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Posts Tagged ‘self-dual’

We take a 7-circuit classical labyrinth and number the single circuits from the outside inwards. “0” stands for the outside, “8” denotes the center. I take this two numbers into the circuit sequence, although they are no circuits. As start and end point they help to better understand the structure of the labyrinth.

Ariadne's thread in the 7-circuit labyrinth

Ariadne’s thread in the 7-circuit labyrinth

The circuit sequence is: 0-3-2-1-4-7-6-5-8

Everybody which already has “trampled” Ariadne’s thread (the path) in the snow knows this: Suddenly there is no more place in the middle, and one simply goes out. And already one has created a walk-through labyrinth. This is possible in nearly all labyrinths.

Then maybe it looks like this:

Ariadne's thread in a walk-through labyrinth

Ariadne’s thread in a walk-through labyrinth

If one wants a more compact labyrinth, one must change the shape. The internal circuits become, in the end, a double spiral. We can make either two separate ways or join them. So we will get a bifurcation.

Just about:

The 7-circuit walk-through labyrinth

The 7-circuit walk-through labyrinth

We will get the following circuit sequence if we take the left way or the fork to the left:
0-3-2-1-4-7-6-5-0

Now we take first the right way or the fork to the right, then the circuit sequence will be:
0-5-6-7-4-1-2-3-0

Because the two rows are written among each other, they simply can be add up together (without the first and the last digit):
8-8-8-8-8-8-8

This means: If I go to the left, I am in the original labyrinth, if I go to the right, I cross the complementary one.

The complementary labyrinth of the 7-circuit labyrinth

The complementary labyrinth of the 7-circuit labyrinth

It has the circuit sequence 0-5-6-7-4-1-2-3-8.

Or said in other terms: The walk-through labyrinth contains two different labyrinths, the original one and the complementary one.

The 7-circuit labyrinth is self-dual. Therefore I only get two different labyrinths through rotation and mirroring as Andreas has described in detail in his preceding posts.

How does the walk-through labyrinth look if I choose a non self-dual labyrinth?

I take this 9-circuit labyrinth as an example:

A 9-circuit labyrinth

A 9-circuit labyrinth

Here the boundary lines are shown.
On the top left we see the original labyrinth, on the right side is the dual to it.
On the bottom left we see the complementary to the original (on top), on the right side is the dual to it.
However, this dual one is also the complementary to the dual on top.

The first 9-circuit walk-through labyrinth

The first 9-circuit walk-through labyrinth

The first walk-through labyrinth shows the same way as in the original labyrinth if I go to the left. If I go to the right, surprisingly the way is the same as in the complementary labyrinth of the dual one.

And the second one?

The second 9-circuit walk-through labyrinth

The second 9-circuit walk-through labyrinth

The left way corresponds to the dual labyrinth of the original. The right way, however, to the complementary labyrinth of the original.

Now we look again at a self-dual labyrinth, an 11-circuit labyrinth which was developed from the enlarged seed pattern.

An 11-circuit labyrinth in Knidos style

An 11-circuit labyrinth in Knidos style

The left one is the original labyrinth with the circuit sequence:
0-5-2-3-4-1-6-11-8-9-10-7-12

The right one shows the complementary one with the circuit sequence:
0-7-10-9-8-11-6-1-4-3-2-5-12

The test by addition (without the first and the last digit):
12-12-12-12-12-12-12-12-12-12-12

Once more we construct the matching walk-through labyrinth:

The 11-circuit walk-through labyrinth

The 11-circuit walk-through labyrinth

Again we see the original and the complementary labyrinth combined in one figure. If we read the sequences of circuits forwards and backwards we also see that both labyrinths are mirror-symmetric. This also applies to the previous walk-through labyrinths.

Now this are of all labyrinth-theoretical considerations. However, has there been such a labyrinth already as a historical labyrinth? By now I never met a 7- or 9-circuit labyrinth, but already an 11-circuit walk-through labyrinth when I explored the Babylons on the Solovetsky Islands (see related posts below). Besides, I have also considered how these labyrinths have probably originated. Certainly not from the precalled theoretical considerations, but rather from a “mutation” of the 11-circuit Troy Towns in the Scandinavian countrys. And connected through that with another view of the labyrinth in this culture.

There is an especially beautiful specimen of a 15-circuit Troy Town under a lighthouse on the Swedish island Rödkallen in the Gulf of Bothnia.

A 15-circuit Troy Town on the island Rödkallen

A 15-circuit Troy Town on the island Rödkallen, photo courtesy of Swedish Lapland.com, © Göran Wallin

It has an open middle and the bifurcation for the choice of the way. This article by Göran Wallin on the website Swedish Lapland.com reports more on Swedish labyrinths.

For me quite a special quality appears in these labyrinths, even if there is joined a change of paradigm.

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It is known that there are 8 alternating labyrinths with 1 arm and 5 circuits (see “Considering Meanders and Labyrinths”, related posts, below). Of these, four are not self-dual. These four all are in a relationship to each other via the duality and complementarity (see “The Complementary versus the Dual Labyrinth”, related posts, below). The other four labyrinths are self-dual.

I had already pointed to the relationship between complementary and self-dual labyrinths (see “The Complementary Labyrinth”, related posts, below). Here I want to elaborate on it further. For this purpose I use the same form of diagram I had already used in my previous post (see “The Complementary versus the Dual Labyrinth”). I also use the same numbers of the labyrinths according to the numbering of Arnol’d’s meanders (see “Considering Meanders and Labyrinths”), that underlie them.

Figure 1. Labyrinths 1 and 6

The first of the Arnol’d’s labyrinths, number 1, is self-dual. In the diagram, the dual is situated in the same row, the complementary in the same column with the original labyrinth. The dual of number 1 is again number 1 (what actually is the meaning of selfdual). The complementary of number 1 is number 6. And – of course – is the dual to the complementary again number 6. So in the case of self-dual labyrinths, we only captured two different labyrinths, whereas it were four in the case of not self-dual labyrinths.

Thus, two more labyrinths are still missing. We need another diagram to capture labyrinths number 3 and number 8 (fig. 2).

Figure 2. Labyrinths 3 and 8

And, indeed, these two are complementary to each other. So in self-dual labyrinths, only two different labyrinths are in a relationship to each other.

Here the question arises: Do there also exist self-complementary labyrinths? Up to now we have not yet found such a labyrinth. So let us remember, what self-dual imples. The patterns of the original and self-dual labyrinths are self-covering. In fig. 3 I show what that means. The two patterns in the same row are dual. If we shift them together, we can easily see, what I mean.

Figure 3. Self-dual patterns are self-covering

Thus, self-complementary would imply that the original and complementary pattern would also be self-covering.

Figure 4. Complementary patterns are not self-covering

Fig. 4 shows, that even though there is a certain similarity between these two patterns, they are not self-covering. In my opinion there are no self-complementary labyrinths. This is because vertical mirroring with uninterrupted connections to the entrance and center modifies the sequence of circuits. This, however, woult have to remain unaltered.

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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.

O-D Kretisch

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.

SD Kret

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.

UF 7 Farben

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.

UF Muster Kret

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.

Muster d sd

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 Chartres labyrinth has many qualities. For many it is by far the most perfect and beautiful labyrinth.

The special qualities of the Chartres labyrinth are not recognizable at first glance. One must look at it more exactly and try to understand the structure and the principles of its design.

First let us explore the self-duality.

The layout of the Chartres Labyrinth

The layout of the Chartres Labyrinth

The circuits in the drawing are numbered from the outside inwards as well as from the inside out (and the same way in the next diagram below).

First we determine the path sequence for the way in, from the outside (0) inwards (12):
(0)-5-6-11-10-9-8-7-8-9-10-11-10-9-8-7-6-5-4-3-2-1-2-3-4-5-4-3-2-1-6-7-(12).

Now we number the circuits from the inside (0) outwardly (12) and then we read the path sequence for the way back:
(0)-5-6-11-10-9-8-7-8-9-10-11-10-9-8-7-6-5-4-3-2-1-2-3-4-5-4-3-2-1-6-7-(12).

We ascertain, they are identical. So the Chartres labyrinth is self-dual, sign of its extraordinary quality.

The rhythm and the pattern of movement of a labyrinth is manifested in the path sequence. In the Chartres labyrinth the path sometimes touches two quadrants, sometimes only one. But never three or even four quadrants. The biggest step width is five (from circuits  0-5, 6-11, 1-6, 7-12). Most often, however, the step width is one. The most typical step sequence in the Chartres labyrinth is for me at the beginning 0-5-6-11 and 1-6-7-12 at the end. One immediately steps inside, reaches quite directly the innermost circuit, then oscillates through the whole labyrinth, and reaches unexpectedly the center from the outermost circuit. This expresses the dramaturgy of this special pattern of movement.

Admittedly, the counting is laborious. But walking the Chartres labyrinth needs no counting. The real experience can also made by walking.


What’s about the symmetry?

Let us have a look at the rectangle form and give different colors to the circuits. The entrance is at the bottom right and the center is reached on top left. The circuits are numbered in both directions, i.e. from the outside in and from the inside out. The horizontal distances between S, E, N, W, and S in the rectangular form are of no importance, as it shows the structure (Andreas refers to it as the pattern).

The rectangular diagram

The rectangular diagram

Circuit 6 represents the middle in both directions, this is the reflexion axis. The green and the blue fields are alike. Rotation and shifting shows that they are self-covering. The yellow fields are the connecting elements. They run step-shaped in serpentines.
Andreas calls them Cascading Serpentines  (kaskadierende Serpentinen) . He sees in this an own principle of labyrinth construction. To learn more, have a look at his website (see the link below).

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In an earlier post I have rotated the seed pattern for the Ariadne’s Thread of my demonstration labyrinth and generated 12 different figures. Six of these figures rotate clockwise, the others anticlockwise.

Each labyrinth with five circuits has a seed pattern with 12 ends. Thus, the framework presented in my earlier post can also be used to rotate the seed pattern of other labyrinths with five circuits. I have done this with the core-labyrinth of Rockcliffe Marsh (Arnol’d’s figure 8).

Illustration 1: Rockcliffe Marsh

Illustration 1: Rockcliffe Marsh

Ill. 1 shows the Rockcliffe Marsh labyrinth on the left with its core-labyrinth marked. On the right, the script version of the core-labyrinth is shown.

Illustration 2: Seed Pattern of Rockcliffe Marsh

Illustration 2: Seed Pattern of Rockcliffe Marsh

Ill. 2 compares the seed pattern of my demonstration labyrinth (left figure) with the one of Rockcliffe Marsh (right figure). The seed pattern of Rockcliffe Marsh is made up of 2 similar halves. This is a characteristic of self-dual labyrinths. In my demonstration labyrinth, the figure that results when connecting the end 7 of the seed-pattern with the center (figure 7) is the dual of figure 1. Self-dual means, that the two duals are identic. Therefore, in Rockcliffe Marsh, figure 7 is identic with figure 1. The same holds for figure 2 and 8 and so forth. It is therefore sufficient to only connect the first six ends of the Rockcliffe Marsh seed pattern with the center, as the ends 7 to 12 will simply reproduce the figures 1 – 6. In the seed pattern of Rockcliffe Marsh the ends 7 – 12 therefore were not numbered.

Illustration 3: The three pairs of figures

Illustration 3: The three pairs of figures

Ill. 3 shows the result. The numbers of the figures indicate which end of the seed pattern was connected with the center to generate the figure.

  • First: the number of different figures reduces to six. Three of them rotate clockwise, three anti-clockwise.
  • Second: A closer look reveals that there are only three different figures, each in clockwise and anti-clockwise rotation. These pairs of figures have been arranged on the same line in the illustration (figure 1 and 6, fig. 2 and 5, and fig. 3 and 4).

The reason for this is that the seed pattern of Rockcliffe Marsh not only is made up of 2 similar halves. In addition, each of these halves is symmetric around the dashed line (illustration 4).

Illustration 4: Symmetry

Illustration 4: Symmetry

Self-duality reduces the number of different figures from 12 to 6, the symmetry of the seeds in both halves reduces it further to only three different figures.

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