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## How to make a Wunderkreis, Part 2

The previous post was more concerned with the geometry and the mathematically correct construction of the Wunderkreis in general.

Here is an example of how you can make it less theoretically. Denny Dyke from Circles in the Sand often creates double spirals and the Wunderkreis in his Dream Fields on the beach of Oregon. Denny Dyke has kindly shown me his method.
In the following photos it is explained.

Freehand he scratches the lines in the sand. Hence, the way runs between the lines. The double spiral has three arcs, the surrounding labyrinth has five circuits.

Step 1

Denny begins with the lower part of the double spiral and draws three semicircles. On the left he adds two lines and the turning point, on the right there are three lines and the turning point (step 1).

Step 2

Now he scratches three semicircles for the upper part of the double spiral. The first semicircle begins in the middle of the innermost lower semicircle (step 2).

Step 3

All the other curves are drawn in parallel and equal distance to this arc by connecting all free ends of the existing lines and the turning points. Just the way we do it in the Classical labyrinth. We begin on top and draw four lines on the left side around the double spiral to the right side (step 3).

Step 4

In the same way the two free lines below are connected together (step 4). Having done this the Wunderkreis has quite been completed.

The open lower middle section contains the two entries of the Wunderkreis. On the left side we enter the labyrinthine circuits. On the right side we have the exit out of the double spiral.

The completed Wunderkreis

Denny has marked both accesses and has separated them through the “shoehorn” known from the Baltic wheel.

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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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It is only a labyrinth if we accept walk-through labyrinths as such, since it has two accesses and no middle in which one can remain. I also use the German term “Wunderkreis” and not the translated “wonder/miracle circle”.

I consider it as a real labyrinth and even state that it has older roots than the Cretan labyrinth from the Mediterranean area. The activity with the Babylonian labyrinths brought me to this view, as there is a double spiral in the centre of a typical Wunderkreis. But a spiral alone does not make a labyrinth, meandering patterns are also required.

Some examples:

Wunderkreis of stones

This is a nice specimen laid with stones like the Scandinavian Troy Towns. The way runs between the stones. The entrance lies in the middle below and then branches out. I can go on to the left or to the right. However, I must wander through the whole figure to come out again. In the centre the determining change of course takes place. The two turning points around which the way is led pendulously, lie on the left and on the right side. I move towards the middle or sometimes away of it; sometimes I turn right and sometimes I turn left, as I do in a classical labyrinth.
Two parts constitute the figure: the double spiral with the meander in the middle and the circuits around the two turning points. Which part will be run through first, depends on which way you choose. However, the two parts are not mixed, each element must be run for itself.

The element with the two turning points, which form a triangle in combination with the centre in the double spiral, also appears as own labyrinth type, such as the type Knossos, the Baltic wheel and the Indian labyrinth.

The Baltic wheel also has the second access/exit to the middle which  is very short, however. The real middle is formed by a bigger, empty area. Nevertheless, it is not a Wunderkreis, because the second way alone does not constitute one, but the double spiral in the middle.

Old drawing of the Eberswalde Wunderkreis

In this drawing the paths rather than the walls are shown in black lines. The Wunderkreis was put on first in 1609 and to the quartercentenary in 2009 even a coin was designed.

Coin for the quartercentenary

Here the design looks a little bit different, nevertheless, the course of the path is the same as in the drawing. In the meantime, a Wunderkreis from paving-stones was put on again in Eberswalde. Not on the Hausberg like in 1609, but on the Schützenplatz.

The new Eberswald Wunderkreis

Another historical Wunderkreis is passed down from Kaufbeuren.

A similar Wunderkreis has been put on in 2002 in the Jordanpark again.

The 2002 restaured Kaufbeuren Wunderkreis

The Transylvanian Saxons brought new insights to the use of the Wunderkreis with the celebration of the march through it. The original Zeiden Wunderkreis still exists in today’s Romania. The Zeiden community have carried on the traditions round the Wunderkreis here in Germany so that we have learned more about that labyrinth.

Drawing of the Zeiden Wunderkreis

The lines here illustrate the way and first turn to the right. They also do not branch out, but run apart. Thus we can assume that the external circuits were traversed first and then the double spiral.

At quite a different place the following temporary Wunderkreis was built in July 2015 : At low tide on the beaches of Bandon in Oregon (USA).

Dream Field at Face Rock on the beaches of Bandon, Photo © Courtesy of Pamela Hansen

Since 2014 Denny Dyke and his team have put on new creations under “Circles in the Sand” in the Dream Field Labyrinths. Besides, he often uses the double spiral and the Wunderkreis which is particularly suitable for these as it is a walk-through labyrinth. It does not depend on the external shape, a Wunderkreis can also be angular.

Now we can look at the most important features of the Wunderkreis in a sort of a blueprint. Here we have the limitation lines (walls) in black. We see four termini. The two entries are arranged side by side.

The walls of the Wunderkreis

If we color the paths in different colours we can recognize better the essential components of this type of labyrinth. There are two different areas. If we enter through the left entrance we first surround the two turning points in the lower area in a pendular movement changing direction on every side. The way on the right leads into the double spiral.

The paths of the Wunderkreis

The initial movement in a processional labyrinth first leads around the outermost circuits. In the double spiral the most important change of course takes place and leads out from there again.
The Wunderkreis was often used for competitions and even served as a sort of racetrack. Maybe the name can be traced back to this use as well.

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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.

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:

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

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).

## Meander and Labyrinth – Why Straighten-out?

In my last post I have pointed out the close relationship between the figures of Arnol’d and the patterns of labyrinths. These figures are very similar, but they are not equal. One has to straighten-out Arnol’d’s figures to get to the patterns of the labyrinths. But what does this mean: “straighten-out”? I want to show this with the example of one of Arnol’d’s figures. I have purposely chosen figure 8 for that.

Illustration 1: double-spiral

Illustration 1 shows figure 8 of Arnol’d in its original orientation (first image) and rotated by a quarter of a circle (second image). The third image shows the same figure as the second, although reproduced on the computer and a bit more rounded. As can be seen, the curve comes from above left and winds itself inwards anticlockwise with narrowing radii.  It intersects the straight line twice. At the point where it intersects the straight a third time, the curve changes to clockwise direction and winds itself outwards. This curve in fact presents as a double-spiral, that intersects the straight five times.

Now, what happens, if we straighten-out this figure?

This is shown in the next illustration.

Illustration 2: double-spiral-type meander

The double-spiral is dissected along the straight line and both halves are shifted horizontally. Next, the pairs of points that were generated by this separation are connected with horizontal lines (dashed in the figure). These lines in fact represent the circuits of the labyrinth. Separating the double-spiral and inserting connection lines leads to the pattern of the labyrinth. Here lies the relationship between the intersection points in Arnol’d’s curves and the circuits of the labyrinth.

In addition, this example illustrates very well the difference between a double-spiral and a double-spiral-type meander. Remember that Arnol’d’s figure 8 corresponds with the meander Erwin found best suited for labyrinths. This meander is in fact a double-spiral-type meander, and this holds for all of Erwin’s types of this meander (type 4, 6, 8, etc.). Separating Arnol’d’s figure and inserting connection lines transforms the continuous movement of the double-spiral into a stepwise progression from one circuit to the next. That is what makes the difference between the double-spiral and the double-spiral type meander that can be found in the patterns of labyrinths.