How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 3

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 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

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 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.

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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 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 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 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.

Related Posts

• Does the Chartres Labyrinth hide a Troy Town, and if so, what does it look like? 
• How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 1
• How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 2

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How to Draw a Man-in-the-Maze Labyrinth / 10

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.

Related posts:

1. How to Draw a Man-in-the-Maze Labyrinth / 9
2. How to Draw a Man-in-the-Maze Labyrinth / 6

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How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 2

Quite simply: By leaving out the barriers in the minor axes. I have already tried this with the Chartres labyrinth (see related posts below). But is that also possible with any other Medieval labyrinth?

In part 1 I had made it for the type Auxerre. Now I take the type Reims which is also self-dual like Chartres and Auxerre. And again I take the complementary version. All examples are presented in the concentric style.

The Reims labyrinth

 

Here the original with all lines and the path in the labyrinth, Ariadne’s thread. The barriers in the upper minor axis are identical with those in the type Chartres, the barriers in the horizontal axis are different from Chartres, as well as the arrangement of the turning points in the main axis below the center.

The Reims labyrinth without the barriers

The barriers are left out. When drawing the path I had to discover that four lanes cannot be included. These are the both outermost and the both innermost tracks (1, 2, 10, 11). Hence, I have anew numbered the circuits and there remain only 7 circuits instead of the original 11. However, this also means that by changing the Reims  Medieval labyrinth into a concentric Classical labyrinth through this method not an 11 circuit labyrinth is generated, but a 7 circuit.

The circular 7 circuit labyrinth

This is an up to now hardly known and not so interesting labyrinth. Since one enters the labyrinth on the first circuit and arrives at the center from the last. The path sequence is very simple: 1-2-3-4-5-6-7-8, a simple serpentine pattern.

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Now we turn to the complementary labyrinth:

The complementary Reims labyrinth

The complementary labyrinth is generated by mirroring the original one. The upper barriers remain, right and left they run differently and in the main axis, the turning points shift. The entrance into the labyrinth changes to the middle (lane 9) and the entrance into the center is from further out (lane 3).

The complementary Reims labyrinth without the barriers

As with the original four lanes can not be inserted (1, 2, 10, 11). Hence, a 7 circuit labyrinth arises again. I have anew renumbered the lanes and have drawn the labyrinth anew.

Then thus it looks:

The circular 7 circuit labyrinth

The labyrinth is entered on the 7th lane, the center is reached from the first lane. The path sequence is: 7-6-5-4-3-2-1-8. This labyrinth does not belong to the historically known labyrinths. However, it has already appeared in this blog (see related posts below).

The surprising fact is that again no 11 circuit Classical labyrinth could be generated through the transformation. Rather two 7 circuit labyrinths.

Related Posts

• How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 1
• Un- / interesting Labyrinths
• The Six Very Interesting Labyrinths with 7 Circuits
• How to Draw Eight 7 Circuit Labyrinths
• How to make a Circular 7 Circuit Classical Labyrinth and Seven New (up to now unknown) Circular 7 Circuit Labyrinths

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How to Draw a Man-in-the-Maze Labyrinth / 9

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.

Related posts:

1. How to Draw a Man-in-the-Maze Labyrinth / 8
2. How to Draw a Man-in-the-Maze Labyrinth 

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How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 1

Quite simply: By leaving off the barriers in the minor axes. I have already tried this with the Chartres labyrinth (see related posts below). But is that also possible with every other Medieval labyrinth?

As an example I have chosen the type Auxerre that Andreas showed here recently. This labyrinth is self dual as are Chartres and Reims, therefore of special quality. And they all have a complementary version.

The Auxerre labyrinth

Here the original with all the lines and the path in the labyrinth, Ariadne’s thread. The barriers in the minor axes are identical with those of the Chartres type. There is only another arrangement of the turning points (the lanes 4, 5, 7, 8) in the middle of the main axis.

The original Auxerre labyrinth without the barriers

The barriers are omitted. When drawing Ariadne’s thread, I found that four tracks could not be inserted. Hence, I have anew numbered the circuits and there remain now 7 circuits instead of the original 11. However, this also means that by changing this Medieval labyrinth into a concentric Classical labyrinth through this method no 11 circuit labyrinth is generated, but a 7 circuit.

The 7 circuit circular Cretan labyrinth

If one looks more exactly at it, one recognises the well-known path sequence: 3-2-1-4-7-6-5-8. We got a Cretan labyrinth in concentric style.

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Now we turn to the complementary labyrinth:

The complementary Auxerre labyrinth

The complementary labyrinth is generated by mirroring the original one. The upper barriers remain, right and left they run differently and in the main axis, the turning points shift. The entrance into the labyrinth changes to the middle (lane 9) and the entrance into the center is from further out (lane 3).

The complementary Auxerre labyrinth without the barriers

As with the original, four lanes can not be inserted (4, 5, 7, 8). Therefore, the result is again a 7 circuit labyrinth. I renumbered the lanes and have redrawn the labyrinth.

This is how it now looks like:

The complementary 7 circuit circular Cretan labyrinth

The labyrinth is entered on the 5th lane, the center is reached from the 3rd lane. The path sequence is: 5-6-7-4-1-2-3-8. This labyrinth is not one of the historically known labyrinths. But it showed up in this blog several times (see related posts below). Because it belongs to the interesting labyrinths among the mathematically possible 7 circuit labyrinths.

The surprising fact is that no 11 circuit Classical labyrinth could be generated through the transformation. But for that  the 7 circuit Cretan labyrinth. Therefore we can say that the heart of the Medieval Auxerre labyrinth is the Cretan (Minoan) labyrinth as it is in the Chartres labyrinth.

Related Posts

• The Complementaries to the Three Very Interesting Historical Labyrinths with 4 Arms and 11 Circuits
• The Dual Labyrinth
• The Complementary Labyrinth
• The Round Classical Labyrinth
• How to Draw Eight 7 Circuit Labyrinths
• How to Make a Circular Classical 7 Circuit Labyrinth and Seven New (up to now unknown) 7 Circuit Labyrinths
• The Heart of the Chartres Labyrinth is the Classical Labyrinth

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How to Draw a Man-in-the-Maze Labyrinth / 8

Labyrinths With Multiple Arms

Until now, almost exclusively labyrinths of the basic type (Cretan type) have been implemented in the Man-in-the-Maze style. All one-arm labyrinths can be drawn in this style (see related posts 2, below). But is this also possible in labyrinths with multiple arms? I have tried this out with the most famous labyrinth with multiple arms, the Chartres type labyrinth. And it works. I have already shown the result in January (see related posts 1). In order to arrive there, a prolonged process was needed. In the following I will describe the detailed steps.

Jacques Hébert† has shown on his website (see further links 1, below), that a one-arm labyrinth exists, which has the same seed pattern as the main axis of the Chartres type labyrinth. He had derived this from the enigmatic labyrinth drawing (fig. 1) contained in a medieval manuscript.

Figure 1. Enigmatic Labyrinth Drawing from a Manuscript Compiled 860-862 by Heiric of Auxerre

For this, he had deleted the hand drawn figures indicating the side-arms and closed the gaps where the walls delimiting the pathway were left interrupted. He had named the labyrinth after learned Benedictine monk Heiric of Auxerre who had compiled this manuscript in about 860 – 862.

Figure 2. Labyrinth Named after Heiric of Auxerre by Jacques Hébert

The website of Hébert is no longer active any more. Thanks to a note by Samuel Verbiese we can now find it again in The Internet Archive (see further links 2). Erwin also has introduced this type of labyrinth in this blog (see related posts 3).

The Heiric of Auxerre labyrinth is ideally suited as a starting point. It is quasi the Chartres type as a one-arm labyrinth. So let us first transform this labyrinth into the MiM-style (fig. 3).

Figure 3. The Heiric of Auxerre Labyrinth in the MiM Style

The seed pattern of this labyrinth has 24 ends as have all seed patterns of labyrinths with 11 circuits. So we need an auxiliary figure with 24 spokes for the transformation into the MiM-style.

Next, the side-arms have to be included. A first attempt can be made by retrieving the barriers. This can be achieved by inserting 3 additional spokes for each side-arm as shown in fig. 4.

Figure 4. Insertion of the Side Arms

Thus, the auxiliary figure is extended from 24 to 33 spokes. The result is shown in fig. 5.

Figure 5. Labyrinth of the Chartres Type…

This now looks quite decently like a MiM labyrinth. However, upon a closer view it reveals as unsatisfactory. Fig. 6 shows the reasons why.

Figure 6. … in a Hybrid Style

This labyrinth is of a hybrid style. While the main axis is formed in the MiM-style, the side-arms, however, are in the concentric style. The turning points of the pathway (red arcs in the figure) on the main axis are aligned along the circles of the auxiliary figure. On the side-arms, however, they are aligned along the spokes. What is characteristic for the MiM-style is the seed pattern of the main axis. The figure looks much like a labyrinth in the MiM-style because the main axis with it’s 24 of 33 spokes dominates the whole picture.

Therefore, if we want to implement a labyrinth with multiple arms in the MiM-style, we must also transform the side-arms into the MiM-style. For this it is necessary to really understand and consequently adopt

• how the seed pattern is organised in the MiM-style
• and correspondingly, how the pieces of the pathway traversing the arms have to be designed.

More about this in following posts.

Related posts:

1. Our Best Wishes for 2018
2. How to Draw a Man-in-the-Maze Labyrinth
3. Does the Chartres Labyrinth hide a Troy Town….

Further Links:

1. Website by Jacques Hébert
2. The Internet Archive

 

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How to make a 3 Circuit Chartres Labyrinth

By halving a 7 circuit labyrinth in labyrinthine logic, as it was successful for the 5 circuit Chartres labyrinth.

The 7 circuit Chartres labyrinth

The 4th circuit cuts the labyrinth in two parts. Then I receive an external (circuits 1 – 3) and an internal labyrinth (circuits 5 – 7). Both are identical in its path sequences. Even if the “barriers” are at different places.

Two 3 circuit Chartres labyrinths

The path sequence defines the type: 3-2-1-2-3-2-1-4. It is identical for both versions. This 3-2-1-4 reminds very much of the smallest possible labyrinth: the Knossos labyrinth (and of the meander).
If I leave out the barriers, I receive this labyrinth. This once again shows the quality of the Chartres labyrinth.

To make the layout more appealing, I can arrange the barriers in steady distances, in a way make a labyrinth with three arms.

The 3 circuit Chartres labyrinth (Petit Chartres)

This is the smallest possible version of a Chartres labyrinth. And there are just two barriers possible for it. Otherwise it does not work. Also three are not possible, but with four barriers it works aganin.

How should one name this type now? I suggest Petit Chartres because it is a sort of a basic element of the Chartres labyrinth. Still other names are conceivable.

It am speaking here about the type and not about the style. The petals in the middle and the lunations around the perimeter belong to the style.

To create this labyrinth is possible in a variety of ways, not necessarily in the manner described. For more read also the related posts below.

There is even a copyrighted labyrinth of this kind: The Story Path©. Warren Lynn and John Ridder of Paxworks have developed it and call the style “3-circuit-triune”. I do not know how they have found it.

Related Posts

• The Classical 3 Circuit Labyrinth Type Knossos
• A 3-Circuit 3-Axle Labyrinth
• The Heart of the Chartres Labyrinth is the Classical Labyrinth

Further Link

• The Story Path©

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