center from the innermost circuit (d). The dual of an interesting labyrinth always is an interesting labyrinth too, the dual of an unintersting is always uninteresting labyrinth too.

Figure 3 shows the labyrinths corresponding to the patterns in their basic form with the walls delimiting the pathway on concentric layout and in clockwise rotation. Presently, I am not aware of any existing examples of a dual (b), complementary (c) or complementary-dual (d) to the Ravenna type labyrinth (a).

Figure 3. The Relatives of the Ravenna-type Labyrinth – Basic Forms

From these basic forms it can be well seen that it seems justified to classify the complementary and complementary-dual labyrinths as uninteresting. The outermost (labyrinth c) and innermost (labyrinth d) respectively walls delimiting the path appear disrupted. Therefore labyrinths c and d seem less perfect than the original (a) and dual (b) labyrinths, where the pathway enters the labyrinth and reaches the center axially.

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By rotating or mirroring one will get dual and complementary labyrinths of existing labyrinths. Or differently expressed: Other, new labyrinths can be thereby be generated.
So I have three more new labyrinths as I can make a complementary one from a new dual labyrinth and I can make a dual one from a new complementary, which are identical. (For more see the Related Posts below).

Seen from this angle I have examined the still introduced 21 Babylonian Visceral Labyrinths in Knidos style and present here the variations most interesting for me. Since not each of the possible dual or complementary examples seems noteworthy.

Many, above all complementary ones, would begin on the first circuit and lead to the center on the last, which is yet undesirable.

Leaving out trivial circuits also will generate new labyrinths. This applies to the last two ones. If you compare the first and the last example you see two remarkable labyrinths: The first with 12 circuits and the last with 8 circuits, but using the same pattern.

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Among the one-arm labyrinths we have not found any pairs of uninteresting labyrinths complementary to each other (see related posts, below). In labyrinths with multiple arms, however, such pairs do exist, at least if we consider labyrinths as uninteresting in which the path enters on the outermost circuit or reaches the center from the innermost circuit. This is shown in the following example.

Figure 1. Complementary, Uninteresting Labyrinths

Labyrinth a has 2 arms and 3 circuits. The pathway enters on the outermost circuit. Therefore it is an uninteresting labyrinth. The path also reaches the center from the outermost circuit.

The complementary of it, labyrinth b, is also an uninteresting labyrinth. In this, the path enters the labyrinth on the innermost circuit and also reaches the center from the innermost cirucit.

So far, this is nothing special. But in this labyrinth we can observe another special feature. This can be seen, if we also view the two duals of these labyrinths. This is shown in the already familiar manner in figure 2.

Figure 2. The Dual and the Complementary Labyrinths are the Same

The dual (b) to the original labyrinth (a) ist the same as the complementary (c). The dual (d) to the complementary (c) is the same as the original (a). The two labyrinths that are dual-complementary to each other are the same.

Now this is not valid for all pairs of complementary uninteresting labyrinths. However, other labyrinths exist, in which this is also the case. In figure 3 I show two such examples of labyrinths and their patterns (only originals). In these labyrinths also, the complementary and the duals are the same.

Figure 3. Other Labyrinths with this Property

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Among all one-arm labyrinths with up to and including 7 circuits, there are no two uninteresting labyrinths complementary to each other. The reason for this is that in such labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit (see related posts, below). However, there exist uninteresting labyrinths with more than 7 circuits in which this is not the case.

In order to show this, I begin with the example of the 11-circuit Cakra-Vyuh labyrinth (see related posts). Figure 1 shows this labyrinth and the pattern of it.

Figure 1. The 11-circuit Cakra Vyuh Labyrinth

 

As can be seen, the pathway enters the labyrinth on the first circuit and reaches the center from the innermost circuit. So, the outer- and innermost circuits can simply be cut-off (grey lines in the right image). This then results in a labyrinth with 9 circuits, in which the pathway does not enter on the outermost circuit and doesn’t either reach the center from the innermost circuit. The pattern of this labyrinth is shown in figure 2.

Figure 2.The Pattern of the Uninteresting Labyrinth with 9 Circuits

Because we removed the grey circuits, the course of the pathway in the remaining pattern is from top right to bottom left. If we want to show the pattern in the usual form, we have to mirror it horizontally. This does not affect the pattern itself nor the labyrinth related to it, except for the labyrinth changing its rotational direction (see related posts).

Even though the pathway of this labyrinth enters on the 3rd circuit and reaches the center from the 7th circuit, this is an uninteresting labyrinth. This, because it is made up of two elements of the type Knossos on circuits 1 – 3 and 7 – 9 (indicated with brackets in the right image) and three internal trivial cirucits 4, 5, 6 between them (indicated with dashes). Although this labyrinth is uninteresting, it is self-dual.

Parenthesis: This labyrinth has similarities with the well known basic type (former: Cretan type) labyrinth. However, the basic type is a very interesting (that is interesting and self-dual) labyrinth.

Figure 3. The Pattern of the Basic Type Labyrinth

As shown in figure 3, this is also made-up of two elements of the type Knossos. However, between these there is only one circuit. And this is by no means trivial as it is needed to connect the two elements. But adding further circuits in the shape of a serpentine will result in an uninteresting labyrinth.

Let us get back to the uninteresting labyrinth with 9 circuits. How does the complementary labyrinth look like? Is it may be also an uninteresting labyrinth?

Figure 4. The Two Complementary Labyrinths with 9 Circuits

In order to generate the complementary, we mirror the original labyrinth vertically and let the connections with the environment and the center uninterrupted. Then the pathway enters on the 7th circuit and reaches the center from the 3rd circuit. The three trivial internal circuits are still recognizable. However, they are enclosed by the axial pieces of the pathway that lead into the labyrinth and to the center. So they are nested one level deeper. Therefore, this is no more an uninteresting, but an interesting, and, as it is self-dual, a very interesting labyirnth.

Thus it seems, that also in larger one-arm labyrinths there are no pairs of uninteresting labyirnths that are complementary to each other.

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