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Posts Tagged ‘Sequence of Circuits’

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).

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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.

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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).

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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.)

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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.

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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.

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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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