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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.

### 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 he could find on this blog, maybe would like to have an overview.

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

For a better view

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## Variations on the Wunderkreis

In my last posting I had introduced a method to draw the Wunderkreis. Besides, it was always about the boundary lines. However, the path (Ariadne’s thread) in the labyrinth can also be drawn with this method slightly changed then.

And of course numerous variations with differently many circuits can be generated for the double spiral and the labyrinthine windings.

Square Wunderkreis

Here in abstract once again the method:

• I begin in the middle
• Arc upwards from the left to the right, jump to the left, arc downwards
• Path: Arc downwards, immediately following an arc upwards (closed line, like a recumbent “S”)
• Repeat this  as often as desired (on the right side there must always be two free ends which point down)
• Then draw around the whole, beginning on the left, an odd number of curves (at least 3, until as much as you want)
• Path: Extend both most internal lines down (maybe connect them)
• Connect the free line ends on every side in loops
• Boundary lines: Extend both most internal lines on every side inside the innermost loop

Sorry, this was a little longer. Maybe it is easier to understand the text together with the drawings. The different colours should help also. Best you try it yourself.

The labyrinth will be mirrored if one draws the first arc to the other direction.
One recognises the representation of the path by the fact that there are only two, perhaps only one line end (how it is also for the other types of labyrinths). If one sees four free line ends, the boundary lines are shown. Nevertheless, in the Wunderkreis the lines do not overlap as we see that in the classical labyrinth.

I have chosen known Wunderkreise as examples for the simplistic representation of the respective alignments.
In the related posts below you may find them all. As well as the step-by-step instruction.

Related Posts

## The Complementary Labyrinth

If we turn the inside out of a labyrinth, we obtain the dual labyrinth of it. The dual labyrinth has the same pattern as the original labyrinth, however, the pattern is rotated by a half-circle, and the entrance and the center are exchanged. This has already been extensively described on this blog (see related posts, below).

Now, there is another possibility for a relationship between two labyrinths with the same pattern. In this kind of relationship, the pattern is not rotated, but mirrored vertically. Also – other than in the relationship of the duality – the entrance and the center are not exchanged. At this stage, I term this relation between two labyrinths the complementarity in order to distinguish it from the relationship of the duality.

Here I will show what is meant with the example of the most famous labyrinth.

This labyrinth is the „Cretan“, „Classical“, „Archetype“ or how soever called alternating, one-arm labyrinth with 7 circuits and the sequence of circuits 3 2 1 4 7 6 5, that I will term the „basic type“ from now on.

Figure 1.The Original Labyrinth

Figure 1 shows this type in the concentric style.

The images (1 – 6) of the following gallery (figure 2) show how the pattern of the complementary type can be obtained starting from the pattern of the original type.

Image 1 shows the pattern of the basic type in the conventional form. In image 2 this is drawn slightly different. By this, the connection from the outside (marked with an arrow downwards) into the labyrinth and the access to the center (marked with a bullet point) are somewhat enhanced. This in order to show, that when mirroring the pattern, the entrance and the center will not be exchanged. They remain connected with the same circuits of the pattern. In images 3 til 5 the vertical mirroring is shown, divided up in three intermediate steps. Vertical mirroring means mirroring along a horizontal line. Or else, flipping the figure around a horizontal axis – here indicated with a dashed line. One can imagine, a wire model of the pattern (without entrance, center and the grey axial connection lines) being rotated around this axis until the upper edge lies on bottom and, correspondingly, the lower edge on top. In the original labyrinth, the path leads from the entrance to the third circuit (image 3). With this circuit it remains connected during the next steps of the mirroring (shown grey in images 4, 5 and 6). After completion of the mirroring, however, this circuit has become the fifth circuit.The path thus first leads to the fifth circuit (image 6) of the complementary labyrinth. A similar process occurs on the other side of the pattern. In the original labyrinth, the path reaches the center from the fifth circuit. This circuit remains connected with the center, but transforms to the third circuit after mirroring.

Figure 3: The Complementary Labyrinth

In the pattern of the complementary labyrinth we can find a type of labyrinth that has already been described on this blog (see related posts). It is one of the six very interesting (alternating) labyrinths with 1 arm and 7 circuits. That is to say the one with the S-shaped course of the pathway.

So, what is the difference between the dual and the complementary labyrinth?

Let us remember that the basic type is self-dual. The dual of the basic type thus is a basic type again.

The complementary to the basic type is the type with the S-shaped course of the pathway.

By the way: In this case, the dual to the complementary is the same complementary again, as also the complementary of the basic type is self-dual (otherwise it would not be a very interesting labyrinth).

This opens up very interesting perspectives.

Related posts:

## How to make a Wunderkreis or a Baltic Wheel

The Wunderkreis and the Baltic Wheel are compound labyrinths which are constructed from curves around different centres. The two lower turning points are proper for the “labyrinthine” circuits, those in the middle for the double spiral.

A Baltic wheel has a bigger, empty center and a short second exit. This is already a double spiral, yet without more twists. Both accesses are normally separated by an own intermediate piece, a sort of shoehorn.

The pattern for the layout is the same one for both labyrinth types. The tool to produce the layout is also the same. The number of the circuits in all can be different, nevertheless.

Here it is only about the method. The geometrically correct construction is another thing again. There are already several posts in this blog about that.

There is no seed pattern like we have it for the well-known classical labyrinth. However, there is a basically very simple method to draw such a labyrinth or to lay it directly with stones or to scratch it in the sand.

A step-by-step instruction should show it. The boundary lines of the labyrinth are drawn, the path runs between the lines.

Step 1

Step 1: I draw half a curve upwards, from the left to the right.

Step 2

Step 2: I jump a little bit to the left, make a curve downwards to the left, walk round the first curve and land to the right of the preceding curve.
This would already be the center of the Baltic Wheel or the middle of the smallest possible Wunderkreis.

Step 3

Step 3: Nevertheless, the double spiral should become bigger. Hence, I jump again a little bit to the left at the end of the first curve in green, make an other curve downwards to the left and walk again round the preceding curves.
Thus I could continue any desired. There must be left on the right side, however, always two free curve ends. With that the double spiral would be finished inside the Wunderkreis.

Step 4

Step 4: Now I must add at least three semi-circular curves round the previous lines.
If I want to have a bigger labyrinth, I can add more lines in pairs. There must however be an odd number of curves.
In our example we now have on the left side three free line ends, and on the right side five.

Step 5

Step 5: Now I connect on every side the innermost and the outmost lying free line in such a manner that in between an access is possible. This is to be continued (here only on the right side) so long as on every side only one single line end is left.

Step 6

Step 6: The both on every side lying free line ends are extended forwards. They represent the both lower turning points.
The labyrinth is finished.

Finally we will check out if the drawing is correct. We go in between the lines, turn to the right or to the left and must come again to the starting point. If not, something must be wrong.

Best try it out yourself, with a pencil on a sheet of paper. Wishing you success.

Related Posts

## The Labyrinth by Al Qazvini

An interesting labyrinth is reproduced in the book of Kern (fig. 200, p. 119)°. A drawing by Arabian geographer Al Qazvini in his cosmography completed in 1276 is meant to show the ground plan of the residence of the ruler of Byzantium, before the large city of Constantinople was built up.

This non-alternating labyrinth has 10 circuits and a unique course of the pathway. I will show this using the Ariadne’s Thread and the pattern. In my post “From the Ariadne’s Thread to the Pattern – Method 2” (see related posts, below), I have already described how the pattern can be obtained. When deriving the pattern I always start with a labyrinth that rotates clockwise and lies with the entrance from below. The labyrinth by Qazvini rotates in clockwise direction, however it lies with the entrance from above. Therefore I rotate the following images of the labyrinth by a semicircle so that the entrance comes to lie from below. So it is possible to follow the course of the pathway with the Ariadne’s Thread and in parallel see how this is represented in the pattern.

Four steps can be distinguished in the course of the pathway.

Phase 1

The path first leads to the 3rd circuit. The entrance is marked with an arrow pointing inwards. In the pattern, axial sections of the path are represented by vertical, circuits by horizontal lines. The way from the outside in is represented from above to below.

Phase 2

In a second step, the path now winds itself inwards in the shape of a serpentine until it reaches the 10th and innermost circuit. Up to this point the course is alternating.

Phase 3

Next follows the section where the pathway leads from the innermost to the outermost circuit whilst it traverses the axis. In order to derive the pattern, the labyrinth is split along the axis and then uncurled on both sides. As the pathway traverses the axis, the piece of it along the axis has to be split in two halves (see related posts below: “The Pattern in Non-alternating Labyrinths”). This is indicated with the dashed lines. These show one and the same piece of the pathway. In the pattern, as all other axial pieces, this is represented vertically, however with lines showing up on both sides of the rectangular form and a course similarly on both sides from bottom to top.

Phase 4

Finally the pathway continues on the outermost circuit in the same direction it had previously taken on the innermost circuit (anti clockwise), then turns to the second circuit, from where it reaches the center (highlighted with a bullet point).

Related Posts:

°Kern, Hermann. Through the Labyrinth – Designs and Meanings over 5000 Years. Munich: Prestel, 2000.

## How to make a Babylonian Wunderkreis

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

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, 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

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 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.

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

Related Posts

## Sequence of Circuits – Conclusion

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

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: