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Completion of the Seed Pattern

Two more steps are still needed in order to bring the Chartres-type labyrinth into the Man-in-the-Maze style. First, the seed pattern has to be completed.

We already have the seed pattern for the walls delimiting the pathway, but still without the pieces of the pathway that traverse the axes. These are still represented as pieces of the Ariadne’s Thread (fig. 1).

Figure 1. Seed Pattern and Pieces of Path Traversing the Axes

 

The labyrinth should be represented entirely by the walls delimiting the pathway. For this, the walls around the pieces of the path traversing the axes have to be completed (fig. 2).

Figure 2. Completion of the Walls Delimiting the Pathway – 1

We begin from the outside to the inside and first draw the walls around the outermost of these pieces of the pathway.

As a next step we add the walls delimiting the next inner pieces of the pathway (fig. 3).

Figure 3. Completion of the Walls Delimiting the Pathway – 2

As one can see, in each step, for each piece of the path, 2 or 4 for spokes have to be prolonged inwards, which are then connected with an arc of a circle.

And so we continue until all pieces of the path traversing the axes are enveloped by walls delimiting them (fig. 4).

Figure 4. The Final Seed Pattern for the Walls Delimiting the Pathway

This results in the complete seed pattern for the walls delimiting the pathway. In the center of the seed pattern and where the path traverses the axes there exist areas that are not accessible. This is quite analogue with the seed patterns in alternating labyrinths in the MiM-style, in which the center is not accessible either.

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Quite simply: By leaving out the barriers in the minor axes. I have already tried this with the Chartres labyrinth some years ago. And in the last both posts on this subject with the types Auxerre and Reims. You can read about that in the related posts below.

Today I repeat this for the Chartres labyrinth. Here the original in essential form, in a concentric style.

The Chartres labyrinth

The Chartres labyrinth

The original with all lines and the path in the labyrinth, Ariadne’s thread. The lunations and the six petals in the middle belong to the style Chartres and are left out here.

Now without the barriers in the minor axes.

The Chartres labyrinth without the barriers

The Chartres labyrinth without the barriers

All circuits can be included in the labyrinth originating now, differently from the types Auxerre and Reims. The path sequence is: 5-4-3-2-1-6-11-10-9-8-7-12. We have eight turning points with stacked circuits. It is self-dual. That means that the way out has the same rhythm as the way in.

But this 11 circuit labyrinth is quite different from the more known 11 circuit labyrinth, that can be generated from the enlarged seed  pattern.
Since this looks thus:

The 11 circuit labyrinth made from the seed pattern

The 11 circuit labyrinth made from the seed pattern

The path sequence here is: 5-2-3-4-1-6-11-8-9-10-7-12. We have got four turning points with embedded circuits. It is developed from quite another construction principle than the Chartres labyrinth. However, it is self-dual.


Now we turn to the complementary labyrinth.

The complementary labyrinth is generated by mirroring the original. Then thus it looks:

The complementary Chartres labyrinth

The complementary Chartres labyrinth

The entry into the labyrinth happens on the 7th circuit, the center is reached from the 5th circuit. The barriers are differently arranged in the right and left axes, the upper ones remain. It is self-dual.

Without the barriers it looks thus:

The complementary Chartres labyrinth without the barriers

The complementary Chartres labyrinth without the barriers

The transformation again works, as it does for the original. The path sequence is: 7-8-9-10-11-6-1-2-3-4-5-12. Also this labyrinth is self-dual.

We confront it with the complementary labyrinth, generated from the seed pattern.

The complementary 11 circuit labyrinth made from the seed pattern

The complementary 11 circuit labyrinth made from the seed pattern

The path sequence on this is: 7-10-9-8-11-6-1-4-3-2-5-12.
Contrarily to the original this type did not show up historically.

So we have created two completely new 11 circuit labyrinths from the Chartres labyrinth, which look different than the 11 circuit labyrinths that can be developed from the seed pattern.

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Traversing the Axes

In alternating labyrinths with multiple arms the pathway does not traverse the main axis. However, it must traverse each side-arm (see below: related posts 1). How then have the axes to be traversed in the MiM-style? Let me first remember that I have already transformed a non-alternating labyrinth with one arm into the MiM-style (see related posts 2). From this it can be seen what happens when the pathway traverses the axis (figure 1).

Figure 1. Labyrinth of the St. Gallen Type in the MiM-Style

At the places where the pathway traverses the axis, the innermost circle is interrupted. The pieces of the pathway traversing the axes, and only these, in the MiM-style pass through the center of the seed pattern. In all alternating one-arm labyrinths the innermost circle is closed. The center of the labyrinth lies outside of it in any case.

Now for the Chartres type labyrinth in the MiM-style, in each side-arm several pieces of the pathway have to be passed through the middle. From the seed patterns it can clearly be seen, where the side-arms are traversed. These are the gaps between the pieces of arcs where the innermost circle is interrupted. Let us have a look at the firs side-arm in detail (figure 2). The seed pattern of this side-arm lies in west quadrant (highlighted in black).

Figure 2. The Seed Pattern of the First Side-arm

The purpose is to transform the pieces traversing this side-arm into the MiM-style (figure 3).

Figure 3. The Pieces of the Path Traversing the Axis

As everybody knows, the pathway in the Chartres type labyrinth first leads along the main axis to the 5. circuit, makes a turn at the first side-arm, returns to the main axis on circuit 6 and from there reaches the innermost 11th circuit. On this circuit it follows half the arc of a circle whilst it traverses the first side-arm. Then it makes a turn at the second side-arm. From there it returns on the 10th circuit to the main axis whilst passing the first side-arm again. The pathway also traverses the first side-arm on the 7th, 4th and 1st circuit. The pieces of the pathway on the outer circuits enclose those on more inner circuits and outermost piece of the pathway on circuit 1 encloses all others.

Figure 4 shows what happens with the pieces of the pathway traversing the axis (colored in red, the color of the Ariadne’s Thread), when the side-arm is transformed from the concentric into the MiM-style.

Figure 4. Transformation from the Concentric into the MiM-style

The left image shows the side-arm split and slightly opened. The course of the pieces of the path is still quite similar as in the base case from bottom up or top down. However, all pieces of the pathway bend to the opposite direction. In the central image the original course is hardly recognizable any more. Both halves of the side-arm are widely opened. The pieces of the path sidewards come in to the one half and leave from the other half of the side-arm. Between the two halves of the side-arm their course is in vertical direction. The pieces of the pathway on inner circuits enclose the pieces more outwards. The innermost piece on circuit 11 encloses all others. Next, there is only a slight change from this to the right image. All the pieces of the pathway and the seed pattern are transformed into a shape so that they lie between (pieces of pathway = pieces of the Ariadne’s Thread) and on (seed pattern for the walls delimiting the pathway) the spokes and circles of the MiM-auxiliary figure.

Figure 5 shows all three side-arms with all pieces of the pathway traversing the arms in the MiM-style.

Figure 5. All Traverses of Axes

The west and east side-arm have five each, the north side-arm has three pieces of the path traversing the axis. Therefore in the center of the MiM-auxiliary figure additional auxiliary circles are needed to capture the paths traversing the axes. For this, five auxiliary circles are required. And also the spokes have to be prolonged further to the interior. This is because the walls delimiting the pathway (black) all come to lie on the auxiliary circles and spokes. Near the center the distances between the spokes are continually narrowed. Therefore the innermost auxiliary circle must have a certain minimum radius for the walls and the pathways not to overlap each other.

Now we have all elements together we need to finalize the Chartres type labyrinth in the MiM-style.

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The Seed Patterns

In order to transform a labyrinth with multiple arms into the Man-in-the-Maze (MiM) style, also the side-arms must be appropriately transformed (see related posts 1, below). So let us first have a look at what happens when the main axis is transformed. This can be done using the one-arm labyrinth of Heiric of Auxerre. Because this has the same seed pattern as the main axis of the Chartres type labyrinth.

First, the seed pattern is obtained (fig. 1).

Figure 1. Seed Pattern of the Labyrinth by Heiric of Auxerre

It is not important to draw an exact copy (left image). What counts is that the structure is clearly recognizable. The seed pattern consists of vertical and horizontal lines and of dots. It is aligned to the central wall delimiting the pathway (central image). The seed pattern now has to be transformed in such a way, that it fits to the auxiliary figure of the MiM-style (see related posts 2). For this purpose it has to be aligned to a circle of the auxiliary figure or, respectively, to be bent over such an auxiliary circle. The effect of this should be that the central piece of the wall delimiting the pathway lie on the auxiliary circle and the horizontal lines and dots emanate radially from the circle. For this, the seed pattern can be split along the central wall and divided into two halves (right image).

Next, both halves will be bent over an auxiliary circle (fig. 2).

Figure 2. Transformation into the MiM-Style

For this, both halves are opened to a wide angle such that they can be aligned to the auxiliary circle (left image). Then they are bent over the circle and fitted together again on top (right image). Please note that for this process, two pieces of the central wall delimiting the pathway have to be prolonged (dashed lines). Otherwise when transforming the vertical central lines to the semi circles, two gaps on the central circle would remain, one opposite the entrance to the labyrinth and one opposite to the center.

Now we apply the same procedure to the four arms of the Chartres type labyrinth (fig. 3).

Figure 3. The 4 Seed Patterns of the Chartres Type Labyrinth

First we have to obtain the seed patterns of all four arms. In order to facilitate the illustration I choose a labyrinth with a strongly enlarged center and copy the seed patterns of the four arms. Then I shift each of the seed patterns towards the center. In order to transform them into the MiM-style all four seed patterns have to be aligned to one of the circles of the auxiliary figure. For this, they are split into two halves, just the same as previously twith the seed pattern of he one-arm labyrinth.

In a next step the seed patterns are opened wider in such a way that they can be bent over the auxiliary circle (fig. 4).

Figure 4. Their 8 Halves Opened Wide

Then, all eight halves are aligned to the auxiliary circle, i.e. their straight shapes are bent to an arc of a circle (fig. 5).

Figure 5. Aligning the 8 Halves to the Auxiliary Circle

Note again that on the seed pattern of the main axis, two pieces of the central wall delimiting the pathway have to be added in order to complete the transformation into the circular form. This is only necessary in the main axis as on this axis the entrance to the labyrinth and the access to the center are situated. In the seed patterns of the side-arms there is no need for that. The result of the whole process is shown in fig. 6.

Figure 6. The 4 Seed Patterns in the MiM-Style

A much larger auxiliary circle is needed, as not 2, but 8 halves of 4 seed patterns have to be bent over.

The seed pattern of the main axis lies in the south quadrant. It has, similar with the seed pattern of the Heiric of Auxerre type labyrinth, 24 ends.

The seed patterns of the left / upper / right side-arms lie in the west / north / east quadrants. These seed patterns all have two ends less than the seed pattern of the main axis, i.e. 22 ends each.

Thus, the number of spokes needed for the auxiliary figure of the Chartres type labyrinth in the MiM-style, can be calculated. It corresponds with the total number of all ends, i.e. 24 + 3*22 = 90 spokes.

The former outer ends of the seed patterns lie now on the places marked with the small squares in south, north, and slightly above the horizon in east and west. At these places, in each seed pattern its two own halves are connected to each other.

The former inner ends of the seed patterns, however, connect with the inner ends of each neigbouring seed pattern. These connections are situated at the places marked with dashed lines.

One more thing remains to be noted. The inner arc of the circle of the seed pattern of the main axis is formed by an uninterrupted line. This represents the central wall delimiting the pathway. The labyrinths of the Heiric of Auxerre type as well as of the Chartres type are alternating labyrinths. This means, the pathway doesn’t traverse the axis (type Heiric of Auxerre) / main axis (type Chartres). This is different in the side-arms. The pathway always has to traverse a side-arm somehow. Otherwise it would not be possible to design labyrinths with multiple arms at all. The places where the pathway traverses the side-arms are clearly recognizable as gaps where the inner circular line is interrupted.

What this implies for the design of the labyrinth will be shown in the next post.

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In part 1 (see Related Post below) about the simplified seed pattern I only have spoken of the enlargement of labyrinths.

The seed pattern

But of course the number of circuits also can be reduced by this way. This is possible for all labyrinths built from this seed pattern, as well as for all containing this pattern. I would like to call them compounded labyrinths.

For me this are the Indian Labyrinth, the Baltic Wheel and the Wunderkreis. They all have only two turning points, however, the middle is formed in each case differently.
The Indian Labyrinth (Chakra Vyuha) contains a spiral, the Baltic Wheel has a big empty middle and a second access, the Wunderkreis contains a double spiral and also has the second access.

Here the Indian Labyrinth which can be generated through a seed pattern contained in a triangle:

The Indian Labyrinth

The Indian Labyrinth

The Indian Labyrinth with two more circuits:

The enlarged Indian Labyrinth

The enlarged Indian Labyrinth

Here the Baltic Wheel. The middle section is constructed in a special way. But the circuits round the two turning points can be increased or decreased in pairs.

The Baltic Wheel

The Baltic Wheel

The Baltic Wheel with two less circuits:

The downscaled Baltic Wheel

The downscaled Baltic Wheel

The Wunderkreis has a double spiral in the middle section. The double spiral can have more or less windings (not shown here). But the typically “labyrinthine” circuits round the two turning points can be influenced as mentioned above.

The Wunderkreis

The Wunderkreis

The Wunderkreis with two less circuits:

The downscaled Wunderkreis

The downscaled Wunderkreis

In the quoted statements I would like to show that there is a “technology” through that one can influence the size of a labyrinth.

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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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In the last post I have introduced the eleven-circuit Cakra Vyuh Labyrinth. Even though the seed pattern has a central cross and also can be easily drawn freehand, it is not a labyrinth in the Classical style. In fig. 1 I show the seed pattern in different variants.

CaVy_SP_var

Figure 1. Variants of the Seed Pattern

Image a shows the original seed pattern, image b the seed pattern in the Classical style, image c in the Concentric style, and image d in the Man-in-the-Maze style.

This figure clearly shows that the original seed pattern deviates from the Classical style. It is true that this seed pattern has a central cross as for instance the Cretan labyrinth also. However in the Cakra Vyuh seed pattern, from this cross further junctions branch off.

This is different in the Classical style. The Classical style consists of verticals, horizontals, ankles and dots. For this, no central cross is required. This page illustrates well, what I mean (left figure of each pair). If a seed pattern includes ankles these lie between the cross arms and do not branch off from them.

The four images in fig. 1 in part look quite different one from each other. So how do I come to the assertion that they are four variants of the same seed pattern? Let us remember that these figures show seed patterns for the walls delimiting the pathway. Now let us inscribe the seed patterns for the Ariadne’s Thread into these figures (fig. 2).

CaVy_SPab

Figure 2. With the Seed Pattern for the Ariadne’s Thread Inscribed

At first glance this looks even more complex. However, if we focus on the red figures, we will soon see what they have in common.

CaVy_SPa

Figure 3. Seed Pattern for the Ariadne’s Thread

The seed pattern represents a section of the entire labyrinth. More exactly, it is the section along the axis of the labyrinth. The turning points of the pathway align to the axis. This can be better seen on the seed pattern for the Ariadne’s Thread compared with the seed pattern for the walls delimiting the pathway.

In all four seed patterns, turns of the pathway with single arcs interchange with turns made-up of two nested arcs. This constitutes the manner and sequence of the turns and is the basic information contained in the seed pattern. In the four seed patterns shown, the alignment of the turns may vary from circular (image a, image d) to longisch, vertical, slim (image b, image c). The shape of the arcs is adapted to the shape of the walls delimiting the pathway. However in all images it is a single turn in alternation with two nested turns.

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