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## Complementary and Self-dual Labyrinths

It is known that there are 8 alternating labyrinths with 1 arm and 5 circuits (see “Considering Meanders and Labyrinths”, related posts, below). Of these, four are not self-dual. These four all are in a relationship to each other via the duality and complementarity (see “The Complementary versus the Dual Labyrinth”, related posts, below). The other four labyrinths are self-dual.

I had already pointed to the relationship between complementary and self-dual labyrinths (see “The Complementary Labyrinth”, related posts, below). Here I want to elaborate on it further. For this purpose I use the same form of diagram I had already used in my previous post (see “The Complementary versus the Dual Labyrinth”). I also use the same numbers of the labyrinths according to the numbering of Arnol’d’s meanders (see “Considering Meanders and Labyrinths”), that underlie them.

Figure 1. Labyrinths 1 and 6

The first of the Arnol’d’s labyrinths, number 1, is self-dual. In the diagram, the dual is situated in the same row, the complementary in the same column with the original labyrinth. The dual of number 1 is again number 1 (what actually is the meaning of selfdual). The complementary of number 1 is number 6. And – of course – is the dual to the complementary again number 6. So in the case of self-dual labyrinths, we only captured two different labyrinths, whereas it were four in the case of not self-dual labyrinths.

Thus, two more labyrinths are still missing. We need another diagram to capture labyrinths number 3 and number 8 (fig. 2).

Figure 2. Labyrinths 3 and 8

And, indeed, these two are complementary to each other. So in self-dual labyrinths, only two different labyrinths are in a relationship to each other.

Here the question arises: Do there also exist self-complementary labyrinths? Up to now we have not yet found such a labyrinth. So let us remember, what self-dual imples. The patterns of the original and self-dual labyrinths are self-covering. In fig. 3 I show what that means. The two patterns in the same row are dual. If we shift them together, we can easily see, what I mean.

Figure 3. Self-dual patterns are self-covering

Thus, self-complementary would imply that the original and complementary pattern would also be self-covering.

Figure 4. Complementary patterns are not self-covering

Fig. 4 shows, that even though there is a certain similarity between these two patterns, they are not self-covering. In my opinion there are no self-complementary labyrinths. This is because vertical mirroring with uninterrupted connections to the entrance and center modifies the sequence of circuits. This, however, woult have to remain unaltered.

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## The Epprechtstein Labyrinth in the Fichtelgebirge (Germany)

Finally, I got around to visiting this unusual labyrinth from granite ashlars in the Fichtelgebirge.

You may reach it over the street from Kleinschloppen to Kirchenlamitz. There is a parking place opposite the restaurant Waldschmiede in the district Buchholz and directly behind it lies the labyrinth.

Willi Seiler from Wunsiedel, a former professional schoolteacher in the technical school for stone processing in Wunsiedel had the idea of the labyrinth. The construction works were carried out after the plans of architect Peter Kuchenreuther from Marktredwitz in 2009.

The labyrinth is from type Roman sector labyrinth with a meander in every quadrant and has 5 circuits. It is put on squarely and has the dimensions 34 x 34 m. The middle is a square of 6 m sides length with a 5-m-high obelisk, where Hermann Kern’s famous words: “In the labyrinth you will not get lost. In the labyrinth you will find yourself. In the labyrinth you will not meet the Minotaurus. In the labyrinth you will meet yourself.” are chiseled.

The ways and the granite bolders are each about 1.20 m wide. The higher ashlars in the middle and around are about 1.20 m high, the smaller ones inside from 60 to 80 cm. In every quadrant there is a small loophole to leave the way which amounts to 400 m after all. The middle contains the obelisk, some wooden benches and the ground is covered with a paved labyrinth showing the paths enlargedin black stones as it were a negative of the “big” labyrinth.

The layout

The middle enlarged:

The middle

Behind the labyrinth a small hill is raised from which one can overlook the whole area. Several boards of information to the geology, fauna, granite quarrying in the Fichtelgebirge among other things as well as to the idea of the labyrinth are put up on the site.

Information board

Service station for spirit and soul

Service station for spirit and soul

Labyrinths still are in the world since millenniums in the most different forms. After Ancient Greek myth the first labyrinth was built by Dädalos for king Minos on Crete as a prison for the Minotauros. In the antiquity it is often shown as a square built by windings of meanders. The Christians pervaded this ancient motive with new sense. In many old churches labyrinths drawn on the ground with black and white stones show with their unpredictable bends the human life with all its scrutinies, delays and complications, while in the middle, the aim, waits heavenly Jerusalem.

The labyrinth is always purposeful and not a maze, how frequently is falsely presumed.

„The construction plan of the labyrinth is conceivably simple. It has an entrance and a way which leads in numerous bends to a middle. One can go through it fast without having found out something. Then the way through the labyrinth is not more than just a leisure activity or a sportive act. Who crosses, however, the way with a spiritual feeling, who embarks on a journey consciously and with alert soul, will attain a place of self-encounter and self-discovery.“ Uwe Wolff

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## How to Make a Wunderkreis, Part 1

A Wunderkreis is a double spiral, surrounded by a simple labyrinth with two turning points.

We begin in the centre with the double spiral. One  semicircle below and one semicircle above the horizontal line would suffice as a minimum. Many more semicircles could be added to enlarge the double spiral. Here we make three arcs which we name A, B and C. The lower ones are drawn around M1 as the centre, the upper ones are arranged around M2 as the centre and shifted to the right.

Step 1

Then we add three arcs on the left side. They are drawn in a triangular sector around the midpoint M1. We number the circuits from the outside with 1, 2 and 3. Circuit 3 will finally form the entrance.
The turning and midpoint M3 for the lower semicircle lies concentric between the both external circuits 1 and 2.

Step 2

Now we go to the right side. Here two arcs more than on the left side are necessary, that means a total of five. Again we number the circuits from the outside inwards from 1 to 5. The circuit 5 will later lead to the exit.
The turning point M4 lies concentric between the four circuits 1 to 4. In the lower middle section two semicircles are traced around that midpoint M4.

Step 3

Now the upper semicircles are completed around the midpoint M2. There are four semicircles (and circuits) more on each side than at the beginning.

Step 4

The Wunderkreis is usually entered through the labyrinthine circuits on circuit 3 and left through the double spiral on circuit 5. The path sequence then is as follows: 3-2-1-4-C-B-A-A-B-C-5.
The path sequence 3-2-1-4 forms the basis of the meander, as connoisseurs know, as in the Knossos labyrinth.

Now we choose more circuits and apply the abovementioned principles to the construction. Through that Wunderkreise with a varied number of circuits can be generated. We can add circuits to the double spiral one by one, to the labyrinth we have to do it in pairs.
On the right side two circuits more are necessary than on the left. The lower turning points (M3 and M4) must lie concentric between the even-numbered left or right circuits. In the following example we have 6 circuits on the left and 8 on the right side.

If we know how many circuits for a Wunderkreis we want, we can lay both lower turning points on a line and determine the middle for the double spiral (M1) in a triangle. Entrance and exit can also be arranged  side by side without any space.

Nevertheless we can begin, while marking out, with the definition of the middle M1 and also determine the adjustment of the main axis (vertical line). The remaining centres M3 and M4 can afterwards be fixed in that triangle.

The main dimensions

Best of all we consider the measurements as units, so either “metre” or “yard” or “step width” or something similar. Then we can also scale all dimensions.
The smallest radius begins with 1 unit and then gradually grows by 1 from arc to arc. Then the biggest radius has 12 units. The boundary lines add themselves on 407 units, the whole way through the Wunderkreis reaches 362 units.

The completed Wunderkreis

In this example the Wunderkreis has four circuits more than in the other at the top of the page and no space between entrance and exit. This area is formed quite differently in the historical Wunderkreise. Sometimes the paths are joined together, sometimes they run apart.

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## The Babylonian Labyrinth

I have already written about the Babylonian visceral divination labyrinths and tried to prove their relationship with the labyrinth. They date to the Middle Babylonian and Neo-Babylonian time (ca. 1500 to 500 BC).

However, there are even older labyrinth representations from Old Babylonian time (ca. 2000  to 1700 BC) which look quite differently than the visceral labyrinths and which can probably be taken for the ancestors of the labyrinth.

The Swedish historian of Babylonian mathematics and cuneiform script expert Jöran Friberg has studied the Babylonian mathematical  tablets of the Norwegian Schøyen Collection in detail and has documented that in 2007. He calls the following figures labyrinths and tries to prove that.

In the journal Caerdroia 42 Richard Myers Shelton has written extensively on the subject of the Babylonian Labyrinths. Most of my information I got from him. Here it is a matter for me of founding in what the relationship with the labyrinth consists.

One must take therefore the following representations as the oldest labyrinths known so far.

Here a rectangular labyrinth labelled MS 3194 in the Schøyen Collection:

The rectangular labyrinth MS 3194, source: Schøyen Collection

We do not know anything about the purpose of this figure. It could have served quite philosophical or mathematical considerations.

In what does the relationship with the labyrinth exist now?

We must look at it more exactly. Richard Myers Shelton could reconstruct the lines on the clay tablet perfectly and therefore I can present a colored drawing of the entire figure.

The rectangular Babylonian labyrinth

The thin black lines limit the ways. These are the free space between the lines. There are two open entries to the rectangle. One entrance lies roughly in the middle of the left side, the other one opposite on the right. The way from the left is highlighted in ochre, from the right in green. In the middle they meet and change the direction. The one way is leading in, so to speak, and the other out.

There are no forks or dead ends. The whole, long and winding path must be accomplished. The entire rectangle is crossed.

The layout shows a certain, but not quite successful symmetry. The last laps round the center remind a double spiral. The other circuits are intertwined in the shape of meanders.

We have thus an unambiguous, doubtless and purposeful way through a closed figure, as we know it from a “true” labyrinth.

Then there is still a square labyrinth labelled MS 4515. Here the colored drawing:

The square Babylonian labyrinth

Maybe it should represent a town? As we know that from other labyrinths. With gates, bastions, walls?

Amongst the Babylonian tablets is another one with geometrical illustrations. Jöran Friberg calls them mazes. They are quite sure not.

One could consider these lines as labyrinthine finger exercises. Some are difficulty to reconstruct. So, Friberg and Shelton come to different results.

There are two rows with four fields in which a rotationally symmetric closed path runs without beginning and end through four sectors. All areas are mostly touched, sometimes there are inaccessible places. One is reminded of the Roman sector labyrinths many centuries later.

The tablet MS 4516, source: Schøyen Collection

Here the drawings of two fields:

The first field on top left

The fourth field on bottom left (reconstructed)

Clearly one recognises the meander, the symmetrical arrangement and the alignment of the paths between the black lines.

Much later similar representations on the silver coins of Knossos are found:

Swastika meander on a coin, 431-350 BC / source: Hermann Kern, Labyrinthe, 1982, fig. 49 (German edition)

The right “ingredients” for a labyrinth, namely meander and spiral were already known in Old Babylonian times. The idea of a confusing, winding, nevertheless unequivocal way in a restricted space with rhythmical movement changes can have originated from there.

We can push back the time for the origin of the labyrinth some hundred years later to the time about 1800 BC. At first it was the idea of a walk through labyrinth. The further development happened in Middle to New-Babylonian times in the intestinal labyrinths with also two entries, yet unambiguous way.

Since 1200 BC we know the Cretan labyrinth with only one entry and the end of the path in the center. We could call this a way in labyrinth whereas the Babylonian labyrinth is a way through labyrinth.

Till this day have remained walk through labyrinths in the type of the  Baltic wheel and the Wunderkreis (wonder circle). We recognise them as real labyrinths, although they also have two entrances and do not end in the middle.

The Kaufbeuren Wunderkreis

More information is to find about the Babylonian labyrinths in an excellent article by Richard Myers Shelton in Jeff Sawards Caerdroia 42 (March 2014), and in a new article from him in Caerdroia 44 (April 2015) about the Transylvanian Wunderkreis.

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## The Babylonian Visceral Labyrinth, Part 2

Via Facebook  I have found this modern walk through labyrinth:

Drawing by kind permission of © Sergej Likhovid

The drawing is sketched for a labyrinth by Sergej Likhovid, that was structured in an abandoned swimming pool in Odessa (Ukraine). See more about the project in a news article in the Further Links at the bottom. Besides, it is a sector labyrinth and uses the meander. And with that we get onto the subject of the post:

In the history of the labyrinth the meander plays a big role. The meander can be traced back till the Neolithic Age. So the meander is much older than all up to now known labyrinth figures (on the tablet of Pylos in 1200 B.C.). When was the first combination meander – labyrinth? The connection with the labyrinth can be presumably proved now till the Babylonian time (about 1800 B.C.).

In the 1st part I have already introduced the labyrinth from fig. 5 of the Near East clay tablet VAT 9560 in Weidner’s article. The tablet is dated by him based on the attributed cuneiform inscriptions to the time about 1000 B.C.

The visceral labyrinth VAT 9560, fig. 5 (Ariadne’s thread)

On this representation of the path’s structure (the so called Ariadne’s thread) one can recognize very nicely the meander in the middle.

Here the geometrically correct representation of the limitation lines:

The visceral labyrinth VAT 9560, fig. 5 (the lines)

In this drawing the basic pattern can be read. It has an amazing resemblance with that for the Indian labyrinth, nevertheless, is a little bit differently constructed.

In Weidner’s script there is still fig. 4 of the tablet VAT 9560. Though the figure is incomplete, however, it shows clearly an access on the top left and the end in the middle:

The visceral labyrinth VAT 9560, fig. 4

The both lines on the right side can be reconstructed unambiguously, and the completed figure shows a labyrinth:

Drawing of the complete visceral labyrinth VAT 9560, fig. 4

Here the graphics in a geometrically correct manner:

Graphics of the visceral labyrinth VAT 9560, fig. 4

The comparison of the different labyrinths from fig. 5 and fig. 4 shows within the triangle in the geometrically correctly drawn representations an identical pattern. And this is identical again with a quite known basic pattern, namely of the Indian labyrinth (also called Chakra Vyuha). Read more about the Indian Labyrinth on Related Posts at the bottom.

The seed pattern for the Indian labyrinth

Only the connection of the dots and lines is a little bit differently for the walk through labyrinth after fig. 5. For the Indian labyrinth (and the one of fig. 4) one begins in the triangular seed pattern on top and makes the first curve down to the next line end below on the right side. And then one connects all the further line ends and dots in usual manner as for the classical labyrinth in parallel arcs to the first curve. For the walk through labyrinth after fig. 4 one also begins on top, pulls the first curve, nevertheless, to the second line end. The rest is constructed again as usual.

The Indian labyrinth is still known in other variations. Here an illustration from Hermann Kern’s book:

The Indian labyrinth, source: Hermann Kern, Labyrinths (2000), fig. 607, p. 287

The Indian labyrinth is very old, but the origin is not so easily to prove. Who has discovered the basic pattern for it, to my knowledge is unknown, may presumably have occurred in newer time.

To my conviction one may consider the Babylonian labyrinths as genuine labyrinths, even if most of them are walk through labyrinths. They follow a different paradigm than our usual Western notion of a single path ending at the center. Nevertheless, we can count them to the real labyrinths, like we do it with the Baltic wheel and the Wunderkreis of Kaufbeuren, as well as with many other contemporary creations.

In the meantime I could find about 50 different walk through and intestinal labyrinths from Babylonian time. Whether a mutual influence under these different cultural spheres existed, is uncertain, and which is now the oldest historically manifested labyrinth, is not yet proved.

However, another example of a divination labyrinth from Mesopotamia from about 1800 B.C. could outstrip the clay tablet of Pylos from 1200 B.C. On the website of Jeff Saward I found a picture of it (more on the Links below). Here a drawing of it:

The Mesopotamian divination labyrinth from 1800 B.C.

It is certainly not comparable directly with the classical labyrinth, nevertheless, a closer look at it is worthwhile and shows the relationship to the labyrinth figure.

Following graphics with the representation of the lines, the normally hidden path (Ariadne’s thread in Red) in a geometrically correct way:

Graphics of the Mesopotamian divination tablet from 1800 B.C.

It looks quite differently than we would have expected. However, it has only one entrance and an end in the middle. Though the middle is below, but here ends the way. The path spirals upwards in serpentines and turns down through a meander.

The way is unequivocal, fills the whole space, have no forks and dead ends, must be absolved completely, leads to a goal – and turns back to the outside. Even if the lines would be open in the middle below, the diagnosis “Labyrinth” would be kept up.

… To be continued

More information about the Babylonian clay tablets can be found in an excellent article from Richard Myers Shelton in Jeff Sawards Caerdroia 42 (March 2014).

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## The Babylonian Visceral Labyrinth, Part 1

This fading artwork of Denny Dyke on the beach of Bandon, Oregon shows double spirals, knots and a walk through labyrinth with a meander in the middle.
Is this something new or are there some historical ancestors?

Dream-Field from Denny Dyke on the beach of Bandon, Oregon. Photo courtesy of Amber Shelley-Harris

One of the first pictures in Hermann Kern’s book “Labyrinths” shows the so-called . It is on a clay tablet from presumably Middle- to Neo-Babylonian time (from 1100 to 600 B.C.) in the Near East Museum of Berlin (Vorderasiatisches Museum Berlin) under the number VAT 744. It shows the intestines of a sacrificial animal with the drawing as a pattern for the ancient practice of extispicy.
For Hermann Kern this is not a labyrinth, but a double spiral with changing direction in the middle. Also spirals, meanders and knots are no labyrinths. These are not in the strict sense, but they are elements in labyrinths.

The Berlin Labyrinth

The Near East archeologist and Assyriologist Ernst Friedrich Weidner has 1917 written about that in an article under the title “Zur babylonischen Eingeweideschau, zugleich ein Beitrag zur Geschichte des Labyrinths” (translated: “About the Babylonian extispicy; at same time a contribution to the history of the labyrinth”) in the “Orientalistische Studien” (see link below, on the pages 191-198).

Diagram tablet of intestines VAT 984

He sees in these intestinal drawings an extraordinary close relationship to the labyrinth drawings of the Aegean culture (as on the jug of Tragliatella) and the Troy Towns of Northern Europe.

The jug of Tragliatella

But he didn’t prove this relationship. However, thus is not done so easily. Therefore a closer look to the tablets in Weidner’s writ is worthwhile. Only an analysis of the alignment of the paths shows the resemblance.

First the double spiral:

A double spiral

There are two entrances / exits. Both paths (colourfully marked) meet in the center where the direction of the movement changes. The alignment corresponds to a meander.

The alignment of the >Berlin Labyrinth<:

The path in the Berlin Labyrinth

Entrance and exit are placed side by side. There are three turning points where the path changes direction. But it is not a double spiral, because there would the direction change only once.

Following a drawing with the original, Ariadne’s thread and the walls in geometrical correct shape:

Drawing of the Berlin Labyrinth

By the way, the labyrinth can be quite simply drawn, even if the description sounds complex. It refers to the right lower drawing.

• I draw two straight inclined lines, meeting in a center point (in blue, dashed)
• Inside the left half  I draw around this center point in steady distances eight semicircles (in black), the both outside only partially
• Now the right side:
• I connect the 3rd and 5th curve end (counted from above on the left) with the 4th curve end as a center with a semicircle (in cyan)
• I connect the 1st and 3rd curve end (counted from the middle) with the 2nd curve end as a center with a semicircle (in green)
• I continue with three other semicircles (in green) in parallel distance
• The last three semicircles (in brown) have as center the first curve end below the intersection of the two blue auxiliary lines
• Three semicircles have in common an already “occupied” curve end point: the 3rd and 5th from the left on top, the 3rd from the right below
• Eight arcs on the left side of a common line and seven arcs on the right side of it generate the “Berlin Labyrinth”
• The “fontanel” as an empty space is relatively big

The relationship to a classical labyrinth is yet not so good to recognize. But you may still guess that it could be a labyrinth.

Another figure from Weidner’s script fits better:

The Near East clay tablet VAT 9560

There are two entrances / exits and four turning points.

In the graphic we look at every way separately:

Graphics of the Near-East clay tablet VAT 9560

Though the alignment of the turning path is spiral-shaped, nevertheless, it is no double spiral. The circuits swing about two turning points. One time directly and another time embedded around the turning point of the other way. Two circuits of a path thereby also run side by side. In the middle the paths meet and are connected through a meander with each other. One path is leading in and one out.
Every path for itself looks like a labyrinth. Hence, we have two labyrinths intertwined together who are connected through a meander. The paths are unequivocal and purposeful, change commuting the direction and have no branchings or dead ends. They fill out the whole interior and must be followed entirely. All that what Hermann Kern demands for a labyrinth.

Following the path in a Babylonian visceral labyrinth in a geometrically correct shape:

Ariadne’s thread in a Babylonian visceral labyrinth

Following the “walls” in a geometrically correct drawing:

The Babylonian visceral labyrinth

This labyrinth has even a seed pattern. Who finds it? (More about that in a later posting).

There is no end of the path in a clearly defined center as we now (in the Western world) are accustomed. It is a path not leading to a center, but through it. It shows a quite different meaning of the labyrinth. It comes from a quite different culture and served other purposes. It matches rather the motto: The way is the aim.
Even if we do not recognize that as a “full-value” labyrinth, one must see it as a precursor of the “true” labyrinth.
We have two paths in the Baltic Wheel. The Wunderkreis of Kaufbeuren has even a branching and a meander in the middle. We accept, in the meantime, also other creations as walk-through or processional labyrinths.

However, I have found in Weidner’s script something else very interesting: A visceral labyrinth with only one way ending in a center. It can be drawn with a already known seed pattern. More about that in a later posting.

… To be continued

Further Links

More information about the Babylonian clay tablets can be found in an excellent article from Richard Myers Shelton in Jeff Sawards Caerdroia 42 (March 2014).

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## Seed Pattern and Pattern

In several previous posts I have shown, that different variants can exist for a certain labyrinth or seed pattern.

Illustration 1. Variants of the Same Seed Pattern

In Ill. 1 I again show some variants of the seed pattern for the Ariadne’s Thread of my demonstration labyrinth. This same seed pattern can be drawn e.g. with a circular, elliptic, petal-shaped or rectangular outline. The outline figure is only an auxiliary figure. The seed pattern itself is formed by the system of lines within this outline figure. Depending on the shape of the outline figure, also the orientation and rounding of the seeds may somewhat differ. However, they are always ordered the same way. On top left one (not-nested) turn, on bottom left two nested and on the right three nested turns. Which variant of the seed pattern is best suited depends on the purpose for which it is used.

In this post I want to show the relationship between the seed pattern and the pattern. For this purpose, the rectangular variant is best suited. The seed pattern can be transformed to the pattern in a few steps.

Illustration 2. From Seed Pattern to Meander

The left figure of ill. 2 shows the rectangular variant of the seed pattern. This is also shown as baseline in grey in the right figure. As a first step, the right half of the seed pattern is shifted against the left (shown in red), until it comes to lie on the other side of the left half.

Illustration 3. From Meander to Pattern

The result of this shift is a meander. It is one of Arnol’d’s figures. This meander is in a next step straightened-out, as has already been shown here. For this, the right half of the seed pattern is shifted somewhat further to the left. The ends opposite each other are then connected with lines.

Illustration 4. Pattern

The result of this process is shown in ill. 4. Apparently, in transforming the meander to the pattern, the first and most important step is the horizontal straightening-out. By this the situation of the circuits in the pattern are made apparent. Next, one can easily straighten-out the axial segments and finalize the pattern.

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