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## Circuits and Segments

In my last post I have shown the sequence of segments in labyrinths with multiple arms. This is unambigous. But as a disadvantage it does not indicate directly which circuit is encountered by the pathway.

Now it is also possible to keep the partition in segments but only number the circuits. This allows to indicate directly in the sequence of segments, which circuit is visited by the pathway. Thus the same number may repeatedly occur in this sequence. This works well in many cases but may also leed to problems. In the labyrinth I had shown in my last post the problem does not occur. Therefore I will illustrate it here with an other example. For this I chose the labyrinth by Valturius as this is a small, understandable example (Fig. 1).

Figure 1. Labyrinth by Valturius. Source: Kern 2000, fig. 315, p. 179.

This labyrinth from a military manuscript by Robertus Valturius of the 15th century has three arms and four circuits. (Please note, that the arms are not proportionally distributed. This, however, has no influence here. I therefore use a proportional distribution for reasons of simplicity.)

Figure 2. Numbering of the Segmente: Left Image by Segment, Right Image by Circuit

Figure 2 shows in the left image the partition and numbering by segments I had already used in my last post. The right Image shows the same partition of segments although numbered by circuits only. As the labyrinth has four circuits, there are 12 segments.

The labyrinth by Valturius is alternating. However there exists a non-alternating labyrinth with the same level sequence. And this brings us back to the problem.

Figure 3. Sequences of Segments Numbered by Segments

Figure 3 shows the alternating labyrinth by Valturius (left image) and the non-alternating variation (right image). They show two different courses of the pathway. These are also correctly represented by the two different sequences of segments. Both sequences of segments are similar for the first 9 segments: 1 4 7 8 5 2 3 6 9 … The sequences of the three last segments, however, are different. In the labyrinth by Valturius the sequence continues with segments ……… 12 11 10. On the other hand, the sequence of segments in the non-alternating variation is ……… 10 11 12.

If, however, we number the segments by circuits, we lose the uniqueness.

Figure 4. Sequences of Segments Numbered by Circuits

Figure 4 shows the same labyrinths as fig. 3. But with their segments numbered by circuits. Both variants have the same sequence of segments 1 2 3 3 2 1 1 2 3 4 4 4. So here we can always identify in the sequence of segments, which circuit is encountered by the pathway. However, for the same sequence of segments there may exist multiple (in this case two) different courses of the pathway. The same problem occured already in the level sequence of one-arm labyrinths.

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## 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.

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.

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.

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## How to Make a Babylonian Visceral Labyrinth (Umma Labyrinth) from a Wunderkreis

And vice versa: How to make a Wunderkreis from a Babylonian visceral labyrinth.

That’s possible, at least with the Babylonian Umma Labyrinth.

The essentials of a labyrinth ly in the course of the pathway expressed by the level sequence, not the external form or layout. More exactly Andreas calls this the pattern.

The Babylonian Umma Labyrinth

The Umma labyrinth has two turning points surrounded by two circuits each and a meander in the middle. The two entries ly outside. There is only one, unequivocal way through the labyrinth.

The Wunderkreis has a double spiral in the centre and two other turning points with arbitrarily many circuits. Besides, a side has a circuit more than the other. The entries are in the middle section.

A large Wunderkreis

In order to indicate the single developing steps I first transform a “completely developed” Wunderkreis into the smallest possible version.

It looks thus: A meander in the middle and two other turning points with a total of three circuits as to be seen in the labyrinth type Knossos.

The smallest Wunderkreis

To be able to compare this small Wunderkreis to the Umma labyrinth, I lay all centres (at the same time the ends of the boundary lines or the turning points) on a single line. Just as if I folded the triangle built from the turning points.

The compressed Wunderkreis

Both entries are here in the middle section, in the Umma labyrinth they are outside and side by side. Besides, there is one more circuit on the left side. Now I add one circuit to the figure and the entry will change to the outer side on the right as well.

One more circuit

I now turn the second entry to the left side. As a result, the two entries  point in different directions.

The two entries outside

Hence, I turn the right entry completely to the outer side on the left beside the left entry. As I do that geometrically correct, two empty areas appear.

The two entries side by side

Now I extend both entry paths by a quarter rotation upwards and turn the whole figure to the right by some degrees . Thus I receive the complete Umma labyrinth.

The Babylonian Umma Labyrinth

If I want to develop the Wunderkreis from the Umma labyrinth, I must leave out some circuits, turn the whole figure and finally raise the middle part.

The nucleus

The supplements made in the preceding steps are emphasised in colour. The nucleus of the visceral labyrinth contains the Wunderkreis.

Surely the Wunderkreis as we know it nowadays was not developed in this way. There are no historical documents to prove that. However, in my opinion the relationship of both labyrinth figures can be proved thereby. They are not simply spirals or meanders. These elements are rather included and connected in a “labyrinthine” way.

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## Variants of the Cretan Labyrinth

The Cretan is the most frequently encountered type of labyrinth, and so for this type we can find a broad range of individual variants. Here I want to show some examples that are of particular interest for various reasons. Unless stated otherwise, all figures are sourced from the book Through the Labyrinth by Hermann Kern. The details can be found here.

Figure 1. Pylos

This graffito on a clay tablet from Pylos dates from 1200 BCE at the latest and is the oldest securely dated labyrinth. It shows the Cretan-type on a rectangular layout.

Figure 2. Silver coin, Knossos

This figure shows the labyrinth with a concentric layout on a silver coin from Knossos, ca. 190 100 BCE. The center of the labyrinth covers with the middle of its circuits. The axis, however, is somewhat eccentric, as the pathway reaching the center is aligned centrally.

Figure 3. Walahfrid

On this drawing from a parchment manuscript by Walahfrid Strabo (808-849), the labyrinth is shown in full concentric form. The axial wall that connects the innermost with the outermost wall of the labyrinth is aligned centrally with the center.

The following examples show, that variants of the layout are not limited to standard forms, such as circles or rectangles.

Figure 4. Heart labyrinth

This heart-labyrith by Mario Höhn is of the Cretan-type, although with an additional closed circuit at the inside. Not all circuits are in parallel course (as with a supposed 7-lane roundabout). Circuits 7 and 6 are limited to the right heart chamber. Circuit 5 leads to the left chamber, where it is connected with the closed 8th circuit.

Figure 5. Double labyrinth

An other method to generate a heart labyrinth was used by Marty Kermeen and Jeff Saward. They apply a double labyrinth (DL). This is made up of two identic labyrinths (L) that are mirrored horizonally and connected to each other. So the actual labyrinth is one of these two part-labyrinths. This is a Cretan-type projected on a half-hearted layout.

Figure 6. Abhuyumani Tantra

This tantric drawing from Rajasthan, India, 19th century, shows the labyrinth arranged on three quarters of a circle – most of it actually is unrolled to a semi circle. Only the turn from the first to the fourth circuit covers the whole third quadrant. The fourth quadrant is not covered by the figure.

Figure 7. Nîmes

This roman mosaic labyrinth from Nîmes, France, 1st century, has an inconspicious rectangular outline. But, like no other, it shows that the layout of a labyrinth is not only limited to its outline (circle, rectangle, heart, etc.). It is also important to consider how the course of the pathway is organized within this outline form. And this is really tricky. Just try to identify the seed pattern of this labyrinth. The course of the pathway is special in at least three points.

• All circuits do not rotate by a full (360°) but only a 3/4 (270°) circle. This is the same as with the Indian labyrinth described above. It is a sort of a 3/4 labyrinth. However, the layout covers all four quadrants.
• The inner circuits are completely embedded in quadrants 1 and 2. Normally all circuits cover all quadrants.
• Only the outer 4 circuits cover all quadrants.

These shiftings and transformations vary the layout of the labyrinth so that it is barely recognizable.

But what makes me classify all these different examples as Cretan-type labyrinths? What do all these have in common? What defines a Cretan-type labyrinth has been repeatedly described on this blog and elsewhere:

• One-arm labyrinth
• alternating, i.e., the pathway does not traverse the axis
• 7 circuits
• level sequence: 3-2-1-4-7-6-5.

It is important to keep in mind that we are dealing with alternating labyrinths. There exist also non-alternating labyrinths. Only among the alternating labyrinths there is exactly one type of labyrinth for each level sequence. The other way round, this allows us to unequivocally describe each type of an alternating labyrinth by its level sequence.

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