Feeds:
Posts
Comments

Posts Tagged ‘meander’

I was particularly fascinated by the technique of double barriers in Gossembrot’s 7 circuit labyrinths presented in recent posts. This makes possible completely new types of labyrinths. He probably did not “invent” the double barriers, but he was the first to consistently and systematically use them.

How does this technique affect 5 circuit labyrinths?
I tried that and came across a whole new kind of sector labyrinths.
As you know, one sector after another is traversed in these before the center is reached.

The historical Roman labyrinths are divided into three different variants: the meander type, the spiral type and the serpentine type (see the Related Posts below).
The entry into the labyrinth is usually up to the innermost lane. And in all four sectors the structures are the same.
The change to the next sector either always takes place outside or even once inside (or alternately).

Now the new type:

The new sector labyrinth in concentric style

The new sector labyrinth in concentric style

What is so special about that?
Already the entrance: It takes place on the 3rd lane. This does not occur in any historical sector labyrinth. And the entrance into the center is also from the 3rd lane.

Then the structure expressed by the path sequence is different in each quadrant.

Quadrant I:   3-2-1-4-5
Quadrant II:  5-2-3-4-1
Quadrant III: 1-4-3-2-5
Quadrant IV: 5-4-1-2-3

The transitions to the next sector are always alternately.

Nevertheless, the new labyrinth is very balanced and mirror-symmetrical.

Here in a square shape:

The new sector labyrinth in square shape

The new sector labyrinth in square shape

This makes it easier to compare with the previously known Roman labyrinths (see below), which are mostly square.

The difference to these becomes clear especially in the presentation as a diagram. Because this shows the inner structure, the pattern.

The diagram for the new sector labyrinth

The diagram for the new sector labyrinth

Very nice to see are the nested meanders.

But even in Knidos style, this type is doing well:

The new sector labyrinth in Knidos style

The new sector labyrinth in Knidos style

How should one call this type? And who builds one as a walkable labyrinth?

Related Posts

Read Full Post »

In 2017, a commemorative coin dedicated to the Minoan civilization was issued by the Mint of the Central Bank of Greece.
This earliest civilization in Europe can be traced back to the years around 2600 BC. The Minoan civilization got its name from the famous King Minos. The story goes that, with the help of the god of the seas, Poseidon, and a white bull, he came to power and thus gained fame and reverence among his people.

The 50-euro gold coin from 2017 was issued with an edition of 1500 pieces and minted in real gold (999.9 / 1000) in the highest collector quality “polished plate”.

Here is the value side:

Value side: Hellenic Democracy 50 Euro

Value side: Hellenic Democracy 50 Euro

And here the picture side:

Picture side: Minoan Civilization 2017

Picture side: Minoan Civilization 2017

Two nested cross meanders can be seen in a large square around 5 smaller squares.
Here is the structure in a black and white tracing:

Draw up of the picture side

Draw up of the picture side

The black lines form two closed line systems without beginning and end. The white lines have branches and dead-ends, also without access. This is reminiscent of a similar representation on the silver coins of Knossos, which are well over 2000 years older (see related posts below).

Should the representation again symbolize the labyrinth of the Minotaur?

Related Posts

Read Full Post »

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.

Related Posts:

Read Full Post »

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 layout

The middle enlarged:

The middle

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

Information board

 

Service station for spirit and soul

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

Related Posts

Further Links

Read Full Post »

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

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

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

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

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

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

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.

Related Posts

Read Full Post »

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

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 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

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

The tablet MS 4516, source: Schøyen Collection

Here the drawings of two fields:
The first field on top left

The first field on top left

The fourth field on bottom left (reconstructed)

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

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

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.

Related Posts

Further Links

Read Full Post »

Via Facebook  I have found this modern walk through labyrinth:

Walk-through labyrinth with meanders

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)

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)

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 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

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

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

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

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.

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.

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).

Related Posts

Further Links

Read Full Post »

Older Posts »

%d bloggers like this: