On my own behalf

Featured

Welcome to the Labyrinth

The topic of this blog is the labyrinth. Under nearly all aspects, I would like to arouse your interest on the fascinating lines and the meaning of this old object. Being an old surveyor I put my focus on the geometrical shape.
A new post should be published about twice a month. Meanwhile I am accompanied by Andreas Frei as coauthor.

Contents

In a blog the single posts (articles) are disposed in reverse order: the latest posts first, the older ones following. The display of the content is thus different from a website where it is always permanent.

Anyone who is looking for something special about labyrinths or just wants to know what can be found on this blog, maybe would like to have an overview.

I can provide this now and offer it as an own page titled Contents.

The register with the table of Contents is on top of the blog under the header image next to About us.

For a better view

For a better view

Subscribe

If you would like to be constantly informed about new posts, you can also follow this blog by subscribing.
The widget: SUBSCRIBE TO BLOGMYMAZE  is on the sidebar between IN SEARCH OF … and BLOGROLL.
You only need to enter your e-mail address to receive a mail when a new article is posted.

Advertising

This blog uses a free and unlimited offer from WordPress.com. Therefore, advertising is displayed at various points. We ask for your understanding.

Rights of use

Most of the pictures and graphics were created by Andreas Frei and me (Erwin Reißmann), unless stated otherwise, and are provided under license CC BY-NC-SA 4.0.

Triple Barriers – Combinations of Sector Patterns

In the last post, I have identified the sector patterns that can be placed in the first or last sector (see: related posts 1, below). These are shown here once again in figure 1.

Figure 1. Sector Patterns 

In the second-to-last post, the four possibilities were shown, how the pathway can take its course along all side-arms (related posts 2). This provides us with all basic material we need to be able to generate any sector labyrinth with seven circuits using exclusively triple barriers in all side-arms. I therefore also show here all four possibilities again and in addition I indicate how these can be combined with the sector patterns from quadrants A – D for the first or last sector. 

Figure 2 shows, that the first possibility for labyrinths with an even number of axes can begin with one of the sector patterns from quadrant A and can end with one of the sector patterns from quadrant B. 

Figure 2. Begin with Pattern from Quadrant A, End with Pattern from Quadrant B

The second possibility for labyrinths with an even number of axes can begin with one of the sector patterns from quadrant C and end with one frome quadrant D (fig. 3). 

Figure 3. Begin with Pattern from Quadrant C, End with Pattern from Quadrant D

The first possibility for labyrinths with an odd number of axes can begin with one of the sector patterns from quadrant C and end with one from quadrant B (fig. 4). 

Figure 4. Begin with Pattern from Quadrant C, End with Pattern from Quadrant B

The second possibility for labyrinths with an odd number of axes can begin with one of the sector patterns from quadrant A and end with one from quadrant D (fig. 5). 

Figure 5. Begin with Pattern from Quadrant A, End with Pattern from Quadrant D

Thus, for each of the four possibilities for the course of the pathway there are four patterns for the first and four patterns for the last sector. These can be combined to 16 different patterns each. For any even or odd number of axes there exist two possibilities for the course. Therefore, for any number of axes there exist 32 different sector labyrinths made-up exclusively of triple barriers. 

Now, I don’t have the intention to derive all these labyrinths here. However, I want to show with two examples using the first possibility for the course of the pathway, how they are derived. Figure 2 illustrates how a labyrinth with 2 axes is generated. For this, we select one of the sector patterns from quadrant A for the first and one from quadrant B for the last sector and with these we replace the place holders for the labyrinth with two axes from the first row of figure 2. For this purpose I choose the pattern on top left from quadrant A and the one on top right from quadrant B. However, any other two patterns from these quadrants could also be combined. 

Figure 6. Combination for a Labyrinth with 2 Axes, first Sector from Quadrant A and last Sektor from Quadrant B

These two sector patterns now must be supplemented with the connections to the outside, among each other and to the center in order to generate the comprehensive pattern for the sector labyrinth with 2 axes and triple barrier. This pattern and the resulting labyrinth are shown in figure 7. This is one of 16 possible sector labyrinths with a triple barrier and 2 axes for the first possibility of the course of the pathway in labyrinths with an even number of axes. 

Figure 7. The Labyrinth with 2 Axes

An analogous procedure was used to generate a labyrinth with 6 axes. For this, I chose two different sector patterns for the first and the last sector. (fig. 8). 

Figure 8. Combination for a Labyrinth with 6 Axes, first Sector from Quadrant A and last Sektor from Quadrant B

The result of this combination is represented as a pattern and as a labyrinth in figure 9. Again, it is one of 16 possible sector labyrinths with triple barriers and 6 axes and the first possibility for the course of the pathway in labyrinths with an even number of axes. 

Figure 9. The Labyrinth with 6 Axes

Independent of the number of 2, 4, 6, 8 etc. axes, there are always 16 different sector labyrinths with triple barriers for this possibility of the course of the pathway. This number is determined only by the four sector patterns from quadrant A and B each. 

Related Posts

  1. Triple Barriers – Patterns for the First and the Last Sector
  2. Sector Labyrinths with Triple Barriers

 

How to Repair the Mistakes in Historical Scandinavian Labyrinths, Part 1

Richard Myers Shelton advocates in his guest contribution from January 17th, 2021 the thesis that the alleged errors in some historical Scandinavian labyrinths are not at all, but that these labyrinths had a completely different meaning than we assign to them today. So they were deliberately created in this way.

I can understand his train of thought, but still allow myself a different perspective on these labyrinths.
All these labyrinths were laid on the floor with stones. This makes them very susceptible to change. During our Sweden tour in 2007, we, the participants of this trip, organized by Jeff and Kimberly Saward, always carried out small “repair work” on almost all the Troytowns we visited, by bringing slipped stones back into their correct position.

It is therefore easy to imagine that improper procedures or willful “sabotage” have changed the layout over the years. Or that people simply acted in ignorance of the meaning of the labyrinths. These labyrinths were also not laid out according to the well-known basic pattern. A mistake in the layout of the labyrinth could have crept in.
The researchers’ records could also contain errors.

Most of the surviving Scandinavian labyrinths from this period look like what we imagine labrinths to be today.

First a “correct” labyrinth. I use the images from Nigel Pennick: European Troytowns. It is a walk-through labyrinth that I like to call a Wunderkreis. In the center there is a double spiral, the outer circuits wind around two turning points. Depending on which path (here on the left, the 7th circuit) I take first, I come first into the spiral  or in the outer circuits (here by entering on the 5th passage on the right side). In the end I walked through the whole labyrinth. There is no actual center like in the classical labyrinth. The labyrinth can also be mirrored or have more or fewer circuits (as it is in fig. 3).

Fig. 1: A stone labyrinth type Wunderkreis

Fig. 1: A stone labyrinth type Wunderkreis

Now the second Borgo labyrinth. Again, I use the illustration by Nigel Pennick.

Fig. 2: The "faulty" Borgo labyrinth

Fig. 2: The “faulty” Borgo labyrinth

I added the numbering of the circuits to the drawings.
It is an open labyrinth with direct access to the center. This sometimes happens in Scandinavian Troytowns. If you look more closely, however, it is a Wunderkreis with a branch in the path, a so-called walk-through labyrinth.
The inner double spiral (admittedly: very bulged) is formed by the circuits 8 – 15. The outer lanes are formed by the circuits 1 – 9.

So where is the mistake?
Roughly speaking, the Wunderkreis is formed in three stages. First the center and the entire upper part up to two thirds of the entire circumference is built. Then the right and left lower part. In Figure 1 you can see that the 5th (right) and the 7th (left) access lead into the labyrinth.
In the Borgo Labyrinth (fig. 2 and fig. 3) this would have to be the 7th on the left side and the 9th on the right side because the labyrinth is larger and mirrored. On each side I always need an odd number of lines to be connected (as explained in more detail in the 2nd related post below). In our case, four lines have to be moved to create a free end.

And this is how the “correct” labyrinth could look like:

Fig. 3: The "repaired" Borgo labyrinth

Fig. 3: The “repaired” Borgo labyrinth

In my opinion the mistake was made by the person (s) who built this labyrinth, not the rapporteur, Aspelin.
The labyrinth was probably created in a transition period from the classical labyrinth, laid out according to the seed pattern, to the walk-through labyrinth such as the Wunderkreis, which is laid out according to other principles.
Or perhaps the builder (s) really wanted to close the passage to the middle (on the 7th circuit) and lead to the dead end in the fourth circuit, as Richard Myers Shelton suspected?

Related Posts

Additions to the New Year’s Labyrinth 2021

Today I want to give some more information on the New Year’s Labyrinth of this year. In the caption of the figure, I had indicated that it has 6 axes, 7 circuits and symmetrically arranged single barriers, double barriers and a triple barrier and characterized it as self-transpose (see related posts 1, below). Here I want to explain more in detail what that means. 

Figure 1 shows the pattern of the New Year’s Labyrinth. The axes are numbered. The main axis that is represented on both sides in the pattern, bears number 6. This is not a sector labyrinth. It is made up of two similar halves that are mirrored at the 3rd axis. One could also see in this two superordinated sectors that are composed of 3 segments each. 

Figure 1. Pattern of the New Year’s Labyrinth 2021

In fig. 2 the transpose pattern is derived from the pattern (a). For this purpose, first, the pattern has to be mirrored horizontally (against the vertical dashed red line). This results in pattern (b). Mirroring of the pattern interrupts the connections to the exterior (triangle) and to the center (bullet point). The connection lines (grey) point to the wrong direction. In order to reconstruct these connections after the mirroring, second, these two connection lines have to be flipped as indicated with the red arrows. As a result, we obtain in (c) the transpose of pattern (a). What is special in the New Year’s Labyrinth is, that its transpose is the same. Therefore, this labyrinth is referred to as self-transpose. 

Figure 2. Deriving the Transpose Pattern

With its six axes, this labyrinth is well suited for a transformation into the Flower-of-Life style (related posts 2). For this, the Ariadne’s Thread is used, as shown in figure 3. 

Figure 3. New Year’s Labyrinth in the Flower-of-Life Style

Related Posts

  1. Labyrinth for the New Year 2021
  2. Flower of Life – on Track

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