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Posts Tagged ‘complementary’

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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Wishing all visitors of this Blog a Merry Christmas and a Happy New Year!

Christmas tree Labyrinth

The complementary 7 circuit Classical labyrinth as Christmas tree Labyrinth

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There are 42 different one-arm alternating labyrinths with 7 circuits. Among these there is one pair of complementary interesting labyrinths. Now how does it look in pairs of complementary unintersting labyrinths? This question has already been indirectly answered in my last post (see related posts below): There is none! This sounds surprising. Therefore I address it further here. The 42 labyrinths form 21 complementary pairs. One of it is composed of 2 interesting labyrinths. We also know there are 22 interesting labyrinths. So the other 20 pairs are made up of an interesting and an uninteresting labyrinth each. Therefore no possibility remains for a pair with two complementary uninteresting labyrinths. What is the reason for that?

As we have seen, only in alternating labyrinths with an odd number of circuits it is possible to derive a complementary (see related posts). In such labyrinths the pathway always enters on an odd-numbered ciruit and also reaches the center from an odd-numbered circuit. Further, in one-arm labyrinths the pathway cannot enter the labyrinth on the same circuit from which it reaches the center.

In uninteresting labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The complementary is derived by mirroring. By this, the outermost is transformed to the innermost circuit and vice versa. If in an original labyrinth the pathway enters on the first circuit, it is an uninteresting labyrinth. In the complementary the path will enter on the innermost circuit. Thus the complementary is not an uninteresting labyrinth, unless the path would reach the center from the innermost circuit. This, however is not possible, as it already enters the labyrinth on this circuit. The original is an unintersting, but the complementary an interesting labyrinth. The other alternative would be that the path in the original labyrinth reached the center from the innermost circuit. But then in the complementary it would reach the center from the outermost circuit what is not an unintersting labyrinth. Therefore the complementary could only be an unintersting labyrinth, if the path would enter it on the outermost circuit. This, however is impossible, as the path reaches the center from this circuit.

These results are only valid for one-arm labyrinths with up to 7 circuits. In labyrinths with mulitiple arms, the pathway may reach the center from the same circuit on which it enters the labyrinth. Thus, for example it could enter the original labyrinth on the first circuit and also reach the center from the first circuit. This would consitute an uninteresting labyrinth. In the complementary, the pathway would then enter the labyrinth on the innermost circuit and also reach the center from the innermost circuit, what again would qualify for an uninteresting labyrinth. In one-arm labyriths with more thean 7 circuits the definition of what constitutes an uninteresting labyrinth can be extended. In these cases trivial circuits can be added not only at the outside or inside of smaller interesting labyrinths (what generates uninteresting labyrints) but also on central circuits between other interesting elements at the inside and outside of the labyrinth, what also may generate uninteresting labyrinths.

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