The theme of this blog is the labyrinth in almost all aspects. About twice a month a new post should appear.
Meanwhile, Andreas Frei from Switzerland is my co-author.

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Where does a labyrinth belong? And what relatives does it have? How do I actually sort the related labyrinths in a group? What kind of relationships are there? Or: How do I find the related ones in a group?

If I want to know something more, I first take an arbitrary labyrinth and generate the further relatives of a group by counting backwards and completing the numbers of the circuit sequences. It doesn’t matter whether I “catch” the basic labyrinth by chance or any member of the group.

As an example, I’ll take the 11 circuit labyrinth chosen as the second suggestion in my last post. Here it can be seen in a centered version in Knidos style:

The level sequence is: 0-7-2-5-4-3-6-1-8-11-10-9-12. The entrance to the labyrinth is on the 7th circuit, the entrance to the center is from the 9th circuit. This is the reason to name it 7_9 labyrinth.

By counting backwards (and swapping 0 and 12), I create the transpose labyrinth to it: 0-9-10-11-8-1-6-3-4-5-2-7-12.

The entrance to the labyrinth is on the 9th circuit, and the entrance to the center is on the 7th circuit.

Now I complete this circuit sequence 9-10-11-8-1-6-3-4-5-2-7 to the number 12 of the center, and get the following level sequence: 0-3-2-1-4-11-6-9-8-7-10-5-12. This results in the corresponding complementary version.

Now a labyrinth is missing, because there are four different versions for the non-self-dual types. The easiest way to do this is to count backwards again (so I form the corresponding transpose version) and get from the circuit sequence 0-3-2-1-4-11-6-9-8-7-10-5-12 the circuit sequence: 0-5-10-7-8-9-6-11-4-1-2-3-12. Alternatively, however, I could have produced the complementary copy by completing the digits of the path sequence of the first example above to 12.

The entrance to the labyrinth is made on the 5th circuit, and the entrance to the center is made from the 3rd circuit.

Now I have produced many transpose and complementary copies. But which is the basic labyrinth and which the dual? And the “real” transpose and complementary ones?

Sorting is done on the basis of the circuit sequences. The basic labyrinth is the one that starts with the lowest digit: 0-3-2-1-4-11-6-9-8-7-10-5-12, in short: the 3_5 labyrinth, i.e. our third example above.

The next is the transpose, the 5_3 labyrinth, the fourth example above.

This is followed by the dual, the 7_9 maze, which is the first example above.

The fourth is the complementary labyrinth, the 9_7 labyrinth, the second example above.

The order is therefore: B, T, D, C. This is independent of how the labyrinth was formed, whether by counting backwards or by completing the circuit sequences.

To conclude a short excerpt from the work of Yadina Clark, who is in the process of working out basic principles about labyrinth typology:

Groups

Labyrinths related by Base-Dual-Transpose-Complement relationships

Any labyrinth in a group can be chosen as the base starting point to look at these relationships, but the standard arrangement of the group begins with the numerically lowest circuit sequence string in the base position.

All that remains now are the last two enigmatic Icelandic labyrinths. These are the drawings of two identical labyrinths from the National Museum of Reykjavik, NMI 3135 (Fig. 6) and NMI 5628 (Fig. 7) in the guest post by Richard Myers Shelton.

First I bring them into the geometrically correct form I am used to here.

The labyrinths look very similar. One is simply the other, each mirrored, so they are identical.

Both have 11 circuits and a larger center, but it is not possible to reach it. And there are only dead ends, but not all of them can be reached either. There is a branching for this, similar to the Wunderkreis. The way via circuit 8 leads to 10 and ends here. The way via circuit 6 leads via 2 and 4 to 3 and ends there. I do not reach the end of 4 and 9 at all. The center can only be reached if I would make a hook directly after entering the labyrinth.

The thicker black lines (= the stone settings) form the uninterrupted line, Ariadne’s thread. But without any beginning or end, different from the Dritvík labyrinth. Presumably, the purpose of these labyrinths lies in the stone settings and not in the path between the lines, as we know it otherwise from all other labyrinths from this time and in this region? But which one should it be? A prison for the spirits or trolls? A gateway to the underworld or the otherworld? A monument to a guardian spirit? For rituals or for magic?

Now my explanation: None of the above. Only the attempt to make once another labyrinth. One with 11 circuits, which are numerous in this region. Most of them are based on the extended seed pattern. But mathematically, there are over 1000 possibilities for an 11 circuit labyrinth, as Tony Phillips has calculated.

The sequence of circuits must always consist of a series of even and odd digits. And the entrance to the labyrinth must be on an odd circuit. In addition, the four dead ends must be replaced. A boundary line may end here in each case, but not a path. So they become turning points.

Now my two suggestions for how the labyrinths could be redesigned:

First I drew an 11 circuit labyrinth according to the extended seed pattern with the cross, four double angles and four points (not shown here). I then numbered the circuits from the outside to the inside and then derived the sequence of circuits: 0-5-2-3-4-1-6-11-8-9-10-7-12. I read this backwards and thus got the sequence of circuits for the transposed labyrinth, namely: 0-7-10-9-8-11-6-1-4-3-2-5-12. With this again, I constructed the Knidos style labyrinth shown here. By the way, the complementary one looks exactly like this, because the basic labyrinth according to the seed pattern is self-dual. So here, from the entrance, I first go to the 7th circuit and from the 5th circuit, I enter the center. So we would have a complementary 11 circuit labyrinth in front of us, just like it was the attempt in the 15 circuit Borgo labyrinth.

The second proposal can be developed from a shifted seed pattern:

For this I take a cross, draw one angle at the top of each side and three angles at the bottom of each side. The points come again into the four corners (not shown here). The level sequence is then: 0-7-2-5-4-3-6-1-8-11-9-12. From this I construct the labyrinth shown here in the Knidos style. The three other relatives of this labyrinth I get then with the methods described in detail in this blog by Andreas by counting backwards and supplementing the circuit sequences. This would give us again three additional new suggestions

However, since there are over 1000 other theoretical possibilities, we ultimately do not know what the authors of the Icelandic labyrinths had in mind and what ideas guided them.