The Level Sequence in One Arm Labyrinths

In an earlier post „Type or Style / 6“ (see related posts, below) I had already mentioned the level sequence. And I had stated two reasons for why I do not use it for naming types of labyrinths.

  • Among the one-arm labyrinths only in alternating labyrinths there exists exactly one type of labyrinth for each level sequence. If we also consider non-alternating labyrinths, in which the pathway traverses the axis, there can exist multiple courses of the pathway for the same level sequence.
  • In labyrinths with multiple arms the level sequence may rapidly increase to a length and complexity that is difficult to memorize.

Here I want to address the first issue further. I do this because there is a very simple solution for it. In one-arm labyrinths every circuit is represented by one number. In real practice only few of the larger labyrinths will have more than 15 – 17 circuits. Most one-arm labyrinths have a markedly smaller size. Therefore these labyrinths could be quite simply be named with their level sequence. But there remains the problem with the ambiguity. Erwin had elaborated on it in his post “The Classical 7 Circuit Labyrinth with Crossed Axis“ (see related posts, below). I will illustrate it here and use some figures of Erwin’s post.

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Figure 1. Level Sequence 3 2 1 4 7 6 5

In Figure 1 three labyrinths with the level sequence 3 2 1 4 7 6 5 are shown. The first image shows the alternating Cretan type, the second and third images show non-alternating labyrinths with the same level sequence. In the second image, the pathway traverses the axis when changing from the 1st to the 4th circuit. In the third image it traverses the axis from the 4th to the 7th circuit. (There is an other labyrinth with the pathway traversing the axis twice, first from the 1st to the 4th and second from the 4th to the 7th circuit). We thus are here presented with the only one alternating and several non-alternating types of labyrinths with the same level sequence.

Now there is a simple solution, to take account of this in the level sequence. For this it has to be considered, that the single numbers (not numerals) of the level sequence are separated. This separation can be obtained in different ways, using blanks, commas, semicolons etc. These separators, however, can also be used to indicate how the path will continue on the next level. Therefore we could e.g. define: if the path changes direction from the former to the next circuit, we will separate the numbers with a vertical slash. If, on the other hand, the path continues in the same direction and thus traverses the axis, we separate with a hyphen. This enables us to specify the level sequence so that it is unique also in non-alternating labyrinths. I show this in figure 2 using the images from figure 1.

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Figure 2. Level Sequence with Separators


Here we see for each labyrinth the unique level sequence with separators. The sequence of numbers is the same 3 2 1 4 7 6 5 in all three labyrinths. However, whereas in the alternating Cretan type all numbers are separated by slashes (as the path always changes direction when progressing from one circuit to an other), the level sequence in the second labyrinth is written with a hyphen between 1 and 4, and the level sequence in the third image with a hyphen between 4 and 7.

Indeed, the notation can be even simplified by separating with blanks and using hyphens only to indicate where the pathway traverses the axis. The level sequences would then be written as follows:

for the  1st image: 3 2 1 4 7 6 5
for the  2nd. image: 3 2 1-4 7 6 5
for the  3rd image: 3 2 1 4-7 6 5

What matters is that in the level sequence it is indicated where the path traverses the axis. With this specification it is now possible to give a unique level sequence to each course of the pathway and thus a unique name to each alternating and non-alternating type of labyrinth.

Related posts

The Seed Pattern in Labyrinths with Multiple Arms

The seed pattern is an extract of the axis of the labyrinth without the circuits. A seed pattern can also be drawn for labyrinths with multiple arms.

Figure 1. The Arms

Figure 1. The Arms

Fig. 1 shows this with a one-arm and a two-arm labyrinth compared. The one-arm labyrinth is of the Cretan type, the two-arm labyrinth is one of my own designs. For reasons of simplicity I chose the representation with the Ariadne’s Thread.

In labyrinths with multiple arms a separate seed pattern has to be extracted for each arm. Of course, these two parts belong together. This should become directly evident.

The seed pattern for the Ariadne’s Thread is drawn with an auxiliary line that delimits the layout of the seed pattern (see related posts below). This auxiliary line can be used to graphically connect the two part-seed patterns.

Figure 2. Connecting the Part-Seed Patterns

Figure 2. Connecting the Part-Seed Patterns

Figure 2 shows how we can prodeed for this. In one-arm labyrinths the center lies beyond the seed pattern. And, strictly speaking, it always has to be indicated, where the center of the labyrinth is situated. In labyrinths with two arms the seed pattern of the side-arm is situated beyond the center opposite to the seed pattern of the main axis of the labyrinth. In a seed pattern of a labyrinth with multiple arms, the center is fixed by the situation of the arms relative to each other. And thus it comes to lie within the seed pattern. With the auxiliary line the sub seed patterns for the Ariadne’s Thread can easily be connected in the shape of an “8”. This can be drawn freehand in one line. By doing so, we have performed a variation of the original circular or elliptic form to a petal-shaped form. This, however, is a minor variation and does not affect the seed pattern itself.

Figure 3. Completing the Seed Pattern for the Ariadne's Thread

Figure 3. Completing the Seed Pattern for the Ariadne’s Thread

As fig. 3 shows, the seed pattern of a multi-arm labyrinth is completed exactly the same way as a one-arm labyrinth seed pattern (see related posts). The ends of the seeds nearest to the centre are connected first. By this, the innermost circuit is generated. Next, the ends nearest to the first circuit are connected the same way, and so forth. And so, one circuit after another is added from the inside out. The only difference to a one-arm labyrinth is, that in a multiple-arm labyrinth multiple segments have to be generated for each circuit. In a two-arm seed pattern, for each circuit, four ends have to be connected with two sections of circuits between the two arms.

There are two notable differences in the shape of the seed pattern of the main axis and of the side arms.

  • The seed pattern of the main axis has two ends more than the seed patterns for the side arms. This is due to the fact that the entrance of the labyrinth and the access to the center lie on the main axis.
  • Usually we consider alternating labyrinths where the path does not traverse the main axis (although there are some notable exceptions). In these labyrinths it is indispensable that the path traverses the side arms. Otherwise it would not be possible to reach the area opposite the side-arm and it thus would be impossible to generate the side arm at all.
Figure 4. Completing the Seed Pattern for the Walls

Figure 4. Completing the Seed Pattern for the Walls

Of course, what is valid for the Ariadne’s Thread works with the walls of the labyrinth too. The seed pattern for the walls, however, looks more complicated and less elegant. The part-seed patterns of the two arms are not graphically connected, as the seed pattern for the walls traditionally is drawn without an auxiliary line. The Ariadne’s Thread is the simpler graphical representation of both, the labyrinth and the seed pattern of it.

Relatecd posts:

Seed Pattern – Shifting the Center / 3

Once again I come back to the shifting of the center. This time I want to show the result of rotating the seed pattern of the Näpfchenstein labyrinth.

Illustration 1: The Näpfchenstein labyrinth and seed pattern

Illustration 1: The Näpfchenstein labyrinth and seed pattern

Ill. 1 shows this labyrinth and its seed pattern for the Ariadne’s Thread.

Rotating this seed pattern results in two figures. In fact this is only one figure, the labyrinth itself, either in clockwise or anti-clockwise rotation. This pair of figures is then repeated five times.

Illustration 2: the two figures

Illustration 2: the two figures

A closer look at the seed pattern shows the reasons. The seed pattern is not only composed of 2 similar halves, but made up of six identic sixth parts. Moreover, each of these sixths is symmetric in itself (Ill. 3). This explains why the two figures differ only in their rotational direction (clock- or anticlockwise).

Illustration 3: symmetry

Illustration 3: symmetry

A seed pattern that is composed of multiple similar elements, after a certain number of rotation steps will look exactly the same as in its original position. The Näpfchenstein seed pattern is very well suited to illustrate this.

Illustration 4: rotation steps

Illustration 4: rotation steps

In ill. 4 we first fix the original position of the seed pattern (grey) in fig. a. Then, we place a copy (black) on top of it (fig. b). In the original position, this copy exactly covers the original. Therefore only the black copy can be seen. Third, let’s rotate the (black) copy by one step and connect the next end of it with the center (fig. c). Now the original is uncovered and becomes visible. In fig. d we rotate the copy one step further, and it completely covers the original again.

After only two rotation steps the seed pattern is self-covering. This, of course, generates the same figure as in the original position.

The seed pattern of Rockcliffe Marsh is self-covering after six rotation steps. The seed pattern of my demonstration labyrinth needs a full circle rotation.

The number of rotation steps needed for a seed pattern to be self-covering corresponds with the number of figures that can be generated with it. With the seed pattern of my demonstration labyrinth, 12 figures, with the one of Rockcliffe Marsh 6 figures, and with the Näpfchenstein seed pattern, 2 figures were generated. It has to be kept in mind that the same figure in clockwise and anti-clockwise rotation is counted as two figures. In some seed patterns with an inherent symmetry in each element, the number of different figures may be further reduced.

Related Posts:

Seed Pattern – Shifting the Center / 2

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.

Related posts