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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Tutorial on How to Draw a Classical 7 Circuit Labyrinth in Knidos Style

Here are detailed step-by-step drawing instructions for the construction of a geometrically-mathematically correct labyrinth.

The specifications are as follows: The unit of measurement for the distance between the lines in the axis is 1 m. The diameter of the center should be four times this distance, hence 4 m. The entrance to the labyrinth and the center are aligned with the central axis.

Details on the Knidos style can be found in this article.

Figure 1: First, the center point M1 of the labyrinth is determined. Starting from here, the main axis (vertical line) to the entrance of the labyrinth below is drawn. Then a parallel line is drawn as an auxiliary line at a distance of 1.50 m and an auxiliary circle with a radius of 3 m is drawn in M1. Using an arc, the midpoint M2 is then constructed at the intersection of these auxiliary lines on the right side below.

Fig. 1

Figure 2: The midpoint M3 is constructed by cutting two radii with a radius of 4 m around M1 and M2 to the left of the central main axis.

Fig. 2Figure 3: First the straight lines M1-M2 and M1-M3 are lengthened, then seven circular arcs are drawn around M1 as the center with the radii 2.5 m to 8.5 m. This is Ariadne’s thread, the axis of the pathways, for the labyrinth.

Fig. 3Figure 4: Circular arcs with the radii 0.5 m and 1.5 m are drawn around M2 and M3 up to the ends of the corresponding previously constructed circular arcs. The right circular arc with a radius of 1.5 m only goes up to the intersection with the horizontal construction line and then leads as a straight line to the center M1.

Fig. 4Figure 5: A parallel line is drawn as an auxiliary line at a distance of 1.5 m to the left of the central axis. An auxiliary circle with a radius of 4 m is drawn around M3 as the center point and intersected with the vertical auxiliary line. This creates the center point M4.

Fig. 5Figure 6: The three open arches to the left of the extended line M1 – M3 are connected with the radii 2.5 m, 3.5 m and 4.5 m to the line M3 – M4.

Fig. 6Figure 7: Around M4 as the center point, two curved pieces with the radii 0.5 m and 1.5 m are drawn, the radius 1.5 m only up to the horizontal construction line to M4. From here a straight line connects to the entrance of the labyrinth at the bottom.

Fig. 7Figure 8: Two auxiliary circles with a radius of 4 m are drawn around the center points M2 and M4 and the new center point M5 is constructed to the right of the central axis at the intersection of the same.

Fig. 8Figure 9: In the new sector, the free curved end pieces on the right side are connected with a radius of 2.5 m to 5.5 m to the line M2 – M5 or its extension.

Fig. 9Figure 10: Around M5 as the center, two semicircles with a radius of 0.5 m and 1.5 m are constructed. The complete Ariadne thread for the labyrinth is now drawn.

Fig. 10Figure 11: Parallel to all previous arches, the boundary lines of the labyrinth are now constructed at intervals of 0.5 m. Starting with R 1 m up to R 9 m for the outermost ring. With this all lines for the labyrinth are complete and can be used for different representations of the labyrinth in different variants.

Fig. 11For example here with the same widths for the boundary lines. The Ariadne thread is the free space between these lines:

The Classical 7 circuit labyrinth aligned to the central axis in Knidos style

The Classical 7 circuit labyrinth aligned to the central axis in Knidos style

Here again the previous drawing steps are summarized in a single design drawing, which can be scaled as required.

The design drawing

The design drawing

Here you might view, print or download it as a PDF file.

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The Complementary (transposed) Classical 7 Circuit Labyrinth in Knidos Style

I described this type in the concentric style in my last post (see related posts below). Today it is about the representation of the transposed labyrinth in Knidos style.

The path sequence is: 5-6-7-4-1-2-3-8. The special thing about it is that one enters the labyrinth on the 5th circuit, and the center on the 3rd. circuit.

The walls and Ariadne's thread

The walls and Ariadne’s thread

And yet this type can be aligned to the central axis. This is only possible by editing in the Knidos style.

I come back to the original labyrinth using the same method that I used to get to the complementary type: I add the difference to the last digit (the goal) to the row of numbers in the path sequence. So:
5-6-7-4-1-2-3-8
3-2-1-4-7-6-5-8
8-8-8-8-8-8-8-8
This is then the original, well-known classical (Cretan) labyrinth.

What does the Knidos style actually mean?
By this I mean, above all, that the labyrinth has a larger center than just the width of a path, that it is as compact as possible and, above all, that it is developed from the path sequence and not from the basic pattern for the boundary lines (the walls). So it is Ariadne’s thread, the path in the labyrinth, that determines the construction. And this must be geometrically correct with constant path widths, elements that are as round as possible and as few “spaces” as possible.

Here in another graphic:

The transposed labyrinth in Knidos style

The transposed labyrinth in Knidos style

Here are the drawing instructions for a kind of prototype to be scaled for the axis dimension of 1 m.

The design drawing

The design drawing

Here you might view, print or download it as a PDF file.

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New 5 Circuit Labyrinths with Double Barriers

In dealing with the double-barrier technique in recent posts, I found this installation of Mark Wallinger’s Labyrinths on the London Underground:

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

The special feature of this is that two double barriers are located next to each other in the upper part of the central axis. In the routing chosen by him you move at the transition from the 2nd to the 3rd quadrant first away from the center.

I’ve changed that so much that you would “experience” a movement to the center in a walkable labyrinth.

This is what it looks like:

A new labyrinth in concentric style

A new labyrinth in concentric style

I have also moved the side double barriers and this makes the routing in all quadrants also different. So a new type of labyrinth is born.

Here in Knidos style:

A new centered sector labyrinth in Knidos style

A new centered sector labyrinth in Knidos style

Why not as a two-parted labyrinth?

A new two-parted 5 circuit labyrinth

A new two-parted 5 circuit labyrinth

The left part has the path sequence: 3-4-5-2-1 and the right part: 5-4-1-2-3, so there are two 5 circuit labyrinths in it.

And here again in Knidos style:

A new two-parted and centered 5 circuit labyrinth in Knidos style

A new two-parted and centered 5 circuit labyrinth in Knidos style

The remarkable thing about this type is that both the entry into the labyrinth in the 3rd lane takes place, as well as the entry into the center.

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