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Posts Tagged ‘path sequence’

To say it more exactly, here I relate to the 21 row-shaped visceral labyrinths, still known from some of the previous articles (see Related Posts below).

The appearance is defined by the circuit or path sequence. With that one can construct the different and new labyrinth types (here 21). To this I use the once before presented method to draw a labyrinth (see below).

The path and the limitation lines are equally wide. The center is bigger. The last piece of the path leads vertically into the center. All elements are connected next to each other without sharp bends and geometrically correct. There are only straight lines and curves. This all on the smallest place possible. All together makes up the Knidos style.

Look at a single picture in a bigger version by clicking on it:

I think that by this style the movement pattern of every labyrinth becomes especially well recognizable. With that they can be compared more easy with the already known labyrinths.

Remarkably for me it is that only one specimen (E 3384 v_6) begins with the first circuit. And the fact that many directly circle around the middle and, finally, from the first circuit the center directly is reached. Noticeably are also the many vertical straight and parallel pieces in the middle section.

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Here it is about the decoding of the circuit sequences of the row-shaped 21 visceral labyrinths shown in the last article on this subject (see related posts below).

The question is: Can I generate one-arm alternating labyrinths with one center in the middle from them? That means no walk-through labyrinths where the also unequivocal path passes through, but is ending at an aim in the middle.
Maybe one could call them “walk-in labyrinths” contrary  to the “walk-through labyrinths”?

The short answer: Yes, it is possible. And the result are 21 new, up to now unknown labyrinths.

The circuit sequence for the walk-through labyrinth can be converted into one for a walk- in labyrinth by leaving out the last “0” which stands for “outside”. The highest number stands for the center. If it is not at the last place in the circuit sequence, one must add one more number.
This “trick” is necessary only for two labyrinths and then leads to labyrinths with even circuits (VAT 984_6 and VAN 9447_7).

The gallery shows all the 21 labyrinths in concentric style with a greater center.

Look at the single picture in a bigger version by clicking on it:

 

All labyrinths are different. Not one has appeared up to now somewhere. They have between 9 and 16 circuits, the most 11 circuits. They show between 3 and 6 turning points.

In these constellations there are purely mathematically seen 134871 variations of interesting labyrinths, as proves Tony Phillips, professor of mathematics.

There are still a lot of possibilities to find new labyrinths or to invent them.

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The website of Tony Phillips

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Or differently asked: Can I transform a classical labyrinth into a Babylonian visceral labyrinth?

Therefore we should first see the differences; and then the interlinking components.

As an example I start with the best known classical labyrinth: The 7 circuit Cretan labyrinth.

The 7 circuit labyrinth

The 7 circuit Classical labyrinth, on the right the complementary to it

It has a center and an entrance. There is only one way in. In the middle I am at the aim and at the end of the way. To leave I must turn and take the same way in reverse order.

Among the Babylonian visceral labyrinths one can distinguish two main groups. One are more round and devoured into each other, while in others the loops are arranged row-shaped.

Here as an example the labyrinth E3384_r8 on a clay tablet from Tell Barri (Syria) (for more, please see related posts below).

A Babylonisn visceral labyrinth

A Babylonian visceral labyrinth with 10 circuits and two entries

In the visceral labyrinth I have two entries and no real center. Nevertheless, the way leads through all of the loops to the other access. It is a walk-through labyrinth.

The circuits here are numbered from the left to the right, while in the classical labyrinths they are numbered from the outside inwards. “0” stands for the outside, in the classical labyrinth the last figure for the center.

Every labyrinth is designated by a row of numbers, the circuit sequence or the path sequence. This is the order in which the circuits will be run one by one.

The connecting element therefore is the circuit sequence. Hence, we must construct “row-shaped” walk-through labyrinths from the circuit sequence of the classical labyrinths.

At first we take the 7 circuit labyrinth as shown above. We use the circuit sequence and connect the circuits arranged in row accordingly. The second “0” indicates the walk-through labyrinth.
Then this looks as follows:

Das 7-gängige Labyrinth als Eingeweidelabyrinth

The 7 circuit classical labyrinth as visceral labyrinth, on the right the complementary

We make this still for some more classical labyrinths.

Das 3-gängige Labyrinth

The 3 crcuit labyrinth, on the left the original, on the right the complementary to it

The original is developed from the meander and is also called Knossos labyrinth. The right one is developed from the “emaciated” seed pattern. However, is at the same time complementary to the Knossos labyrinth. Under the walk-in labyrinths the visceral walk-through labyrinths.


A 5 circuit labyrinth:

Das 5-gängige Labyrinth

A 5 circuit labyrinth, on the right the complementary

There are still other 5 circuit labyrinths with an other circuit sequence. But, in principle, the process is the same one.

The shown examples were all self-dual labyrinths.


Now we take a 9 circuit labyrinth. There are more variations:

Das 9-gängige Labyrinth

A 9 circuit labyrinth in four variations

And here the corresponding visceral labyrinths:

Die Eingeweidelabyrinthe

The visceral labyrinths


Here the 11 circuit labyrinth with the corresponding visceral labyrinths:

Das 11-gängige Labyrinth

The 11 circuit labyrinth and its complementary

This one is self-dual again. Therefore there is only one complementary version to it.


Here the 15 circuit labyrinth:

Das 15-gängige Labyrinth

The 15 circuit labyrinth and its complementary

This is also self-dual.

If we compare these newly derived visceral labyrinths to the up to now known historical Babylonian visceral labyrinths, we can ascertain no correspondence. Maybe a clay tablet with an identical labyrinth appears somewhere and sometime?

So far we know about 21 Babylonian visceral labyrinths as row-shaped examples in most different variations.

For comparison I recommend the following article with the overview.

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When I dealt with the Knossos labyrinth it has struck me that the seed pattern can be simplified very easily. It can be reduced to three lines and two dots. To draw the labyrinth they are connected just as we do it for the classical labyrinth. For more information please see the Related Posts below.

Now this seed pattern with the two turning points can be extended in a very simple way, just by adding more lines in pairs.
seed pattern

The bigger labyrinths have more circuits, however, maintain her basic structure. And, nevertheless, these are own types, because they have another path sequence than the 7-, 9-, 11-, 15- etc. circuit  classical labyrinths. But they are not known, neither among the historical, nor among the contemporary labyrinths. Because they are too easy? Besides, the lines have quite a special rhythm. A closer look can be worthwhile.
The 3 circuit labyrinth of this type first appeared about 400 B.C. on the silver coins of Knossos:

The Labyrinth type Knossos

The Labyrinth type Knossos

The circuits are numbered from the outside inwards from 1 to 3. The center is marked with 4. The blue digits labels the circuits inside out. The path sequence is 3-2-1-4, no matter which direction you take. Through that a special quality of this labyrinth is also indicated: It is self-dual.

What now shall be the special rhythm? To explain this, we look at a 5 circuit labyrinth of this type:

The 5 circuit labyrinth in classical style

The 5 circuit labyrinth in classical style

The path sequence is: 5-2-3-4-1-6. At first I circle around the center (6) on taking circuit 5. Then I go outwardly to round 2, from there via the circuits 3 and 4 again in direction to the center, at last make a jump completely outwards to circuit 1, from which I finally reach the center in 6.

Here a 7 circuit labyrinth in Knidos style:

7 circuit Labyrinth in Knidos style

7 circuit Labyrinth in Knidos style

The path sequence is: 7-2-5-4-3-6-1-8. It is also self-dual. The typical rhythm is maintained, the “steps” are wider: From 0 to 7, from 7 to 2, and finally from 1 to 8 (the center).

Here a 9 circuit labyrinth in concentric style:

9 circuit labyrinth in concentric style

9 circuit labyrinth in concentric style

The path sequence is: 9-2-7-4-5-6-3-8-1-10. The step size is anew growing. This labyrinth is self-dual again.

This example exists as a real labyrinth since the year 2010 on a meadow at Ostheim vor der Rhön (Germany):

9 circuit labyrinth in concentric style at Ostheim vor der Rhön (Germany)

9 circuit labyrinth in concentric style at Ostheim vor der Rhön (Germany)

To finish we look at a 11 circuit labyrinth in square style:

11 circuit Labyrinth in square style

11 circuit Labyrinth in square style

The path sequence is: 11-2-9-4-7-6-5-8-3-10-1-12. And again self-dual.

I think, the method is clear: We add two more lines more and we will get two circuits more. So we could continue infinitely.
The shape of the labyrinth can be quite different, this makes up the style. The path sequence shows the type. And for that kind of labyrinth we always have only two turning points.

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Andreas recently brought here a posting to the Jericho labyrinth St. Gallen which probably shows the historically first crossing of the main axis in a classical labyrinth.

I have written about that already. But now I would like to do it once again. Because for me it seems to be an element to design the labyrinth which I have not seen anywhere.

I became aware of that when I have no longer drawn the labyrinth from a seed pattern, but from the path sequence. And, besides, have noted that there also are different possibilities to connect the lines.

Using the example of the classical 7 circuit labyrinth I will explain this once again. How many possibilities to cross the axis are there, and how does it look like?

First the original labyrinth, however round and with a bigger middle.

The classical 7 circuit labyrinth

The classical 7 circuit Labyrinth

The last path sequence into the middle lies on the vertical main axis. The entrance lies on the left side of the main axis, leads to the third circuit, and turns to the left at first. The entry into the middle takes place from the fifth circuit from the right side, and faces the entrance.


How often can I now traverse the axis?
At two positions: From the first to the fourth circuit, and from the fourth to the seventh circuit. This can happen at each position alone or at both positions together. The result are three variations.

Here the first version:

Crossing the axis from the 1st to the 4th circuit

Crossing the axis from the 1st to the 4th circuit

By crossing the main axis from the first to the fourth circuit I do not change the direction of movement as in the original labyrinth. I am still turning to the left in the fourth circuit.
However, thereby I also reach the middle from the left side, so to speak I have laid this entry on the other side of the main axis.
The main entrance slides a little further to the left, and the two lower turning points also move to the left.


The second version:

Crossing the axis from the fourth to the seventh circuit

Crossing the axis from the fourth to the seventh circuit

Here the change from the first to the fourth circuit remains like in the original, however, from the fourth to the seventh circuit I maintain the “spin”.
The entry into the middle is executed from the left side as it is in the original. However, the main entrance slides to the right side. Both lower turning points are shifted to the right.


The third version:

Crossing the axis from the first to the fourth, and from the fourth to the seventh circuit

Crossing the axis from the first to the fourth, and from the fourth to the seventh circuit

The vertical main axis is crossed twice as in the previous versions, now together.
Through that the  lines are displaced considerably. Everything moves to the left. The main entrance lies again on the left side, the entry into the middle is made from the right side.


Here the path as Ariadne’s thread:

Ariadne's thread in familiar manner

Ariadne’s thread in familiar manner

Ariadne's thread with the axis crossed twice

Ariadne’s thread with the axis crossed twice

 

 

 

 

 

 

 

 

 

 

Maybe one can dismiss that as unnecessary? It would be nice, nevertheless to try it out in practice. Above all how it feels to experience another change of direction than in the original.

Maybe the opportunity arises in a big sandtable exercise? On a sandy beach for example? Where one can simply scratch the lines into the sand, and allows the flood to erase them out leniently.

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Sometimes you will get a diamond-shaped, empty element in the middle part of a Knidos labyrinth which is formed normally by a cross. This happens when all the paths have the same width and the walls are aligned to them. This form arises because the four turning points form a square.
A shift of the rhomb may also result if one brings into line the entrance axis (of the path into the labyrinth) and the entry axis into the center of the labyrinth with the main axis of the labyrinth figure. In the “twisted labyrinth” I have demonstrated this already once (see related posts below).

If one wants to give a certain shape to this “empty form”, one can play with the position of the turning points. I have done this to get the “form-fitting” labyrinth. All elements are arcs, however, the four turning points do not lie any more in a square.

The suggestion for this labyrinth dates to the logo sketched by the Swiss artist Agnes Barmettler with the woman in the labyrinth for the public women’s places.

The logo for the public women's places

The logo for the public women’s places

Such a labyrinth can be drawn nicely, but is hard to build, above all as a big labyrinth. Hence, I have tried to develop the shape for this labyrinth with geometrical elements only. Some imagination is asked of course. Anyway, the “empty space” offers creative leeway.

The "form-fitting" Knidos Labyrinth

The “form-fitting” Knidos Labyrinth

The following layout drawing for a sort of prototype shows the geometrical qualities in detail.

Who looks accurately and compares to the original Knidos labyrinth, recognises that one segment less arises. The turning point below on the right is laid in the lengthening of the line from the midpoint of the center and the upper right turning point. The usually narrow “cake piece” is thereby lost in this area.

The drawing

The drawing

Who would like to build such a labyrinth, is invited warmly to it. The drawing contains all that is neccessary. Indeed, one can also use other parametres, because the labyrinth is scaleable. The underlying values are based on the dimension between axes of 1 m. This means that all details changes proportionally (are reduced or extended). If one wants, e.g., a half as big labyrinth, one multiplies all measures by 0.5. One can even proceed in such a way with “crooked” numbers to get the desired result.

Here you may see, print, save, copy the PDF file of the drawing

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Recently I became aware of this illustration:

The Rauner Library from Hanover (USA) acquired recently a specially remarkable manuscript. The manuscript is a copy of the Taj Torah produced in Yemen c. 1400-1450. This is one of only three known Hebrew manuscripts with illustrated carpet pages.

The labyrinth in the Taj Torah

The labyrinth in the Taj Torah (Illustration with kind permission of © Rauner Library)

It is a 6 circuit Jericho labyrinth. The walls are drawn with a thick red line and with a black double line. The way into the middle (Ariadne’s thread) is indicated through a banner. With this the structure of the labyrinth is revealed very well. The entrance is on top.

In a great number of manuscripts the city of Jericho is shown as a labyrinth or in the center of a labyrinth. This tradition is proved in different cultural spheres from the 9th up to the 19th century. The labyrinth type used for the Jericho Labyrinth is from the Classical 7 circuit (Cretan) on to the Chartres labyrinth.

Under them are also some 6 circuit labyrinths with 7 walls which are probably based on the Jewish tradition of the seven walls around the city of Jericho. They show an advancement of the labyrinth form.

The oldest known Jericho labyrinth with the same alignment as in the Taj Torah can be seen on a page of a Hebrew Old Testament which was completed by Josef von Xanten in 1294.

That is the reason why Andreas Frei name this type as “von Xanten”.

The Jericho Labyrinth (type von Xanten) in a Hebrew Bible

The Jericho Labyrinth (type von Xanten) in a Hebrew Bible from 1294 / Source: Hermann Kern, Labyrinths, fig. 225

In the book of Hermann Kern one can find other examples of this type. One is found in the Farhi Bible from the 14th century.

The Jericho Labyrinth (type von Xanten) in the Farhi Bible

The Jericho Labyrinth (type von Xanten) in the Farhi Bible from 1366-1383 / Source: Hermann Kern, Labyrinths, fig. 227

On a Hebrew scroll from the 17th century is this drawing. Here, as well as in the upper labyrinth, the entrance is at the bottom. The first turn goes to the right.

The Jericho Labyrinth (type von Xanten) on a Hebrew scroll

The Jericho Labyrinth (type von Xanten) on a Hebrew scroll from the 17th century / Source: Hermann Kern, Labyrinths, fig. 229

The essential of a labyrinth figure can be shown through a geometrical construction in  a drawing.

The Jericho Labyrinth type von Xanten

The Jericho Labyrinth type von Xanten

On the following drawings the labyrinth is mirrored, the path first turns to the left. The path sequence, numbered from the outside inwards (in green), is 0-3-4-5-2-1-6-7. If one numbers the paths from the inside outwardly (in blue) the path sequence will be 0-1-6-5-2-3-4-7.

The original Jericho Labyrinth type von Xanten

The original Jericho Labyrinth type von Xanten

If one draws a labyrinth according to this order, one receives the dual labyrinth. This looks here differently than the original one. This shows that this type of labyrinth is not selfdual as for example the Cretan 7 circuit labyrinth. One obtains a new labyrinth with another structure.

For more information about that context I recommend the website of my co-author Andreas Frei about the pattern (in German). Link >

The dual Jericho Labyrinth type von Xanten

The dual Jericho Labyrinth type von Xanten

Here is the alignment of the original Labyrinth depicted as a rectangular diagram:

The pattern of the Jericho Labyrinth (type von Xanten) as diagram

The pattern of the Jericho Labyrinth (type von Xanten) as diagram

In this diagram with the entrance below on the right, and the center on the top on the right too, the path structure can also be presented very nicely. It is simply Ariadne’s thread as an uninterrupted line in angular form. Andreas calls this the pattern, and shows it a little bit differently. However, the essential things are identical.

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