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.

Related Posts

How to Make the Related Labyrinths

I use a different method to generate the related labyrinths than Andreas. But I’ll get the same result. This is how we complement each other.

Essentially, I am using the level or path (= circuit) sequence to get the version of a particular labyrinth I want. Also, I am taking the path sequence to construct the labyrinth, not the seed pattern.

The basic labyrinth
Fig.1: The basic labyrinth

I usually number from the outside in (the left digits in blue), additionally here from the inside to the outside (the right digits in green).
The level sequence for the basic labyrinth is here: 0-1-2-5-4-3-6. “0” stands for outside, “6” stands for the center. We have a 5 circuit labyrinth in front of us. “1” to “5” are the numbers of the circuits (paths), hence the path sequence 1-2-5-4-3 (fig. 1).


To create the dual labyrinth, I just use the green numbers on the right side of the basic labyrinth. I determine the level sequence by going outwards from the center. I get 6-3-2-1-4-5-0. Now I draw a labyrinth using this row of digits, going from the outside to the center. But first I replace “6” with “0” and “0” with “6”, I swap inside and outside as it were. The new level sequence is then: 0-3-2-1-4-5-6 (fig. 2).

The dual labyrinth
Fig. 2: The dual labyrinth to the basic labyrinth

The left numbers now indicate the level sequence: 0-3-2-1-4-5-6. If I now read the green numbers on the right side, I of course get the basic labyrinth again.


Now I use another technique to get the transposed labyrinth. I take the path sequence of the dual labyrinth, here: 3-2-1-4-5 and complement all numbers to “6”.
3-2-1-4-5 dual
3-4-5-2-1 transpose
————
6-6-6-6-6
The second line, completed by “0” for the outside and “6” for the center, gives the level sequence for the transposed labyrinth: 0-3-4-5-2-1-6 (fig. 3).

But there is still a different technique to get there: I can read the path sequence from the basic labyrinth backwards and again complete with “0” and “6”.
1-2-5-4-3 base
3-4-5-2-1 transpose
The second line, completed by “0” for the outside and “6” for the center, also gives the level sequence for the transposed labyrinth: 0-3-4-5-2-1-6 (fig.3).

The transpose labyrinth
Fig.3: The transposed labyrinth to the base labyrinth

If I now take the green numbers of the right side, I’ll get the dual of this transposed labyrinth, which is the next, the complementary labyrinth with the level sequence: 0-5-4-1-2-3-6 (fig. 4).


But again there is also the above described technique to get the complementary labyrinth. I take the basic labyrinth and complement the numbers of its path sequence to “6”.
1-2-5-4-3 base
5-4-1-2-3 complement
————
6-6-6-6-6
The second line, completed by “0” for the outside and “6” for the center, gives the level sequence for the complementary labyrinth: 0-5-4-1-2-3-6 (fig. 4).

I can also take the dual labyrinth and read the path sequence backwards, and again add “0” and “6”.
3-2-1-4-5 dual
5-4-1-2-3 transpose
The second line, added with outside and center gives the level sequence for the complementary labyrinth: 0-5-4-1-2-3-6 (fig. 4).

The complement labyrinth
Fig.4: The complementary labyrinth to the base labyrinth

If I now take the green numbers on the right side, I’ll get the dual labyrinth to this complementary labyrinth, namely the transposed labyrinth with the level sequence: 0-3-4-5-2-1-6 (fig. 3).


So we have seen three different ways to transform one labyrinth into another by using the path or level sequence.

However, it only takes two methods to create the appropriate labyrinths. I personally prefer the “transposing” technique and the “complementing” technique.

First we have the basic labyrinth (fig.1). Through transposing the path sequence of the basic labyrinth 1-2-5-4-3 into 3-4-5-2-1 I’ll then get the transposed labyrinth (fig.3).
This transposed labyrinth with the path sequence 3-4-5-2-1 I transform to the dual labyrinth by complementing the path sequence to 3-2-1-4-5 (fig.2).
This dual labyrinth I then transform to the complementary labyrinth by transposing its path sequence 3-2-1-4-5 into 5-4-1-2-3 for the complementary labyrinth (fig.4).
For control purposes, I can transform the basic labyrinth into the complementary by complementing the path sequence 1-2-5-4-3 of the basic into 5-4-1-2-3 (fig. 4) for the complementary.

All of these transformation methods have the same effect as the rotating and mirroring techniques by Andreas.

Related Post

The Related Labyrinths

The subject of closely related labyrinths has been treated on this blog already repeatedly. When deriving the complementary labyrinth, I have encountered the group of the four related labyrinths. This occurred when I derived the complementary of the base labyrinth and then derived the duals of the base and the complement (see: related posts 1, below). As a fourth labyrinth there indirectly resulted the dual of the complement what is nothing else than the transpose. 

Richard Myers Shelton, however, has published this group of closely related labyrinths already earlier°. In his article, he introduced the concept of the transpose. He derived the transpose of the base labyrinth and subsequently derived the duals of the base and the transpose. Thus, as a fourth labyrinth there indirectly resulted the dual of the transpose what is the same as the complementary labyrinth. 

We are thus presented with two different representations of the same situation. In the following, I want to examine this further and in addition will present a third version of the same situation. I will again base my considerations on the labyrinth with one axis and five circuits I had already used when deriving the complementary labyrinth (related posts 2). This is the second of the eight different alternating labyrinths with one axis and five circuits and the simplest labyrinth for which a complete group of four closely related labyrinths exists. This is the “base” labyrinth and has the three relatives, the “dual”, the “transpose”, and the “complement” as shown in fig. 1.

Figure 1. The four Related Labyrinths
Figure 1. The four Related Labyrinths

In fig. 2 I show, how we can directly derive the other three labyrinths from the base labyrinth. For this, the pattern of the labyrinth is used. The pattern is obtained from the Ariadne’s Thread in the rectangular form (related posts 3). 

In the first line, the derivation of the dual pattern is shown. For this purpose, the base pattern is rotated by 180 degrees. The connections to the outside (triangle) and to the center (bullet point) are interrupted. After rotation, these ends are reconnected, but the connections are exchanged.

The second line shows how the transpose pattern is derived. For this, the base pattern is mirrored horizontally (against the vertical). Again, the connections to the exterior and the center are interrupted. After mirroring, the ends point to the wrong direction. They have to be flipped to be reconnected with the exterior and the center. These two operations mirroring and flipping are combined in icon symbolizing the transpose operation. 

In the third line, the derivation of the complementary pattern is shown. This is obtained by vertically mirroring the pattern (mirroring against the horizontal). Similarly, the connections to the exterior and the center are interrupted and after mirroring, the ends point to the wrong direction. They have to be flipped to be reconnected with the exterior and the center. These two operations mirroring and flipping are combined in icon symbolizing the complement operation. 

Figure 2. Direct Derivation of the Related from the Base Labyrinth
Figure 2. Direct Derivation of the Related from the Base Labyrinth

These are the three operations that can be used to derive the dual, the transpose and the complementary labyrinth from the base labyrinth. A direct application of two of these operations is sufficient to indirectly effect the third one. Thus, the group of the four closely related labyrinths can be represented in three different ways. More about this in the next post. 

° Shelton, Richard Myers. 2015. „Wayland’s New Labyrinths“ Caerdroia 44, 44-55.

Related Posts:

  1. The Complementary versus the Dual Labyrinth
  2. The Complementary Labyrinth
  3. From the Ariadne’s Thread to the Pattern

The “Mistakes” in Historical Scandinavian Labyrinths

This guest post was kindly contributed by Richard Myers Shelton when a conversation was developing over a previous article. His contribution:

Whose Mistake?

The complement of the classical 7-course labyrinth is highlighted in Erwin’s recent post “The Complementary Classical 7 Circuit Labyrinth” (20 September 2020). The term “complement” is due to Andreas; see his post of 2 July 2017. The complement of a labyrinth visits the courses in reverse order: the complement of the classical labyrinth, for example, traces the courses in the order (5 6 7 4 1 2 3), just the reverse of the standard classical order (3 2 1 4 7 6 5). Photos of a modern example of this design on the banks of the Rhine near Duisburg are featured in Erwin’s post.
My comment on the post pointed out that “complementary classicals” are not unknown historically. The complement of the 15-course classical labyrinth was reported near Borgo (modern Porvoo, Finland, some 50 km east of Helsinki) by Johan Reinhold Aspelin in a letter to the Berlin Society for Anthropology, Ethnology and Prehistory. (The letter is included in the report of the Society for 17 Nov 1877, in the Zeitschrift für Ethnologie, vol. 9, 1877, pp. 439–442.)
This is one of two stone labyrinths near Borgo mentioned in Aspelin’s letter; the other is a straight-forward Baltic-style labyrinth with a central spiral and separate exit path. Both are illustrated in the letter, and are reproduced in Figs 2 and 3 of Nigel Pennick’s “European Troytowns” (http://www.cantab.net/users/michael.behrend/repubs/et/pages/pennick.html). The complementary classical at Borgo is also illustrated in Fig. 125, page 148, of Matthews’s Mazes and Labyrinths.

Figure 1: Aspelin’s drawing of Borgo

Figure 1: Aspelin’s drawing of Borgo

But the Borgo complement has a peculiarity, easily visible in Aspelin’s drawing: while it does trace the two meanders in reverse order (inner first, then outer), the path does not lead to the center. Instead it dead-ends in the outer meander: the outermost course 1 is connected to course 7 instead of course 6; course 4 therefore has no escape, forcing it to dead-end. In fact, the labyrinth incorporates a thick wall of rocks, completely isolating the left side from any access to the center. The difference can be seen by comparing the two level charts below: one for the complement of Classical-15, one for Borgo.

Figure 2: Level chart for complementary Classical-15

Figure 2: Level chart for complementary Classical-15

Note: The entrance is at the bottom left, the center is at the top right (unlike as in the diagrams by Andreas or Erwin). Figure 2 is self-dual.

Figure 3: Level chart for the Borgo labyrinth

Figure 3: Level chart for the Borgo labyrinth

In my comments, I characterized the dead-end as a mistake; and Erwin and I exchanged a few emails about whether the mistake was in the labyrinth itself or in Aspelin’s drawing of it. But in fact, several early drawings of labyrinths from Scandinavia show features we commonly think of as mistakes – paths that don’t lead to the center, or branching paths that force the walker to make a choice, or even portions of the path that are completely isolated from the exterior (and sometimes from the center as well).
Another good example of this is Karl Ernst von Baer’s well-known diagram from 1844 of the stone labyrinth on Wier Island in the Gulf of Finland (known today as South Virgin Island). This has a fork in the path, and while one choice leads to the center, the other dead-ends near the perimeter.

Figure 4: von Baer’s drawing of Wier

Figure 4: von Baer’s drawing of Wier

I doubt that these early researchers were being careless in their diagrams. On the contrary, they were earnestly trying to preserve a rapidly vanishing past from oblivion, carefully recording these objects for posterity. I conclude, therefore, that these anomalies were present in the labyrinths themselves. But we ought not to conclude that what we see as anomalies are mistakes. Instead, it is we who make the “mistake”: namely, of assuming that the people who built these labyrinths intended them to be walked as we walk them today; of assuming that any labyrinth that is not walkable that way must be mistaken.
Some of the odd labyrinths probably were mistakes. A pattern that shows up with some frequency is what you get by drawing a classical labyrinth from a seed pattern that forgets to include the four dots inside the four angles.
And it is clear from various accounts that labyrinths in Germany and England were indeed often meant to be walked or run ceremonially or in contests or games, particularly in association with spring-time celebrations of May Day or Easter or Whitsun – and that this custom appeared later in Scandinavia as well.
(In this regard, it is curious that English and German are the only languages whose word for Easter recalls the pagan goddess Ēostre, as recorded in Bede; other languages refer to the Christian nature of the holiday, or to Passover or the end of Lent.)
But the evidence and the stories from Scandinavia (and further east into Estonia and Russia) hint at a darker purpose: many of these devices were probably intended as traps, perhaps inheriting the idea that led the Romans to place labyrinths near entry-ways to ward away evil.
Christer Westerdahl’s article “The Stone Labyrinths of the North” (Caerdroia 43, 2014) lists several contexts where labyrinths would have been intended this way: near graveyards (to keep the dead in their graves), near ancient burial mounds (to hold back their ancient and possibly non-human inhabitants), near gallows (against the vengeful spirits of executed criminals), along coastlines (against trolls or other bad luck seeking to follow the fishing boats, or even to hold ill winds and currents at bay).
I am particularly struck by examples from Iceland. In “The Labyrinth in Iceland” (Caerdroia 29, 1998) Jeff and Deb Saward tell of their quest to locate all the recorded Icelandic stone labyrinths. They found that only one still survives, at Dritvík on Snæfellsnes. When the Sawards saw it, this labyrinth was heavily overgrown, but someone has since restored it as a typical Baltic-style labyrinth. A drawing by Brynjúlfs Jónssonar from around 1900, however, shows a different plan, with four separate paths, some ending in dead-ends and one completely isolated.

Figure 5: Jónssonar’s diagram for Dritvík, ca. 1900

Figure 5: Jónssonar’s diagram for Dritvík, ca. 1900

The Sawards also found three labyrinths carved on old wooden bed-boards preserved at the National Museum of Iceland in Reykjavík. One of these has the 7-course classical design, but two others (NMI 3135 and NMI 5628) share identical plans with isolated paths. In fact, while this shared plan is not the same as the old Dritvík plan (because it has two more courses), it shows the same general arrangement, with a large meander on one side opposite two smaller ones on the other. And in all three cases, the path that dead-ends in the large meander also dead-ends in the center. These features are clearly not haphazard; the same general design principle (the “Icelandic way”?) was at work.

Figure 6: Diagram for NMI 3135

Figure 6: Diagram for NMI 3135

Figure 7: Diagram for NMI 5628

Figure 7: Diagram for NMI 5628

Why include dead-ends or isolated paths at all? The stories seem to indicate that the ordinary classical or Baltic designs were considered effective at slowing trolls down long enough so that a boat could get safely away across the water.
But if the design contained in its very construction the magic of “unwalkable-ness”, it could be even more effective! The design itself becomes imbued with the property of entrapment or imprisonment. In this way, might it not become all the more powerful at holding evil things at bay? It would not just slow them down; it could hold them fast!
To us today this doesn’t seem entirely logical. But sympathetic magic isn’t built strictly on logical analogy alone; our irrational hopes and fears get mixed in as well. Consider the wall in the Borgo labyrinth: The dead-end by itself should have slowed the trolls and ghosts down. Why add that massive wall along the side of the meander?? Logically, this seems entirely superfluous, as the trolls and ghosts can just turn around and retrace their path to get out. But somehow that wall must make the trap seem that much more secure.

— Richard Myers Shelton, 17 December 2020