Sigmund Gossembrot / 6

The Complex Labyrinth

On folio 54 r, finally, is depicted the complex labyrinth shown in figure 1 (see also: related posts, below). In it’s center is written: „laborintus melior inter priores aquia magis errabunda inducens et educens“ – this labyrinth is better than the previous ones, as it is more misleading, leading in and out. This labyrinth has 12 circuits and its’ turns of the pathway are arranged in a confusing order. The number of arms cannot be easily counted.

Figure 1. The Complex Labyrinth on Folio 54 r

The pathway enters the labyrinth from below on the first (outermost) circuit (fig. 2). There it first bifurcates, and one can follow it in both rotational directions (clockwise or anticlockwise). On top of this circuit deviates another piece of the pathway. This then leads further into the labyrinth. Thus, the outermost circuit is designed not unicursally but multicursally as a maze.

Figure 2. The Outermost Circuit

The outermost circuit can be removed (fig. 3). This brings us to an autonomous core-labyrinth with 11 circuits. Additional circuits, however, cannot be simply removed without destroying the core-labyrinth. The core-labyrinth has clearly recognizable a main axis that is oriented to the top and it rotates clockwise.

Figure 3. Core Labyrinth

For a further investigation (in fig. 4) we now rotate the labyrinth, such that the main axis points to the bottom. By this, the labyrinth presents itself in the form we always use as a baseline. The main axis (in a blue frame) has exactly the same shape as the one of the Chartres type labyrinth. The other turns of the pathway are arbitrarily distributed over about the upper 2/3 of the area.

Figure 4. Main Axis

However, in view of the shape of the main axis the idea suggests itself, that also the remaining turns of the pathway could have something to do with the Chartres type. Indeed, three areas can be easily identified (fig. 5). The turns of the pathway inside these trapezoidal areas (red) can be aligned axially.

Figure 5. Side Arms

For this purpose, they need to be shifted along their circuits. Two turns of the path (the innermost of the 1st and 2nd side-arm are almost already in their right place. This is shown in fig. 6. The other ones need to be shifted further. This is illustrated with the red circles and arrows. In their new alignment they indeed result in a Chartres type labyrinth.

Figure 6. Type Chartres

Considered the other way round, we can state, that Gossembrot has derived a multicursal maze from the Chartres type labyrinth. For this, he has dispersed the regular order by shifting the turns of the pathway away from the side-arms and arbitrarily distributing them over the area of the labyrinth. Then he has attached a further circuit at the outside and on this circuit has introduced a multicursal course of the pathway.

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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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