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Posts Tagged ‘Arnol’d’

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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In several previous posts I have shown, that different variants can exist for a certain labyrinth or seed pattern.

KSAF_var

Illustration 1. Variants of the Same Seed Pattern

In Ill. 1 I again show some variants of the seed pattern for the Ariadne’s Thread of my demonstration labyrinth. This same seed pattern can be drawn e.g. with a circular, elliptic, petal-shaped or rectangular outline. The outline figure is only an auxiliary figure. The seed pattern itself is formed by the system of lines within this outline figure. Depending on the shape of the outline figure, also the orientation and rounding of the seeds may somewhat differ. However, they are always ordered the same way. On top left one (not-nested) turn, on bottom left two nested and on the right three nested turns. Which variant of the seed pattern is best suited depends on the purpose for which it is used.

In this post I want to show the relationship between the seed pattern and the pattern. For this purpose, the rectangular variant is best suited. The seed pattern can be transformed to the pattern in a few steps.

KS Umf1

Illustration 2. From Seed Pattern to Meander

The left figure of ill. 2 shows the rectangular variant of the seed pattern. This is also shown as baseline in grey in the right figure. As a first step, the right half of the seed pattern is shifted against the left (shown in red), until it comes to lie on the other side of the left half.

KS Umf2

Illustration 3. From Meander to Pattern

The result of this shift is a meander. It is one of Arnol’d’s figures. This meander is in a next step straightened-out, as has already been shown here. For this, the right half of the seed pattern is shifted somewhat further to the left. The ends opposite each other are then connected with lines.

KS Muster

Illustration 4. Pattern

The result of this process is shown in ill. 4. Apparently, in transforming the meander to the pattern, the first and most important step is the horizontal straightening-out. By this the situation of the circuits in the pattern are made apparent. Next, one can easily straighten-out the axial segments and finalize the pattern.

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In my last post I have referred to the interesting labyrinths by Tony Phillips. But what does this mean: interesting? Tony shows by means of combinatorics that there exist a vast number of one-arm labyrinths. And he subdivides them in uninteresting, interesting and very interesting labyrinths.

  • Accordingly, uninteresting are labyrinths that can be generated by simply attaching additional trivial circuits to smaller labyrinths at the inside or outside.
  • Interesting are labyrinths that cannot be generated this way. This implies that the path of such labyrinths may not enter on the first circuit nor reach the center from the last circuit.
  • Very interesting are the self-dual among the interesting labyrinths.

Now let us have a closer look at what this actually implies. With Arnol’d’s figures in mind, we are already familiar with all labyrinths with one arm and five circuits. So let us classify them according to the above-mentioned criteria.

Illustration 1 shows the uninteresting labyrinths. These correspond with Arnol’d’s figures 1 – 4.

Illustration 1: Uninteresting Labyrinths

Illustration 1: Uninteresting Labyrinths

Figure 1 (Labyrinth Näpfchenstein) consists of a mere series of circuits aligned to each other. Figures 2 to 4 are composed of a smaller Knossos-type labyirnth (black) and additional circuits (grey). In figure 2 (Löwenstein 5b) these circuits are attached at the outside, in figure 4 (Löwenstein 5a) at the inside to the smaller labyrinth. Figur 3 has one additional circuit attached to the out- and inside. Among the uninteresting labyrinths there are also self-dual examples, i.e. figures 1 und 3.

Illustration 2 shows the interesting and very interesting labyrinths. These correspond with Arnol’d’s figures 5 – 8.

Illustration 2: Interesting Labyrinths

Illustration 2: Interesting Labyrinths

Figure 5 and the dual in figure 7 are interesting labyrinths. Figure 6 and figure 8 are self-dual each and thus very interesting labyrinths.

Tony’s classification is very plausible. It provides a qualitative ranking of the labyrinths.

Labyrinths with a pattern of a serpentine from the outside in are uninteresting accoring to this classification. This applies to some historical labyrinths: Tholos, Löwenstein 3, Näpfchenstein, Casale Monferrato. Also uninteresting are labyrinths where the path enters on the first circuit or reaches the center from the last circuit. There are also some historical labyrinth types with this property: Temple Cowley, Löwenstein 5a und 5b, von Xanten, Zikkaron und Cakra-vyuh.

Uninteresting labyrinths are composed of a smaller interesting labyrinth and additional circuits. However, which is the intersting labyrinth that forms part of figure 1 (illustration 1)? Illustration 3 shows three options a – c to address this question.

Illustration 3: The Smallest Labyrinth?

Illustration 3: The Smallest Labyrinth?

Figure 1 is made up of a series of five circuits. A first option would be to consider figure a with one circuit (black) as the simplest labyrinth. However, this contradicts with the definition of a labyrinth provided by Kern (Kern H. Through the Labyrinth. Munich: Prestel 2000, p. 23), as the path does not repeatedly change its direction. The second option is figure b with two consecutive circuits. Here the path does not repeatedly but only once change direction. The third option then, figure c, is made up of three circuits with the path changing its drection twice. This is in full accordance with the defintion by Kern.

My answer to this question is that I consider figure b as the smallest / simplest labyrinth. There exists a historical labyrinth of this type, that lies in the Tholos of Epidauros. Also for option c, a historical labyrinth exists: Löwenstein 3.

The smallest / simplest interesting labyrinth, however, is the historical Knossos-type labyrinth. This has three circuits and a pattern of a single meander in my words, or, in Erwin’s terminology, a labyrinth-suited meander type 4. As this labyrinth is self-dual it is even a very interesting labyrinth.

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In my last post I have pointed out the close relationship between the figures of Arnol’d and the patterns of labyrinths. These figures are very similar, but they are not equal. One has to straighten-out Arnol’d’s figures to get to the patterns of the labyrinths. But what does this mean: “straighten-out”? I want to show this with the example of one of Arnol’d’s figures. I have purposely chosen figure 8 for that.

Illustration 1: double-spiral

Illustration 1: double-spiral

Illustration 1 shows figure 8 of Arnol’d in its original orientation (first image) and rotated by a quarter of a circle (second image). The third image shows the same figure as the second, although reproduced on the computer and a bit more rounded. As can be seen, the curve comes from above left and winds itself inwards anticlockwise with narrowing radii.  It intersects the straight line twice. At the point where it intersects the straight a third time, the curve changes to clockwise direction and winds itself outwards. This curve in fact presents as a double-spiral, that intersects the straight five times.

Now, what happens, if we straighten-out this figure?

This is shown in the next illustration.

Illustration 2: double-spiral-type meander

Illustration 2: double-spiral-type meander

The double-spiral is dissected along the straight line and both halves are shifted horizontally. Next, the pairs of points that were generated by this separation are connected with horizontal lines (dashed in the figure). These lines in fact represent the circuits of the labyrinth. Separating the double-spiral and inserting connection lines leads to the pattern of the labyrinth. Here lies the relationship between the intersection points in Arnol’d’s curves and the circuits of the labyrinth.

In addition, this example illustrates very well the difference between a double-spiral and a double-spiral-type meander. Remember that Arnol’d’s figure 8 corresponds with the meander Erwin found best suited for labyrinths. This meander is in fact a double-spiral-type meander, and this holds for all of Erwin’s types of this meander (type 4, 6, 8, etc.). Separating Arnol’d’s figure and inserting connection lines transforms the continuous movement of the double-spiral into a stepwise progression from one circuit to the next. That is what makes the difference between the double-spiral and the double-spiral type meander that can be found in the patterns of labyrinths.

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In a series of recent posts Erwin has elaborated on meanders in labyrinths. He has found a meander best suited for labyrinths. This particular meander exists in various forms. Erwin refers to these as “type” followed by an even number, e.g. type 4, type 6 and so forth.

Indeed, these types of meanders can be often found in existing labyrinths. However other figures occur in labyrinths too that can be denoted as meanders. Generally a broad range of figures is referred to as meanders. And it seems not so clear what a meander is at all.

So, what is a meander?

On his website, Tony Phillips cites Arnol’d, a Russian mathematician. Independent of the labyrinth, Arnol’d wanted to investigate how many meanders there are. He defines a meander as follows:

  • Connected oriented curve,
  • that does not intersect itself
  • and intersects a fixed, oriented line in several points.

He illustrates this with the example of a curve that intersects the fixed line five times. He found that there exist eight different such curves. I have copied the illustration from Tony’s website, enumerated the 8 curves and display it below.

8 curves

8 curves

It can easily be recognized that these curves are very closely related with the labyrinth. One only has to rotate them by a quarter of a circle in clockwise direction and straighten them out somewhat. This results in the patterns of 8 one-arm labyrinths with 5 circuits each. Therefore, let us have a closer look at these figures.

figure 1

figure 1

This figure represents a serpentine that leads from the entrance to the center of the labyrinth. It is this the pattern of the historical Näpfchenstein labyrinth. This labyrinth is self-dual.

figure 2

figure 2

This curve depicts the pattern of the labyrinth type Löwenstein 5b. This is the dual of the labyrinth shown in figure 4. Dual labyrinths have the same pattern, although the pattern is rotated by half a circle, and the entrance and center are exchanged. The “Rockery Labyrinth“, designed by Erwin is also of this type.

figure 3

figure 3

This curve includes a pattern of the Knossos type (3 circuits) to which are attached an additional circuit on both, the outer and the inner side. I am not aware of any existing labyrinth of this type. This labyrinth is self-dual.

figure 4

figure 4

This is the pattern of the labyrinth type Löwenstein 5a. It is dual to the labyrinth shown in figure 2. The “Pilgrim Hospices” labyrinth, designed by The Labyrinth Builders is of this type.

figure 5

figure 5

This figure contains the pattern of the labyrinth I use for my investigations and presentations, so to speak my demonstration labyrinth. It has the following properties that are important for this purpose: the path does not enter on the first circuit, it does not reach the center from the last circuit and the labyrinth is not self-dual. The dual to this labyrinth is shown in figure 7.

figure 6

figure 6

This pattern corresponds with a serpentine from the inside out. The path enters along the axis and first encounters the innermost circuit. From there it winds itself out circuit by circuit until it reaches the first (outermost) circuit. Then it is directed axially to the center. Erwin has discovered this type of labyrinth (Chartres 5 classical) by omitting the side-arms of the Compiègne-type labyrinth. This labyrinth is self-dual.

figure 7

figure 7

This is the dual of my demonstration labyrinth shown in figure 5.

figure 8

figure 8

This curve represents Erwin’s meander best suited for labyrinths. It is a type 6 meander. This is the pattern of the core-labyrinth of the historical Rockcliffe Marsh labyrinth. Rockcliffe Marsh is a very special labyrinth. First it has an unusual layout. The figure is opened along the axis and unrolled to a segment of a circle. Second, it is made up of a core-labyrinth (the inner 5 circuits) that is enclosed by a spiral outside.

Conclusion

Arnol’d’s definition of a meander is closely related with the labyrinth. His curves correspond with the patterns of all one-arm labyrinths in which the pathway does not cross the axis. The number of intersections between the curve and the fixed line corresponds with the number of circuits in the labyrinth. This was demonstrated in detail for labyrinths with 5 circuits.

  • Thus there exist 8 different patterns for a labyrinth with one axis and five circuits with the pathway not crossing the axis.
  • With an increasing number of circuits, the number of different pattern increases dramatically. E.G. there are 14 possible patterns for a labyrinth with 6 and 42 for a labyrinth with 7 circuits.
  • According to Arnol’d’s definition, all 8 figures are meanders. If we follow Erwin’s definition, only figure 8 is a meander suited for labyrinths.
  • If we adopt the definition of Erwin, we will capture the most common and essential labyrinths. However, we will also miss a broad range of existing and potential patterns of labyrinths.
  • If we adopt the definition of Arnol’d, every pattern of a one-arm labyrinth, in which the way does not cross the axis is referred to as a meander. This definition seems too broad and can be further differentiated.

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