Posts Tagged ‘sequence of segments’

The notation using coordinates is consistent, understandable and works well in alternating and non-alternating one-arm and multiple-arm labyrinths. However it has a particular property. Whereas in multiple-arm labyrinths the number of segments is obtained by multiplying the number of arms with the number of circuits, this is not sufficient in one-arm labyrinths. They necessitate a partition in two segments per circuit. And thus the sequence of segments has the same length in one-arm and two-arm labyrinths with the same number of circuits.

I will show this here with the example of a two-arm labyrinth with 7 circuits.

This is a labyrinth I had designed during the course of my studies on the labyrinth of the Chartres type and its further developments.

According to the number of arms and circuits, this labyrinth has 14 segments. The corresponding sequence of segments is:

Now let us remember the sequences of segments in th one-arm labyirnths from the last post. For comparison I show here the sequence of segments of the basic type labyrinth.

This also has 14 numbers and thus has the same length as our two-arm labyrinth.

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With the coordinates for segments from the last post (see related posts below) we have now found an understandable notation for the sequence of segments of labyrinths. Here it seems important to me to add that such coordinates can also be used for one-arm labyrinths. I will show this with the examples for which I had already shown the sequences of circuits (see related posts). For this, each circuit has to be divided into two segments.

Partitioning of Circuits in Segments

Next we write the sequences of segments for the three examples and also compare them straightaway with their sequences of circuits.



A unique notation for one-arm labyrinths can also be achieved, if we can write two different numbers on the same circuit, one for each side of the axis. For this, the circuits have to be partitioned into two segments. This allows us to write unique sequences of segments for alternating and non-alternating labyrinths. Also it is possible to use the same form of notation in one-arm and multiple-arm labyrinths. However, this notation will always need 14 coordinates for each one-arm labyrinth with 7 circuits. This is clearly more digits than are needed for the sequences of cirucits with separators.



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At the end of the last post (see related posts) we were left with the following problem. If we number the segments consecutively, we obtain a unique seqence of segments. However it can not be directly seen in the sequence of segments which circuit is encountered by the pathway. If we number the segments by circuits, the sequence does indicate which circuit is encountered. However it then looses the uniqueness.

Now there is a possibility to combine the numbering. That means to write a number for the circuit first, then a separator and then a number for the segment. In the example of the labyrinth by Valturius this looks as follows (fig. 1).


Figure 1. Numbering by Circuits and Segments


The labyrinth has four circuits and three arms, and thus also three segments per circuit. The first number indicates the circuit, the second indicates the segment. This numbering provides some kind of coordinates for the various segments.

Let us now write the sequences of segments for the alternating and non-alternating labyrinths from the last post using this numbering.


Figure 2. Sequences of Segments of the Alternating and Non-alternating Variants

Both variants have their own unique sequences of segments. In each element of the sequence of segments it can be identified which circuit and which segment is encountered by the path. Such a sequence of segments can be easily generated and memorized. A shortcoming of this numbering is that each element is composed of two figures and a separator. Furthermore the elements must be clearly separated from each other. Therefore this sequence of numbers requires more digits and more space.

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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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There is now a new labyrinth at this extraordinary and historically significant place.

In the church Mariä Schutz a labyrinth was built during the three-year period of renovation and rebuilding on the area of the Vogelsburg.
Father Bernhard Stühler, hospital chaplain of the Juliusspital, initiated it. Architect Stephan Tittl from the office SequenzSieben Würzburg made the architectural design of the church and delivered the layout. During the inauguration of the project turned out, that Sr. Hedwig Mayer, prioress of the Augustinusschwestern on the Vogelsburg, always had wished a labyrinth.

The new labyrinth

The new labyrinth

It’s a newly created sector labyrinth with 5 circuits. In the middle is a bowl-shaped pitch circle to divert the direction. The dividing bars form a cross and are arranged symmetrically.
The diameter amounts to 6 m, the middle to 2 m. The ways are 34 cm wide and are marked by a 6 cm wide brass sheet on the terrazzo floor. The way into the center amounts to about 64 m.

One enters the church from the south over an outside stair. On the left hand of the entrance is the labyrinth which is aligned to the west and the east. You enter it from the west, arriving the center, one looks to the east in the direction of the altar and leaves it also again in this direction.

The Oberpflegeamtsdirektor (Chief Administrative Officer) Walter Herbert of the Juliusspitalstiftung (foundation Juliusspital) said on occasion of the inauguration of the altar in May, 2016 to the interior design of the church:

With the elected interior design and with the labyrinth in the ground we would like to offer to every visitor of the Vogelsburg the possibility to find the way to one’s own center, to get back to basics and to find the possibility of steering towards God in the church space.

The segments of the 5 circuits

The segments of the 5 circuits

As Andreas proposed in his last article we can number the 20 segments for the 5 circuits in this 4-armed labyrinth. The sequence of segments can be derived from it for the pathways. Some segments form a connected section which runs through several quadrants. These segments can be marked by brackets. The sequence of segments then looks as follow: 9-5-(1-2-3-4)-8-12-(16-15)-11-(7-6)-10-(14-13) – (17-18-19-20)-21. I write the result a little bit differently than Andreas and still add the center at the end. Inside this labyrinth we have as a specific feature two segments which enclose the full length of a circuit.

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In one-arm labyrinths, each circuit is represented by one number. Therefore it is possible to capture even quite large labyrinths appropriately with the level sequence. In labyrinths with multiple arms, the pathway may repeatedly encounter the same circuit. Various possibilities exist to take account of this in the level sequence. For this, according to the number of arms, the circuits have to be further partitioned to segments. Here I will show a method in which all segments are numbered through.

For this I use an example of a labyrinth that has repeatedly been presented on this blog. It has 3 arms and 3 circuits.


First, each circuit is partitioned to three segments. One segment corresponds with a unit of the pathway between two arms. Next, the segments have to be numbered through. This can be done in different ways. Here I number them from the outside to the inside and one circuit after each other.


Now we can track the course of the pathway through the various segments. This results in the sequence of segments encountered by the pathway. In labyrinths with multiple arms the level sequence thus extends to a sequence of segments.

The sequence of segments of this labyrinth is 7 4 1 2 5 8 9 6 3. The length of this sequence of numbers is a result of the number of circuits multiplied with the number of arms. Thus, for a labyrinth with 3 circuits and 3 arms, 9 numbers are required. Whereas in a one-arm labyrinth with 3 circuits only 3 numbers are needed.

However, besides the numbers no other information is needed. The sequence of segments itself determines where the pathway makes a turn or traverses an axis. In one-arm labyrinths this had to be indicated additionally by use of separators.

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