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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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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.

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

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In Greek mythology, the labyrinth is the place where the Minotaur is hidden and imprisoned. It is therefore not necessarily a real place.
The labyrinth, as we know it today, is highly inappropriate. Because it has an entrance, a clear path and an accessible center.
Thus, on the silver coins from Knossos we also find very different interpretations of the labyrinth. There are meanders and other symbolic representations.
I want to pick out a motif today and take a closer look at it.

I found two examples with the same motif. One on a coin from the Coin Cabinet of Berlin:

Minotaur 420-380 BC

Minotaur 420-380 BC: Coin Cabinet of the Staatliche Museen zu Berlin, object 18218282 obverse

Labyrinth 420-380 BC.

Labyrinth 420-380 BC: Coin Cabinet of the Staatliche Museen zu Berlin, object 18218282 reverse

And one on a coin from the British Museum in London:

Square area meander 500-431 BC

Square area meander 500-431 BC / source: Hermann Kern, Labyrinthe (German edition), 1982, fig. 43

They both represent the same thing. Although the “Berlin” coin seems to be more exact, it contains small errors in two places in the upper area. Two vertical lines collide, where a gap should actually be. This area is more accurately represented on the “London” coin, although the lines there are harder to see.

I made a “final drawing” that shows what the coin maker wanted to show. You can see lines that follow a certain pattern. They are symmetrical, repeating themselves and showing an intricate “path system”. The drawn red thread shows that.
There are four nested paths without beginning and end, but also without entrance. This is not “our” labyrinth but better suited as a prison. The Minotaur would not come out that fast.

The revised area meander

The revised area meander

This could be a hint of the Roman sector labyrinth hundreds of years later.

But it also shows a certain relationship to the Babylonian labyrinth, hundreds of years older and developed in a different culture (see the labyrinthine finger exercises in the post about the Babylonian labyrinth).

Related Posts

Further Link

Sector Labyrinths

At the end I will also transform a sector labyrinth into the MiM-style. What is special in sector labyrinths is, that the pathway always completes a sector first, before it changes to the next. As a consequence of this, the pathway only traverses each side-arm once. Thus it seems, that sector labyrinths may be easier transformed into the MiM-style than other labyrinths with multiple arms. I will use as an example a smaller labyrinth with four arms and five circuits. There exist several labyrinth examples of this type. I have named it after the earliest known historical example, the polychrome mosaic labyrinth that is part of a larger mosaic from Avenches, canton Vaud in Switzerland.

Figure 1. Sector Labyrinth (Mosaic) of Avenches

Figure 1 shows the original of this labyrinth (source: Kern 2000: fig 120, p 88). It is one of the rarer labyrinths that rotate anti-clockwise. On each side of the side-arms it has two nested turns of the pathway and 3 nested turns on each side of the main axis. The pattern corresponds with four double-spiral-like meanders arranged one after another – Erwin’s type 6 meanders (see related posts 2). When traversing from one to the next sector the pathway comes on the outermost circuit to a side-arm, traverses this on full length from outside to inside and continues on the innermost circuit in the next sector.

In order to bring this labyrinth into the MiM-style, first the origninal was mentally rotated so that the entrance is at bottom and horizontally mirrored. By this it presents itself in the basic form, I always use for reasons of comparability. Fig. 2 shows the MiM-auxiliary figure.

Figure 2. Auxiliary Figure

This has 42 spokes and 11 rings what makes it significantly smaller than the ones for the Chartres, Reims, or Auxerre type labyrinths. The number of spokes is determined by the 12 ends of the seed pattern of the main axis and the 10 ends of each seed pattern of a side-arm.

In fig. 3 the auxiliary figure together with the complete seed pattern including the pieces of the path that traverse the axes is shown and the number of rings needed is explained. For this the same color code as in the previous post (related posts 1) was used.

Figure 3. Auxiliary Figure, Seed Pattern and Number of Rings

As here the angles between the spokes are sufficiently wide, it is possible to use all rings of the auxiliary figure for the design of the labyrinth. We thus need no (green) ring to enlarge the center. Only one (red) ring is needed for the pieces of the path that traverse the axes – more precisely: for the inner wall delimiting them –, four (blue) rings are needed for the three nested turns of the seed pattern of the main axis, one ring (grey) for the center, and five rings (white) for the circuits, adding up to a total of 11 rings.

Fig. 4 finally shows the labyrinth of the Avenches type in the MiM-style.

Figure 4. Labyrinth of the Avenches Type in the MiM-Style

The figure is significantly smaller and easier understandable than the labyrinths with multiple arms previously shown in the MiM-style. Overall it seems well balanced, but also contains a stronger moment of a clockwise rotation that is generated by the three asymmetric pieces of the pathway and of the inner walls delimiting these on the innermost auxiliary circle.

Related posts

  1. How to Draw a MiM-Labyrinth / 14
  2. How to Find the True Meander for a Labyrinth

The silver coins of Knossos are quoted again and again when we talk about the labyrinth. They can be found in the major museums of the world.

Last year I was able to see and photograph one of them on a trip to Vienna in the Coin Cabinet of the Kunsthistorisches Museum.

Kinsthistorisches Museum Wien

Kinsthistorisches Museum Wien

The book “Labyrinths” by Hermann Kern shows illustrations of 20 coins from the British Museum in London.

Meanwhile there is a digital interactive catalog of the Coin Cabinet of the Staatliche Museen zu Berlinn, where you can access more than 34,000 coins.

With the search term “Labyrinth Knossos” I found 22, which I can show here under the following license.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Germany License.

The coins cover a period of 425 BC until 12 BC. Shown is mostly the reverse of the coin.

For the interpretation of the representations I have found some interesting information in the description that I quote here (translated from German):

The Cretan town of Knossos has been closely linked to the myth of the Minotaur since antiquity. His mythical dwelling, the labyrinth, was one of the city’s landmarks. However, the depiction of the labyrinth on the Knossos coins came in very different ways, since a real non-existing place had to be shown. The labyrinth is always pictured in supervision, but with different outer shapes and structuring. Only in supervision, the labyrinth can be detected as such.

I highly recommend visiting the digital catalog. There are to find many additional details about the coins. In particular, there is the possibility to look at both sides and to retrieve further information.

Related Posts

Further Links

For Year 2019 we wish you all the best and interesting encounters with the labyrinth.

Labyrinth with 7 Axes, 15 Circuits and 6 Nested Turns at Both Sides of Each Axis, and yet no Sector Labyrinth – Rather the Opposite, Self-dual

 

 

Dipl. Ing. Norbert L. Brodtmann uses the curvy and tortuous path in the Chartres Labyrinth to demonstrate the possibilities of the robot arm technology he has developed. He transforms the straight lines and radii of the path elements for the way in the Chartres labyrinth in Bezier curves, which he draws in inverse kinematics by a robot.

I was able to provide him with the necessary coordinates for the trajectories from my true-to-scale drawings of the Chartres Labyrinth.

Related Post

Related Links (requires Adobe Flash Player)

Further Link

Christmas 2018

Wishing all visitors of this Blog a Merry Christmas and a Happy New Year!

An 11 circuit Christmas tree labyrinth

An 11 circuit Christmas tree labyrinth

Related Posts

Type Reims and Type Auxerre

Using the example of the Chartres type labyrinth it could be shown that also labyrinths with multiple arms can be designed in the MiM-style (see related posts 1, below). Generally, all labyrinths with four arms and 11 circuits require in the MiM-style an auxiliary figure with 90 spokes (see related posts 2). This is determined by the number of arms and of circuits.

The number of rings for the Chartres type labyrinth amounts to 22. This number can vary for different types of labyrinths with four arms and 11 circuits. This number depends on the number of circuits, the depth of nestings of the turning points of the path (see related posts 5) and the number of pieces of the pathway traversing the axes. This is explained more in detail in fig. 1.

Figure 1. Auxiliary Figure of Chartres Type – Rings

For the outer walls delimiting the pathway of the 11 circuits (white), 11 rings are needed. One more ring is used for the center (grey). (This space could also have been saved. My first draft of this labyrinth for the New Year did not contain a separate ring for the center yet – see related posts 3 -. However, in such a case no space would be provided for the center. Therefore in the final design I have added a separate ring for the center). There are turns of the pathway with a maximum depth of two nested turns at the main axis. For this, three rings are required (blue). The side-arms all have only single turns of the pathway. So for them, tow rings would be sufficient. For the pieces of the pathway traversing the side-arms, an additional five rings are needed (red). In order to ensure that sufficient space is left for the pathway at the narrowest point between two walls delimiting the path, two rings in the center of the figure are added that are not even used for the lines of the labyrinth. These just serve to enlarge the center (green) such, that the figure can be reasonably drawn at all. Thus, for all this together, 22 rings are needed.

In order to transform a labyrinth with four arms and 11 circuits into the MiM-style, a considerable effort is required. To draw the figure with sufficient exactness, compass and ruler or a drawing application will be needed. Now once we have designed the figure for the Chartres type labyrinth, we can easily bring certain other types of labyrinths into the MiM-style. Such types must have four arms, 11 circuits, 5 (and less) pieces of the pathway traversing the axes, as well as two (and less) nested turns of the pathway. This is true, among others, for the two other very interesting historical types of labyrinths, the Reims type and the Auxerre type (related posts 4). In order to transform these into the MiM-style, we can start from Chartres type in the MiM-style. All the lines delimiting the pathway on all spokes and auxiliary circles outside the seed pattern can be left unchanged. Only the sed patterns have to be amended at certain places and the connections to the walls delimiting the pathway have to be adjusted correspondingly.

Figure 2. Labyrinth of the Reims Type in the MiM-Style

Fig. 2 shows the Reims type in the MiM-style. The seed pattern of the main axis has two nested turns only in two places next to the entrance and next to the center of the labyrinth. Otherwise there are only single turns of the pathway at the main axis. There are five pieces of the pathway traversing the first and the third side-arm, and three traversing the second side-arm, just the same as in the Chartres type labyrinth. However, the pieces traversing the first and third side-arme are distributed differently over the side-arms than in the Chartres type labyrinth.

Figure 3. Labyrinth of the Auxerre Type in the MiM-Style

In fig. 3 the Auxerre type in the MiM-style is depicted. The seed pattern of the main axis of this type is somewhat different from that of the Chartres type. The seed patterns of the side arms and thus the pieces of the path traversing the side-arms are the same as in the Chartres type.

Other types of labyrinths can be transformed into the MiM-style in the same way too, e.g. the complementary of Reims. These will all be based on an auxiliary figure with 90 spokes and 22 rings.

However, in other types of labyrinths with 4 arms and 11 circuits, this does not work that easy. So, for instance, the complementaries of Auxerre or Chartres on the main axis have also three nested turns of the pathway. Therefore, for the seed patterns of these types of labyrinths, four (blue) rings are needed. The auxiliary figure for these labyrinths has 23 rings. Thus, the center and all eleven circuits would have to be shifted one ring further outwards. In order to draw these labyrinths, again the seed patterns would have to be amended, and the connections appropriately adapted. In addition each piece of a wall delimiting the pathway outside the center would have to be shifted and modified.

I refrain form drawing these types of labyrinths in the MiM-style. Already from the presently available figures it can be seen, that the style clearly dominates the look of the labyrinth and that a quite careful closer view is needed, if we want to identify the differences between these types in this style.

Related Posts:

  1. How to Draw a Man-in-the-Maze Labyrinth / 13
  2. How to Draw a Man-in-the-Maze Labyrinth / 9
  3. Our Best Wishes for 2018
  4. The Complementaries of the Three Very Interesting Historical Labyrinths with 4 Arms and 11 Circuits
  5. How to Draw a Man-in-the-Maze Labyrinth / 5
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