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## How to Draw a Man-in-the-Maze Labyrinth / 11

### 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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## 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.

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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.

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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.

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## Sequence of Circuits – Conclusion

The notation with the coordinates is consistent, understandable and works well in one- and multiple-arm, alternating and non-alternating labyrinths. However, for a labyrinth with three circuits, at least 6 segments are needed (in one- and two-arm labyrinths: number of circuits times two, in all other labyrinths: number of segments times number of arms).

Correspondingly, the sequences of segments rapidly increase in their length with the size of the labyrinth. The Chartres type labyrinth e.g. has 44 segments, as have all other types of labyrinths with 4 arms and 11 circuits.

Here I present the sequence of segments of the Chartres type labyrinth for illustration. This is:

Nevertheless this sequence of segments is a well understandable instruction of how to draw the labyrinth. It reads about like this: Go first to the fifth circuit, walk along the first segment (5.1), then proceed to the 6. circuit and stay in the first segment (6.1). Next, go to the 11th circuit in the first segment (11.1) continue on the same circuit to the 2nd segment (11.2), skip then to the 10th circuit in the 2nd segment (10.2) asf. This also implies that from each coordinate subsequent to the previous it becomes clear, whether the path makes a turn (as from coordinate 5.1 to 6.1) or if it traverses the arm (such as from 11.1 to 11.2). However it is a long and complex series of numbers.

Now there are also various other possibilities to write notations for multiple-arm labyrinths that may have less digits. In any case, the labyrinths first have to be notionally partitiond into segments. However in some notations it is possible to combine multiple segments in one term. I will illustrate this here with the example of a notation for the Chartres labyrinth by Hébert°.

This is a notation comparable with the one presented in the post „Circuits and Segments“, where the segments had been numbered by circuits. In this case, if the pathway passes through multiple segments on the same circuit, the number of the circuit was repeated accoridingly. This, for the labyrinth of Chartres would result in 44 numbers. In the notation by Hébert the length of the sequence reduces to 31 numbers. However, each number must now be written with a prefix. For instance, „-“ indicates, that the following number is written only once, as the path traverses only one segment. A prefix „+“, on the other hand, indicates that the following number would have to be written twice as the path passes two subsequent segments. Thus, different prefixes have to be taken into account. And two prefixes will not be sufficient. Additional prefixes will be required to capture the pathway passing through three, four or more subsequent segments, or to indicate that the arm is traversed whilst the path skips onto another circuit. So while this notation is shorter it is also more difficult to apply. Furthermore it is subject to the weakness already discussed earlier, that, althoug it indicates the circuit, it does not indicate the segment actually covered by the pathway.

Other notations exist as well. I do not address this further here. It should have become clear that the sequences of segments in multiple-arm labyrinths rapidly increase in length and complexity. In most types of such labyrinths the sequence of segments is therefore not suited for giving a name. Just try to imagine to name the labyrinth I had shown in January with its sequence of segments. This labyrinth has 12 arms and 23 circuits and thus 276 segments.

I abstain here from writing down the sequence of segments of this labyrinth. It would fill some 14 – 15 lines.

Conclusion

To conclude, I want to come back to the original question whether the sequence of circuits can be used for giving names to the different types of labyrinths. I had two concerns about this:

• First, in one-arm labyrinths this sequence was not unique. However this problem could be easily solved by adding a prefix „-“ only to those numbers of circuits where the pathway traverses the axis. Therefore in not too large types of one-arm labyrinths the sequence of circuits can be used for naming.
• Second, in multiple-arm labyrinths the sequence will rapidly increase in length. It turned out that in these labyrinths the sequence of segments has to be considered and that this usually becomes either be too long or too complex or both. Therefore I consider it not suited for giving name in multiple-arm labyrinths.

° Hébert J. A Mathematical Notation for Medieval Labyrinths. Caerdroia 2004; 34: 37-43.

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## Type or Style / 12

### Type in Style

For the typology of labyrinths I exclusively use one criterion: the course of the pathway. This becoms best apparent in the pattern. Labyrinths with the same pattern are thus of the same type. From this I distinguish the style. Style can be described as a trailblazing form of the graphical design.

Type and style complement each other. In many labyrinth examples it is possible to indicate the type and the style. However, it is not possible to indicate a style  in every example. At the end I will attribute the labyrinth examples I used in this series to types and, if possible, indicate also the styles of them. A list of the types used is given at the end of this post.

From post Type or Style / 1

Cretan type in the Chartres style

Chartres type in the classical style

From post Type or Style / 3

Cretan type in the classical style

Cretan type in the concentric style

Chalice: There exist historical labyrinths with the same pattern. I therefore name this type Abingdon (not shown in post but mentioned)

Trinity: type of it’s own (type Trinity) in the Chartres style

St. Anthony: type of it’s own (type St. Anthony)

Circle of Peace: type of it’s own (type Circle of Peace)

Santa Rosa: type of it’s own (type Santa Rosa; not shown in post, but mentioned)

From post Type or Style / 4

Chartres 8 circuits: type of it’s own  (type Regensburg; Cretan type with one additional trivial circuit at the inside)

Chartres 8 circuits: type of it’s own (type Charneu in the Chartres style).

Grey’s Court: type Grey’s Court

Ravenna 5 circuits: There exist historical labyrinths with the same pattern. I therefore name this type Compiègne

Chartres, 5 circuits: type of it’s own (type Emendingen)

From post Type or Style / 5

Type Reims

Type Chartres in the Reims style

In order not to overload this post I interrupt here and will present the other types in my next post.

The types:

## Type or Style / 10

### Types and Examples

In my last post I have attributed some labyrinth examples to various styles. Here I will now attribute labyrinth examples to types of labyrinths. I have already described what is a type of labyrinth in post / 6 of this series (see related posts). I refer to single individuals of labyrinths as examples, irrespective of whether they appear as images, drawings, plans, laid-out built labyrinths etc. I will use the three types already described in my previous post and attribute a selected number of examples to each.

Examples of the Cretan Type

There exist a vast number of labyrinth examples of the Cretan Type. This is the type with the most examples. Great differences can be seen with respect to style or layout of the various examples. But all of them are alternating one-arm labyrinths with seven circuits and a level sequence of  3, 2, 1, 4, 7, 6, 5.

Examples of the Reims Type

There are only a very few examples of the Reims type. I know only two historical examples. Therefore in order to make one line complete, I have added a drawing of my own.

Exemples of the Chartres Type

This is completely different for the Chartres type. This is the second most frequent type of labyrinth.

This type is also particularly suited to highlight the difference between type and style. As is known,  there exist a Chartres type as well as a Chartres style. So we just have to compare the labyrinths attributed to the Chartres type here with the examples from my last post (Type or Style /9) that were attributed to the Chartres style.

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