How to sort a Labyrinth Group

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:

11 circuit Classical 7_9 Labyrinth
11 circuit Classical 7_9 Labyrinth

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.

11 circuit Classical 9_7 Labyrinth
11 circuit Classical 9_7 Labyrinth

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.

11 circuit Classical 3_5 Labyrinth
11 circuit Classical 3_5 Labyrinth

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.

11 circuit Classical 5_3 Labyrinth
11 circuit Classical 5_3 Labyrinth

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.

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