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Posts Tagged ‘path sequence’

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:

Das Jericho Labyrinth (Typ von Xanten) als Diagramm

Das Jericho Labyrinth (Typ von Xanten) als Diagramm

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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The oldest known labyrinth figure is the Classical 7 Circuit labyrinth (sometimes also called: the Cretan labyrinth). Its origin is about 1200 B.C. The further development falls in the time of the Roman empire from 165 B.C. till 400 A.D. The general name is Roman labyrinth and there are different types again. They have in common that different sectors (mostly four) are run one after the other.

The Classical 7 circuit labyrinth in square form

The Classical 7 circuit labyrinth in square form

In his book “Labyrinths and Mazes of the World” (published in 2003 by Gaia Books, London) Jeff Saward has described how the development of the Roman labyrinth from the Classical labyrinth is possible. Her I only want to put this across in a few steps.

We begin with the Classical labyrinth in square form.
In the drawings the boundary lines are shown in black. The seed pattern contained therein is emphasized in blue. The ways are put in orange, in the same width as the boundary lines.

The whole figure is reduced to a quarter through a rotation. The vertical parts of half the seed pattern move to a horizontal line.

The quartered Classical labyrinth

The quartered Classical labyrinth

To generate an entire Roman labyrinth from the quartered labyrinth, another two circuits must be inserted in every sector: One around the middle, and one at the outside. In the outer rings one walks to the the next sector, the last path leads to the center.
If one examines exactly the paths, one can recognize that the way is the same as the way back in a Classical labyrinth. Or differently expressed: In a Roman labyrinth one wanders four times the way back of a Classical labyrinth.

The Roman labyrinth

The Roman labyrinth

The path sequence can be understood with the help of the figures.  So one well can see the Classical labyrinth inside the Roman labyrinth.

Even better one recognizes the relationship with the Classical labyrinth in the diagram illustration.

The diagram of the Roman labyrinth

The diagram of the Roman labyrinth

The Roman labyrinth is self-dual like it is the Classical labyrinth. One sees this well in the following graphics. Howsoever the diagram is rotated or mirrored, the path sequence is always the same. Also it plays no role whether one walks in direction to the center or reversed, or whether one fancies the entrance below or on top.

The diagram of the Classical labyrinth in four variants

The diagram of the Classical labyrinth in four variants

There are different historical Roman labyrinth of this kind. The oldest one comes from the second century A.D. and is to be seen on a mosaic in Pont Chevron (France). This is why Andreas Frei calls it type Pont Chevron (see link below).

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For every labyrinth exists a second or dual one. And in special cases the dual one looks like the original one. Then this is a self-dual labyrinth.

These connections should be explained here.

Andreas Frei has done this on his website under the topic “Grundlagen” (basics), to this day only in German. I expressly recommend to take a look at it, there are some meaningful drawings also.

Here again we will see it from the practical side. Hence, it is a continuation of the post from the 1st of September, 2013 about the circular 7 circuit labyrinths. Through the dual labyrinths here we will get six more to add to the seven there. So we will have 13 new labyrinths in all.

How we will reach for that, should be shown step by step. Maybe a little bit awkwardly, but I hope, understandably.

We number all labyrinths from the outside inwards in black. “0” stands for the  outside and “8” for the center. The path sequence, that is the order in which we walk through the circuits to arrive at the center, is noticed on the bottom left in black.
Then we number all circuits once again from the inside outwardly in green. “0” is now the center and “8” is now the outside. We write down the circuits in the order in which we walk them while going backwards from the middle. This path sequence is noticed on the bottom right in green.

Typ 032147658Typ 034765218Typ 034567218Typ 036547218Typ 054367218Typ 056723418Typ 056743218Typ 056741238

As already mentioned, there is to every (original) labyrinth a second (dual) one. And this arises when we interchange inside and outside, when we turn inside out. The path sequence which we will get, is normally different from the one of the original labyrinth.

If it is the same, we speak of a self-dual labyrinth. Then an internal symmetry is given. Or differently expressed: The rhythm and the motion sequence is the same when stepping inside or outside.  In our examples this applies to the first (well-known Cretan) labyrinth, and to the last, a new labyrinth.


The remaining six have another path sequence and, hence, are to be taken for new, different labyrinths.
Here the six new types (click to enlarge, print or save):

Typ 076321458Typ 076123458Typ 076143258Typ 076125438Typ 074561238Typ 076541238

These examples shows that always at first the middle is circled around. After that one moves inside the the labyrinth and finally one enters the center from the 3rd or the 5th circuit.

In the case of the types introduced in the last article the entry into the center was always from the outermost, the first circuit. Here we have the circling around the middle immediately after stepping into the labyrinth.

The motion sequences are completely different.
It would be of interest exploring that by a temporary or even a permanent labyrinth. Worldwide there are still no labyrinths of this kind.
The shape must not necessarily be perfectly circular. It is important only to adhere to the path sequence.
For the rest, they can be as simply build in sand like the types introduced in the below mentioned post.

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In the meantime, I have developed for myself a method to construct a labyrinth by only using the path sequence. I do not apply the seed pattern to do that.

I would like to show this here for a 7 circuit classical labyrinth with the path sequence  0-3-2-1-4-7-6-5-8. It is from an other shape as the usual classical labyrinth and was to be seen in a previous post.

First I draw freehand the path of the labyrinth (Ariadne’s thread) according to the path sequence. To get an other shape, I cross the main axis with the 4th and the 7th circuit. From the sketch I derive the number of the turning points and the construction lines. Then I count the number of the ways between the central point and the turning points which are also centers in the further construction.

The freehand sketch

The freehand sketch

The broad for the paths and the walls is the same with 1 m, this makes a dimension between axes of 2 m. The diameter of the middle is the fourfold dimension between axes. The lengths of the different construction lines are calculated from this details.
I begin with a horizontal line (M2 – M3) and I fix the central point M1 by applying the two distances from M2 and M3. In the same way the other points are defined. (Fig. 1)

The construction lines

The construction lines

Starting from the center M1 I draw auxiliary circles in an interval  of 1 m from the inner to the external diameter. (Fig. 2)

The auxiliary circles

The auxiliary circles

The lines M1 – M2 and M1 – M3 are extended up to the external diameter, also the lines M2 – M4 and M3 – M5. (Fig. 3) They are limiting the circular arcs.

The external arcs

The external arcs

Then the different curved sections are drawn with the help of the path sequence in the centers M2 to M5.
In fig. 4 this are the semicircles around the turning points of the ways 5/6 in M2 and the ways 6/7 in M3. The construction is made through connection with the end points of the bigger external arcs.
In fig. 5 the curved sections of the remaining external arcs are formed around the centers M2 and M3.
In fig. 6 follows the connection of the ways 1/2 around M4 and the ways 2/3 around M5.
In fig. 7 the open, innermost pieces of the way are connected by strictly watching the path sequence. These are curved sections which joins each other without sharp bends.

The inner arcs

The inner arcs

In fig. 8 I turn all curves and lines around the center M1 in such a way that the entry axis for the middle is strictly vertical.

Fig. 9 shows the completed labyrinth. The paths are highlighted in color. The walls have the same broad and are left white. The left blank fontanel is good to recognise. It has another shape as the usual classical labyrinth. It appears when the paths are drawn in constant width.

The 7 circuit Knidos labyrinth

The 7 circuit Knidos labyrinth

The drawing is rather a sort of model or template. The labyrinth can still have got different shapes, can be angular or completely round, right- or left-handed. The paths and the walls can be of different widths, also the middle can be bigger or smaller. It is only important to maintain the alignment of the labyrinth.

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