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## Sequence of Circuits – Conclusion

The notation with the coordinates is consistent, understandable and works well in one- and multiple-arm, alternating and non-alternating labyrinths. However, for a labyrinth with three circuits, at least 6 segments are needed (in one- and two-arm labyrinths: number of circuits times two, in all other labyrinths: number of segments times number of arms).

Correspondingly, the sequences of segments rapidly increase in their length with the size of the labyrinth. The Chartres type labyrinth e.g. has 44 segments, as have all other types of labyrinths with 4 arms and 11 circuits.

Here I present the sequence of segments of the Chartres type labyrinth for illustration. This is:

Nevertheless this sequence of segments is a well understandable instruction of how to draw the labyrinth. It reads about like this: Go first to the fifth circuit, walk along the first segment (5.1), then proceed to the 6. circuit and stay in the first segment (6.1). Next, go to the 11th circuit in the first segment (11.1) continue on the same circuit to the 2nd segment (11.2), skip then to the 10th circuit in the 2nd segment (10.2) asf. This also implies that from each coordinate subsequent to the previous it becomes clear, whether the path makes a turn (as from coordinate 5.1 to 6.1) or if it traverses the arm (such as from 11.1 to 11.2). However it is a long and complex series of numbers.

Now there are also various other possibilities to write notations for multiple-arm labyrinths that may have less digits. In any case, the labyrinths first have to be notionally partitiond into segments. However in some notations it is possible to combine multiple segments in one term. I will illustrate this here with the example of a notation for the Chartres labyrinth by Hébert°.

This is a notation comparable with the one presented in the post „Circuits and Segments“, where the segments had been numbered by circuits. In this case, if the pathway passes through multiple segments on the same circuit, the number of the circuit was repeated accoridingly. This, for the labyrinth of Chartres would result in 44 numbers. In the notation by Hébert the length of the sequence reduces to 31 numbers. However, each number must now be written with a prefix. For instance, „-“ indicates, that the following number is written only once, as the path traverses only one segment. A prefix „+“, on the other hand, indicates that the following number would have to be written twice as the path passes two subsequent segments. Thus, different prefixes have to be taken into account. And two prefixes will not be sufficient. Additional prefixes will be required to capture the pathway passing through three, four or more subsequent segments, or to indicate that the arm is traversed whilst the path skips onto another circuit. So while this notation is shorter it is also more difficult to apply. Furthermore it is subject to the weakness already discussed earlier, that, althoug it indicates the circuit, it does not indicate the segment actually covered by the pathway.

Other notations exist as well. I do not address this further here. It should have become clear that the sequences of segments in multiple-arm labyrinths rapidly increase in length and complexity. In most types of such labyrinths the sequence of segments is therefore not suited for giving a name. Just try to imagine to name the labyrinth I had shown in January with its sequence of segments. This labyrinth has 12 arms and 23 circuits and thus 276 segments.

I abstain here from writing down the sequence of segments of this labyrinth. It would fill some 14 – 15 lines.

Conclusion

To conclude, I want to come back to the original question whether the sequence of circuits can be used for giving names to the different types of labyrinths. I had two concerns about this:

• First, in one-arm labyrinths this sequence was not unique. However this problem could be easily solved by adding a prefix „-“ only to those numbers of circuits where the pathway traverses the axis. Therefore in not too large types of one-arm labyrinths the sequence of circuits can be used for naming.
• Second, in multiple-arm labyrinths the sequence will rapidly increase in length. It turned out that in these labyrinths the sequence of segments has to be considered and that this usually becomes either be too long or too complex or both. Therefore I consider it not suited for giving name in multiple-arm labyrinths.

° Hébert J. A Mathematical Notation for Medieval Labyrinths. Caerdroia 2004; 34: 37-43.

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