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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.

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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.

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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.

Calculating the Relatives of the Classical Labyrinth Type

Now, I want to calculate the relatives of the basic labyrinth type / classical (Cretan) type. Those who are already familiar with the subject, will probably know the result. Nevertheless, I will conduct this calculation consequently once here. Figure 1 shows the step from the base labyrinth to the transpose. 

Figure 1.From the Base Labyrinth to the Transpose
Figure 1.From the Base Labyrinth to the Transpose

The calculation of the complement is illustrated in fig. 2. 

Figure 2. From the Base Labyrinth to the Complement
Figure 2. From the Base Labyrinth to the Complement

These directly calculated sequences of circuits of the transpose and the complement are the same. Now, let’s also derive the sequence of circuits of the dual in fig. 2 indirectly. As we know, this can be achieved by writing the sequence of circuits of the complement in reverse order. 

Figure 3. From the Complement to the Dual
Figure 3. From the Complement to the Dual

This results in the same sequence of circuits as for the base labyrinth. What means nothing else than that the labyrinth of the classical or base type is self-dual. In self-dual labyrinths also the two other relatives of them, the transpose and the complement coincide with each other because these are dual to each other and in this case they are also self-dual. 

Now (except for the “labyrinth” with one circuit) there exist no self-complementary labyrinths (see related posts below). Therefore, each group consists of either 2 or 4 different related labyrinths. This, however, applies only to labyrinths with an odd number of circuits, as will be shown in the next post. 

Related Post:

The Dritvík Labyrinth on the Snæfellsnes peninsula in Iceland

Just a few months ago (in June 2021) Daniel C. Browning, Jr. (alias Ancient Dan) visited the Dritvík Labyrinth on the Snæfellsnes peninsula in Iceland (see the first Further Link below)
I warmly recommend reading this article and the associated first part.

That gave me some new insights into this very special labyrinth. Daniel kindly allow me to show some of his photos and graphics here, for which I am very grateful.

First I show Brynjúlf Jónsson’s 1900 plan of the Dritvík labyrinth, which (for me) is clearer than the one I had from Richard Myers Shelton in his guest post from January 2021.

Brynjúlf Jónsson’s 1900 plan of the Dritvík labyrinth
Brynjúlf Jónsson’s 1900 plan of the Dritvík labyrinth

Jónsson calls it Völundarhús (Wayland‘s house). Hermann Kern also states the Islandic labyrinths as Wayland’s houses. Icelandic parchment manuscripts depicting Wayland’s houses were already in existence in the 14th and 15th centuries. However, they are a hybrid of Troy towns and Medieval labyrinths that look very different from the Dritvík Wayland’s house. The other Nordic stone settings are often referred to as Troy towns, Babylons, Jatulintarha, Jericho, Jerusalem and similar. But these often say something about their meaning.

What does the labyrinth look like today? This is shown by an impressive aerial photo of Daniel from June 2021:

Dritvík labyrinth, restored
Dritvík labyrinth, restored, as it appeared in June 2021 (photo © Daniel C Browning Jr, 2021)

You can see the differences to Jónsson’s drawing very clearly. Especially in the lower right part there are considerable deviations, the two loops became one.

Restored Dritvík labyrinth plan
Restored Dritvík labyrinth plan, created from aerial image (© Daniel C Browning Jr, 2021)

Jeff Saward explored the Dritvík Labyrinth in 1997 (Caerdroia 29 from 1998) and shows a photo of it in his book “Labyrinths and Mazes of the World” and in the Worldwide Labyrinth Locator (see the third Further Link below). Already there it shows the same layout as in 2021. A larger pile of stones in the middle is also noticeable. It’s a bit reminiscent of the Russian Babylons.

He calls it Volunderhus stone labyrinth and classifies it as Classical Baltic type with spiral at the center.

In order to understand the meaning of the Dritvík Labyrinth, it is very helpful to shed light on the cultural and historical background. And Daniel did that in great detail in the first part of his post. Again, I warmly recommend reading it.

At one point it says: Bárðr disappeared under the glacier and became the guardian spirit of the Snaefellsnes peninsula. The labyrinth could also be seen as the gateway to the underworld and as a monument or memory of Bárðr. Definitely as a place with magical meaning. Maybe we could even call it Bárðr’s house instead of Wayland’s house?

The special layout created by the stone setting are ideally suited to this. Because they alone represent an uninterrupted line, as we would expect from a labyrinth. It is shown in a “normal” labyrinth through the actually invisible part of the labyrinth, the path (or Ariadne’s thread). But here through the stones. And as a special feature, there is also the fact that these lines start and end in the middle, not on the outside as usual. As a result, this labyrinth is not as accessible as we are used to. With its dead ends, it could only serve as a trap.

Even the unfortunately failed restoration of 2000, in my opinion, does not change this finding. There is now an entrance with a branch like it is in a Wunderkreis, also a double spiral in the center. But you can’t go back to the entrance. You end up either on the right or on the left in a dead end.
The stone setting alone again forms an uninterrupted line that begins and ends in the center.

Related Posts

Further Links

Calculating the Related Labyrinths

Based on the third arrangement of the labyrinths from the last post (see: related posts, below) the transpose and complementary labyrinths can be directly, and the dual labyrinth can be indirectly calculated quite simply. For this, the sequence of circuits of the base labyrinth is needed. 

Here, I want to exercise this with the example of the labyrinth depicted in fig. 1. 

Figure 1. Labyrinth from the 18. or 19. Century Carved on a Wooden Pillar of the Old Mosque at  Tal, in Northern Pakistan.
Figure 1. Labyrinth from the 18. or 19. Century Carved on a Wooden Pillar of the Old Mosque at Tal, in Northern Pakistan.
Source: Saward, p. 60°

The labyrinth is situated with the entrance on top and in anti-clockwise rotation. First, I redraw it such that the entrance lies at the bottom and the labyrinth rotates clockwise. By this, it presents itself in the form I always use for a comparison of labyrinths. This labyrinth with one axis and 9 circuits will serve as our base labyrinth. Its sequence of circuits is 5 4 3 2 1 6 9 8 7.

Figure 2. Labyrinth of Tal, Redrawn: Base Labyrinth
Figure 2. Labyrinth of Tal, Redrawn: Base Labyrinth

First, we write the sequence of circuits in reverse order

Base: 5 4 3 2 1 6 9 8 7 <—> 7 8 9 6 1 2 3 4 5: Transpose. 

This brings us to the transpose labyrinth (fig. 3).

Figure 3. The Transpose to the Labyrinth of Tal
Figure 3. The Transpose to the Labyrinth of Tal

Then, second, we complete the sequence of circuits of the base labyrinth to one greater than the number of circuits, i.e. to 10. 

Calculation

By this, we obtain the sequence of circuits 5 6 7 8 9 4 1 2 3 of the complementary labyrinth that is shown in fig. 4. 

Figure 4. The Complement to the Labyrinth of Tal
Figure 4. The Complement to the Labyrinth of Tal

And, if we then write the sequence of circuits of the complement in reverse order, we obtain indirectly the sequence of circuits of the dual labyrinth: 

Complement: 5 6 7 8 9 4 1 2 3 <—> 3 2 1 4 9 8 7 6 5: Dual 

The dual labyrinth is shown in fig. 5. 

Figure 5. The Dual to the Labyrinth of Tal
Figure 5. The Dual to the Labyrinth of Tal

Now we can even check this result by adding the sequences of circuits of the transpose and the dual labyrinth. These must add to 10 at each position, as the dual is the complementary of the transpose. 

Calculation

This examination confirms the result. The dual can also be calculated by completing the sequence of circuits of the transpose to 10. However, it is easier to just write the sequence of circuits of the complement in reverse order. 

Thus, we have to know the sequence of circuits of the base labyrinth. Then we write this in reverse order and obtain the transpose. We complete it to one greater than the number of circuits and obtain the complement. And finally we write the sequence of circuits of the complement in reverse order and obtain the dual. 

° Saward Jeff. Labyrinths & Mazes. The Definitive Guide to Ancient & Modern Traditions. Gaia Books: 2003.

Related Posts:

The Rad in the Eilenriede (Hannover) was Originally a Wunderkreis

Since 1932 there is a labyrinth of the Baltic wheel type in the Eilenriede, the municipal forest of Hannover. In the larger center stands a linden tree and it has an additional, direct, short path to the outside. So we consider it to be a walk-through labyrinth. It is one of the last four historical lawn labyrinths in Germany (the others are Kaufbeuren, Graitschen, Steigra).

The Rad in the Eilenriede nowadays
The Rad in the Eilenriede nowadays, photo: Axel Hindemith, public domain

It was previously located on today’s Emmichplatz and was mentioned as early as 1642 in the city chronicle of Hannover. The occasion was a visit by Duke Friedrich von Holstein with his fiancée, Duchess Sophia Amalia von Braunschweig and Lüneburg, to his Hanoverian brother-in-law, Duke Christian Ludwig. He also organized a “tent camp” for the bridal couple in the Eilenriede, the climax of which was the bridal run in the labyrinth.


But what might the labyrinth have looked like back then?
Only now have I come across an old drawing of the Rad from that time in the book “Reise ins Labyrinth” by Uwe Wolff from 2001 in the chapter on German lawn labyrinths (p. 50 – p. 57).

The Rad in 1858
The Rad in 1858, source: “Reise ins Labyrinth” by Uwe Wolff, 2001

At least that’s how it looked in 1858. And presumably (or hopefully) it corresponds to the originally laid out labyrinth.
What is particularly noticeable in the drawing is that the middle is formed by a double spiral. Just like it is in a Wunderkreis. There are also two entrances, sometimes separate, sometimes with a branch.

While researching the Internet, I came across an old postcard with the labyrinth illustration. It shows the Rad from probably before 1932.

The Rad on a postcard
The Rad on a postcard

The drawing looks a bit idealized and has two circuits less than the drawing from 1858. But there is the double spiral in the middle and the two entrances again. And so it corresponds again to a Wunderkreis.

Years ago I wrote about the differences between the Wunderkreis and the Baltic wheel. I recommend reading the related posts below again.
I was particularly interested in the transformation of a Wunderkreis into a Baltic wheel.
And this transformation obviously took place with the Rad in the Eilenride.

Related Posts