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On my own behalf

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

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Among the Wunderkreise there are some variations:

  • Some with two entries like the Zeiden Wunderkreis
  • Some with one access, but a bifurcation such as the Russian Babylons and the examples of Kaufbeuren or Eberswalde
  • Some with a nearly perfect double spiral like the Zeiden Wunderkreis
  • Some with a “pulled apart” double spiral such as the Russian Babylons, the example of Eberswalde and some Swedish and Finnish examples

The Babylonian Wunderkreis

Wunderkreise are compound labyrinths which are constructed from curves around different central points. Both lower turning points are proper for the “labyrinthine” circuits, the ones in the middle for the double spiral.
The double spiral in the Zeiden Wunderkreis is made from two centres lying side by side, and with it a total of only four centres the whole Wunderkreis can be constructed.

Here a Swedish example with a pulled apart double spiral from the book of Hermann Kern:

Petroglyph on the Skarv Island (Sweden)

Petroglyph on the Skarv Island, Source: Hermann Kern, Labyrinthe, 1982, fig. 584 (German edition); Photo: Bo Stiernström, 1976

A geometrically correct construction for a Wunderkreis with pulled apart double spiral requires more centres. Thus I receive for the Russian Babylons a total of six centres.

A sort of prototype with the dimension between axes of 1 m should serve as example. All values are thereby scaleable and differently big labyrinths can be constructed.

Construction elements

Construction elements

Best of all one begins by defining M1. After that one determines the direction of the perpendicular bisectors of the sides, and then constructs step by step the remaining mid points M2 to M6 through building the intersection of the triangle sides from two known points. All thereto necessary measurements are contained in the drawing.

The main dimensions

The main dimensions

The radii refer in each case to the middle axis of the boundary lines. The way runs between these boundary lines and, hence, is the empty space between these lines.

The different radii

The different radii

Here are the above shown components in one drawing as a PDF file to look at, to print or to copy.

Related Posts

The notation with the coordinates is consistent, understandable and works well in one- and multiple-arm, alternating and non-alternating labyrinths. However, for a labyrinth with three circuits, at least 6 segments are needed (in one- and two-arm labyrinths: number of circuits times two, in all other labyrinths: number of segments times number of arms).

Correspondingly, the sequences of segments rapidly increase in their length with the size of the labyrinth. The Chartres type labyrinth e.g. has 44 segments, as have all other types of labyrinths with 4 arms and 11 circuits.

 

 

Here I present the sequence of segments of the Chartres type labyrinth for illustration. This is:


Nevertheless this sequence of segments is a well understandable instruction of how to draw the labyrinth. It reads about like this: Go first to the fifth circuit, walk along the first segment (5.1), then proceed to the 6. circuit and stay in the first segment (6.1). Next, go to the 11th circuit in the first segment (11.1) continue on the same circuit to the 2nd segment (11.2), skip then to the 10th circuit in the 2nd segment (10.2) asf. This also implies that from each coordinate subsequent to the previous it becomes clear, whether the path makes a turn (as from coordinate 5.1 to 6.1) or if it traverses the arm (such as from 11.1 to 11.2). However it is a long and complex series of numbers.

Now there are also various other possibilities to write notations for multiple-arm labyrinths that may have less digits. In any case, the labyrinths first have to be notionally partitiond into segments. However in some notations it is possible to combine multiple segments in one term. I will illustrate this here with the example of a notation for the Chartres labyrinth by Hébert°.

 

This is a notation comparable with the one presented in the post „Circuits and Segments“, where the segments had been numbered by circuits. In this case, if the pathway passes through multiple segments on the same circuit, the number of the circuit was repeated accoridingly. This, for the labyrinth of Chartres would result in 44 numbers. In the notation by Hébert the length of the sequence reduces to 31 numbers. However, each number must now be written with a prefix. For instance, „-“ indicates, that the following number is written only once, as the path traverses only one segment. A prefix „+“, on the other hand, indicates that the following number would have to be written twice as the path passes two subsequent segments. Thus, different prefixes have to be taken into account. And two prefixes will not be sufficient. Additional prefixes will be required to capture the pathway passing through three, four or more subsequent segments, or to indicate that the arm is traversed whilst the path skips onto another circuit. So while this notation is shorter it is also more difficult to apply. Furthermore it is subject to the weakness already discussed earlier, that, althoug it indicates the circuit, it does not indicate the segment actually covered by the pathway.

Other notations exist as well. I do not address this further here. It should have become clear that the sequences of segments in multiple-arm labyrinths rapidly increase in length and complexity. In most types of such labyrinths the sequence of segments is therefore not suited for giving a name. Just try to imagine to name the labyrinth I had shown in January with its sequence of segments. This labyrinth has 12 arms and 23 circuits and thus 276 segments.

 

 

I abstain here from writing down the sequence of segments of this labyrinth. It would fill some 14 – 15 lines.

Conclusion

To conclude, I want to come back to the original question whether the sequence of circuits can be used for giving names to the different types of labyrinths. I had two concerns about this:

  • First, in one-arm labyrinths this sequence was not unique. However this problem could be easily solved by adding a prefix „-“ only to those numbers of circuits where the pathway traverses the axis. Therefore in not too large types of one-arm labyrinths the sequence of circuits can be used for naming.
  • Second, in multiple-arm labyrinths the sequence will rapidly increase in length. It turned out that in these labyrinths the sequence of segments has to be considered and that this usually becomes either be too long or too complex or both. Therefore I consider it not suited for giving name in multiple-arm labyrinths.

° Hébert J. A Mathematical Notation for Medieval Labyrinths. Caerdroia 2004; 34: 37-43.

Related Posts:

World Labyrinth Day 2017

Again you are invited from The Labyrinth Society to celebrate the World Labyrinth Day:

Celebrate the eighth annual World Labyrinth Day (WLD) on Saturday, May 6, 2017!

The Labyrinth Society invites you to ‘Walk as One at 1″ in the afternoon, joining others around the globe to create a wave of peaceful energy washing across the time zones. 

 

Flyer of the TLS

Flyer of the TLS

Most nicely it would be if everybody which is able would walk a labyrinth. But it is also possible, as a substitute to trace a finger labyrinth, to make a labyrinth meditation or to be active labyrinthine in some way.

More here:

If you are looking for a labyrinth near you, maybe you will find one here:

The Babylons are surely related to the widespread Troy Towns of the European North. However, they look a little bit different.
Directly after the entrance there is a bifurcation and therefore it is possible to go on in two directions. And then often there is no real middle, but rather you are headed back in a double spiral.

The Troy Town of Visby (Gotland Island, Sweden)

The Troy Town of Visby (Gotland Island, Sweden), Source: Ernst Krause, Die Trojaburgen Nordeuropas, 1893, fig. 1, p. 4

However, how could they have developed?
Numerous stone labyrinths have survived down to the present day in Fennoscandia. The Babylons are to be found particularly in the eastern area, from Finland up to the Russian Kola Peninsula. Often they are situated near the coast and on islands. The natives of Northern Europe, the Sami, settled here. It is possible that the Babylons deal with the traditional Sami religion.
They have presumably originated from the 13th century on until our times. And they were built in the same way: With stones fist-sized to head-sized laid down on the ground.

However, why do the Babylons look different and do not follow the well-known seed pattern with cross, angles and four dots? Much Scandinavian Troy Towns have eleven circuits and have been laid after the enlarged seed pattern.

The 11-circuit Cretan (Classical) labyrinth with the seed pattern of the cross, the four double angles and the four dots

The 11-circuit Cretan (Classical) labyrinth with the seed pattern of the cross, the four double angles and the four dots, on the right in a round shape

Thereby divergences and variations appeared. This can happen quite easily through this construction method.
Thus there are Swedish Troy Towns with the open cross which enables to take two directions to reach the middle, and to organise a race, e.g. This is why these also often are called “Jungfrudans” or “Jungfruringen”.

9-circuit stone labyrinth (Jungfruringen) at Köpmanholm (Sweden)

9-circuit stone labyrinth (Jungfruringen) at Köpmanholm (Sweden), Source: © John Kraft, Die Göttin im Labyrinth (1997), fig. 7, p. 26 (German edition)

In the seed pattern for this labyrinth double angles only were used in the lower area. So we have 9 circuits.

Here the layout for a 11-circuit labyrinth:

The 11-circuit Cretan (Classical) labyrinth, on the right with open cross

The 11-circuit Cretan (Classical) labyrinth, on the right with open cross

In the report of Budovskiy I found a graphics (from 1973?) by Prof. Kuratov who has carried out a division of labyrinths and wanted probably show how the Babylon developed (see the sketched line in the graphics).

The table of Prof. Kuratov

The table of Prof. Kuratov

In the first column a sort of principle is to be seen. As first the whole Cretan labyrinth. In the second the left-handed spiral, in the third the right-handed spiral, then the double spiral and below circles.
In row Ia we see the Cretan type in different variations.
In row Ib the open cross and a decreasing middle.
In row II a right-handed spiral and the faulty stone setting discovered by Karl Ernst von Baer (1792 – 1876) in 1838 on the island of Wiehr.
In row III the Babylon with the double spiral.
In row IV some multiple-arm labyrinths which remind of the medieval labyrinths.

The open cross occurs several times under the Scandinavian labyrinths. Besides, the empty middle sometimes becomes smaller and then even slides under the two upper turning points. Finally, it is only indicated and then left out completely.

The drawing of John Kraft shows this:

The Troy Town of Nisseviken (Sweden)

The Troy Town of Nisseviken (Sweden), Source: graphic by © John Kraft in Gotländskt Arkiv 1983 on Gotlands trojeborgar, p. 87

I have found in a report about the Babylons on WeirdRussia, beside numerous photos, also this graphic :

Stone setting on the Bolshoi Zayatsky Island

Stone setting on the Bolshoi Zayatsky Island

The middle exists next to nothing. It is rather a niche or a widening of the way. In this area small stone heaps are sometimes stacked up. Should they show the gate to the underworld or the belly of the snake? The ends of the boundary lines are thickened. This is quite easy to make with some more stones.
The labyrinth has changed its meaning, with this its appearance and became the walk-through labyrinth.

Here the layout in geometrically correct form:

 

Babylon Solovki

Babylon Solovki

Presumably most of the Babylons correspond to this shape.

On this photo one can recognise very well the alignment.

There is a graphic with a little “rounder” double spiral in the table of Prof. Kuratov and in Vinogradov’s report which I have still shown in my last post (see below).

There are  obviously some among the Finnish stone settings which look rather so.

Graphics of a Babylon according to Vinogradov

Graphics of a Babylon according to Vinogradov

According to most of the photos the Babylons doesn’t look exactly like this. The entrance is narrower and has a short straight piece.

Actually, one must consider them as a Wunderkreis. Even if they don’t have such a perfect double spiral like the Zeiden Wunderkreis. The Wunderkreise of Kaufbeuren or Eberswalde matches more likely the Babylons.

How could one call this type? In the last post I had suggested: Babylonian Wunderkreis. However, now I tend rather to Sami Wunderkreis because it developed in the cultural area of the Sami and probably was used in the cult of the dead.

Related Posts

Further Links

The notation using coordinates is consistent, understandable and works well in alternating and non-alternating one-arm and multiple-arm labyrinths. However it has a particular property. Whereas in multiple-arm labyrinths the number of segments is obtained by multiplying the number of arms with the number of circuits, this is not sufficient in one-arm labyrinths. They necessitate a partition in two segments per circuit. And thus the sequence of segments has the same length in one-arm and two-arm labyrinths with the same number of circuits.

I will show this here with the example of a two-arm labyrinth with 7 circuits.

This is a labyrinth I had designed during the course of my studies on the labyrinth of the Chartres type and its further developments.

According to the number of arms and circuits, this labyrinth has 14 segments. The corresponding sequence of segments is:

Now let us remember the sequences of segments in th one-arm labyirnths from the last post. For comparison I show here the sequence of segments of the basic type labyrinth.

This also has 14 numbers and thus has the same length as our two-arm labyrinth.

Related posts:

According to Wikipedia there are in all about 35 labyrinths in the Solovetsky Islands in the Onega Bay of the White Sea in the  Arkhangelsk Oblast (Russia), about 500 km to the north of St. Petersburg and 150 km to the south of the polar circle.

The Labyrinth on the Bolshoy Solovetsky Island

The Labyrinth on the Bolshoy Solovetsky Island, Source: Wikipedia, Photo © Vitold Muratov 2013

How old are they, who has built them, what was the purpose? There are many speculations about that (see the Further Links below). I do not want to take part in it.
I only want to find out how they look like, which type of labyrinth they are. I have found enough indications. There are several photos which reveal a part of the labyrinths quite well, unfortunately, not completely.

On the Internet I have found the following graphics from a book published in 1927 by Nikolai Vinogradov (historian, ethnologist and folklorist, 1876 – 1938).

Graphics of a stone setting

Graphics of a stone setting

In Hermann Kern’s book “Labyrinths” I have found the photo of a petroglyph on the island Skarv in the Stockholm archipelago (Sweden), presumably from the 18th/19th century.

Petroglyph on the Skarv Island

Petroglyph on the Skarv Island, Source: Hermann Kern, Labyrinthe, 1982, fig. 583 (German edition); Photo: Bo Stiernström, 1976

Compared to the graphics above the labyrinth is mirrored and the double spiral has a circuit less.

The labyrinths, called Babylons in the local dialect, have been made in the same way as the Scandinavian Troy Towns, probably at the same time and presumably served similar purposes.
Nevertheless, the layout is completely different. There are none of the well-known 11- or 15-circuit Cretan labyrinths which can be made from the enlarged seed pattern.

They belong to the walk-through labyrinths. These have a double spiral in the middle and labyrinthine circuits round two turning points. They can have two accesses or only one, however, with a bifurcation.

The hints, the Babylons could be seen as part of a cult of the dead and would show two snakes winding into each other, well explain the figure. They could also have been put on as a sort of piece of art.

There appear two spirals interlocking into each other. In a geometrical figure with semicircles around different centres they can be constructed as follows:

Blue and red spirals

Blue and red spirals

Both lines can be drawn well in one go and freehand: You will begin in the middle, turn to the right, circling once around, then in a larger turn outwardly from the right side to the left, from there inwards back to the right side. The red line ends her, the blue returns one more time to the left, circling inwards.
When you know how to draw each line, try to draw one in the other. Best begin with the blue line and leave enough space between the lines. Then put the red line in between.
That sounds complex, and it is. But best of all try several times with a pencil on a sheet of paper.

The result should look like thus:

The red spiral inside the blue one

The red spiral inside the blue one

For a labyrinth laid of stones these semicircular or elliptical curves can relatively simple be realised.

Best of all one starts in the middle. There one can arrange most easily the thickening of the ends and the interpieces. Then the remaining lines follow in steady distances.

Step 1 and 2

Step 1 and 2

One makes three semicircles downwards (step 1), and four semicircles upwards (step 2). Thus the double spiral in the middle is built.

Step 3 and 4

Step 3 and 4

Then I add five semicircles on top (step 3). There are five free ends on the left side, and seven on the right. These I elongate to the sloped line at right and at left (step 4).

Step 5 and 6

Step 5 and 6

In step 5 I connect both outermost free ends on the left and on the right side so with each other that in the middle a gap remains for the entrance. In step 6 the remaining free ends are connected parallel to the curves just made before. The innermost free end on each side will be the turning point.

It is noteworthy that the limitation lines do not overlap like they doe in the Cretan labyrinth. In spite of the bifurcation the way through the whole figure is unequivocal and follows the typical “labyrinthine” rhythm.

The construction elements

The construction elements

Even if the Babylons were not put on so geometrically precisely, nevertheless, these geometrical features show the essential internal structure and let them count to the Wunderkreise. I would like to call them Babylonian Wunderkreise to discern them from the Wunderkreise with two accesses side by side like we see that in the Zeidner Wunderkreis.

The Babylons are related to the Babylonian Labyrinths through the double spiral in the middle and the unequivocal way that leads to it, even if there are two opposite entrances.

Related Posts

Further Links

With the coordinates for segments from the last post (see related posts below) we have now found an understandable notation for the sequence of segments of labyrinths. Here it seems important to me to add that such coordinates can also be used for one-arm labyrinths. I will show this with the examples for which I had already shown the sequences of circuits (see related posts). For this, each circuit has to be divided into two segments.

Partitioning of Circuits in Segments

Next we write the sequences of segments for the three examples and also compare them straightaway with their sequences of circuits.

 

 

A unique notation for one-arm labyrinths can also be achieved, if we can write two different numbers on the same circuit, one for each side of the axis. For this, the circuits have to be partitioned into two segments. This allows us to write unique sequences of segments for alternating and non-alternating labyrinths. Also it is possible to use the same form of notation in one-arm and multiple-arm labyrinths. However, this notation will always need 14 coordinates for each one-arm labyrinth with 7 circuits. This is clearly more digits than are needed for the sequences of cirucits with separators.

 

 

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