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).
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 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 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 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.