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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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To say it more exactly, here I relate to the 21 row-shaped visceral labyrinths, still known from some of the previous articles (see Related Posts below).

The appearance is defined by the circuit or path sequence. With that one can construct the different and new labyrinth types (here 21). To this I use the once before presented method to draw a labyrinth (see below).

The path and the limitation lines are equally wide. The center is bigger. The last piece of the path leads vertically into the center. All elements are connected next to each other without sharp bends and geometrically correct. There are only straight lines and curves. This all on the smallest place possible. All together makes up the Knidos style.

Look at a single picture in a bigger version by clicking on it:

I think that by this style the movement pattern of every labyrinth becomes especially well recognizable. With that they can be compared more easy with the already known labyrinths.

Remarkably for me it is that only one specimen (E 3384 v_6) begins with the first circuit. And the fact that many directly circle around the middle and, finally, from the first circuit the center directly is reached. Noticeably are also the many vertical straight and parallel pieces in the middle section.

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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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Here it is about the decoding of the circuit sequences of the row-shaped 21 visceral labyrinths shown in the last article on this subject (see related posts below).

The question is: Can I generate one-arm alternating labyrinths with one center in the middle from them? That means no walk-through labyrinths where the also unequivocal path passes through, but is ending at an aim in the middle.
Maybe one could call them “walk-in labyrinths” contrary  to the “walk-through labyrinths”?

The short answer: Yes, it is possible. And the result are 21 new, up to now unknown labyrinths.

The circuit sequence for the walk-through labyrinth can be converted into one for a walk- in labyrinth by leaving out the last “0” which stands for “outside”. The highest number stands for the center. If it is not at the last place in the circuit sequence, one must add one more number.
This “trick” is necessary only for two labyrinths and then leads to labyrinths with even circuits (VAT 984_6 and VAN 9447_7).

The gallery shows all the 21 labyrinths in concentric style with a greater center.

Look at the single picture in a bigger version by clicking on it:

 

All labyrinths are different. Not one has appeared up to now somewhere. They have between 9 and 16 circuits, the most 11 circuits. They show between 3 and 6 turning points.

In these constellations there are purely mathematically seen 134871 variations of interesting labyrinths, as proves Tony Phillips, professor of mathematics.

There are still a lot of possibilities to find new labyrinths or to invent them.

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Further Link
The website of Tony Phillips

Our Best Wishes for 2018

We hope you had a good start in new year. This year again promises intersting insights into the world of labyrinths.

 

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