Posts Tagged ‘Arnol’d’

Sigmund Gossembrot has used the double-barrier as a new element for the design of labyrinths. His five-arm labyrinth on fol. 51 r (see: related posts 5) and the four-arm labyrinth hidden in the design on fol. 53 v (related posts 4) are made up exclusively of double-barriers in all side-arms. They have 7 circuits and are no sector labyrinths.

In a series of posts, Erwin has introduced new sector labyrinths with four arms, five circuits and double-barriers (related posts 1, 2, 3). He has based these designs on the 8 possible courses the pathway can take in a one-arm labyrinth with five circuits. Sector labyrinths can be obtained by stringing together such courses of the pathway. Theoretically there exist 4096 variations for an arbitrary stringing together of four out of 8 courses of the pathway. Erwin has shown some of them. However, not all did consistently employ the principle of the double-barriers.

Here I will address the question how many sector labyrinths there are with four arms, five circuits and consistently employing double-barriers. I also start with the 8 possible courses of the pathway. These are based on Arnol’ds’ meanders in fig. 1 (related posts 6).

Figure 1. The Meanders by Arnol’d

In fig. 2 I show the patterns that correspond with the meanders. The patterns bear the same numbers as the meanders they were derived from. The left side of the figure shows the patterns of all alternating one-arm labyrinths with five circuits. Each of hese patterns also contains a connection from the outside into the labyrinth (from top left) and a connection to the center (to bottom right). These connections are represented in grey. In order to be used as segments (sectors) in sector labyrinths, these patterns at first have to be considered without the grey connection lines. The question here is the course of the path inside the sector. In a sector labyrinth, multiple such patterns are stringed together. Only the first pattern contains a connection to the outside and only the last a connection to the center. The patterns representing the 8 possible courses in a sector are displayed in the box on the right side.

Figure 2. The Corresponding Patterns – Left Half: Patterns of the One-arm Labyrinths; Right Half: Patterns of the Sectors

Next we intend to string together such patterns of sectors and to generate four-arm labyrinths using exclusively double-barriers. Let us first have a look at such a double-barrier in the labyrinth type Gossembrot 51 r. Figure 3 shows the labyrinth with the Ariadne’s Thread drawn in (red). Except for the one-arm labyrinths, an axis always lies between two segments, is formed by two different segments. Let us choose the double-barrier at the third side-arm. This connects segments III and IV and lies on the outermost four circuits. In the scaled-up section the seed pattern for the walls delimiting the path is drawn-in in blue color. It can be seen, that two nested turns of the Ariadne’s Thread are mirrored symmetrically against the central piece of the wall delimiting the pathway. Four circuits are needed for the double-barrier. In a labyrinth with five circuits, only one circuit remains free for the passage from one sector to the next. From this it becomes clear, that labyrinths with five circuits and using exclusively double-barriers must be sector labyrinths. There is only one circuit for the path to traverse the arms. This implies that the pathway must have completed the previous sector entirely before changing to the next.

Figure 3. The Double-barrier as by Gossembrot

Figure 4 shows the admissible connections between the sectors. (Pro memoria: the lines represent the pattern, i.e. the Ariadne’s Thread in rectangular form). The double-barriers occupy four circuits side by side. So they can lie in two places on circuits 2 – 5 or on circuits 1 – 4. Only connections on the same circuits are allowed, that is, the two options on the outermost (a) or on the innermost (b) circuit. If we would consider to change to another circuit when connecting the segments, as shown in options c or d, this would result in the insertion of an additional axial piece of the path between the halves of the double-barrier and these halves would be shifted by one circuit one against the other. But this is not a double-barrier any more.

Figure 4. Admissible Connections between Sectors

This circumstance strongly limits the range of the possibilities for stringing together the patterns. Figure 5 shows, how the different patterns can be used. The red numbers, characters at the free ends of each pattern indicate with which patterns it can be connected there (number of pattern, E for entrance, Z for center). A four-arm labyrinth has four segments. These are therefore also referred to as „quadrants“.

Figure 5. Possibilities for the Use of the Patterns

  • Two patterns, no. 1 and no. 6 cannot be used at all. With these it is not possible to generate a double-barrier.
  • Four „one-sided“ patterns, that is no. 2, no. 4, no. 5, and no. 7 have only on one side a half of a double-barrier (circled in red). On this side they can be connected with other patterns to double-barriers. It is true that it is also still possible to connect pattern no. 2 with no 5, and pattern no. 4 with no. 7 (not indicated). However, such a connection results in a two-arm labyirinth with one double-barrier only. On the second side of these one-sided patterns the free end lies on the third circuit. There, no double-barrier can be generated. Therefore, on this side only a connection to the entrance or the center is possible. Thus, these one-sided patterns can only be placed next to the main axis. Pattern no. 2 and no. 7 can only be placed in quadrant IV, where they are connected with the center. Pattern no. 2. can further only be connected with no. 8, and pattern no. 7 can be connected with no. 3.
  • Only two patterns, no. 3 and no. 8 can be completed to double-barriers at both sides. And only these can be placed in quadrants II or III. Furthermore, they may also be placed in quadrants I or IV, and thus be connected with the entrance or with the center (not indicated). Patterns no. 3 and no. 8 can be stringed together alternately or they can be connected with other one-sided patterns (pattern no. 3 with no. 5 and no. 7; pattern no. 8 with no. 4 and no. 2).

This provides us with the basis for the generation of the patterns for the sector labyrinths with the double-barriers. We start with the patterns for the quadrants II and III. For these, there are only two arrangements. Pattern no. 8 can be attached to no. 3 (above) or pattern no. 3 can be attached to no. 8 (below). The upper combination can be supplemented with patterns no. 5 or no. 8 towards quadrant I, and with patterns no. 2 or no. 3 towards quadrant IV. The lower combination can be supplemented with patterns no. 3 or no. 4 towards quadrant I, and with patterns no. 7 or no. 8 towards quadrant IV.

With the upper combination of the patterns no. 3 and no. 8 in quadrants II and III, thus, four patterns of labyrinths with four arms, five circuits and a consistent use of double-barriers can be generated. These patterns are shown in fig. 6.

Figure 6. The Patterns with the Combination no. 3 in Quadrant II – no. 8 in Quadrant III

Also with the lower combination of the patterns no. 8 and no. 3 in quadrants II and III, four patterns of labyrinths with four arms, five circuits and a consistent use of double-barriers can be generated. These patterns are shown in fig. 7.

Figure 7. The Patterns with the Combination no. 8 in Quadrant II – no. 3 in Quadrant III

Figure 8 now shows the labyrinths corresponding to the patterns of fig. 6.

Figure 8. The Labyrinths Corresponding with the Patterns of Fig. 6

Finally, figure 9 shows the labyrinths corresponding to the patterns of fig. 7.

Figure 9. The Labyrinths Corresponding with the Patterns of Fig. 7

The question concerning the number of possible labyrinths can be clearly answered:

  • There exist 8 labyrinths with 3 double-barriers, 4 arms and 5 circuits.

Moreover, in addition to this question, we obtain the following findings:

  • Labyrinths with 5 circuits and fully employing double-barriers must be sector labyrinths.
  • Such labyrinths cannot have double-barriers at the main axis. Double-barriers are only possible at the side-arms.

Related Posts:

  1. New 5 Circuit Labyrinths with Double Barriers
  2. A new Generation of Sector Labyrinths
  3. A new type of Sector Labyrinth inspired by Gossembrot
  4. Sigmund Gossembrot / 3
  5. Sigmund Gossembrot / 2
  6. Considering Meanders and Labyrinths

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


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.


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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