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The Two One-arm Labyrinths

Among the nine drawings by Gossembrot are also two one-arm labyrinths (see related posts, below).

The labyrinth on fol. 53 r has 9 circuits (fig. 1). In the center is written: inducens et educens, leading in and leading out. The design of the axis with it’s rhombus shape is eye-catching.
This almost looks a bit like an anticipation of the Knidos style… Furthermore, this is a non-alternating labyrinth. The pathway traverses the axis when changing from the 6th to the 9th circuit. I have highlighted this position in the labyrinth with two dashed red lines. To these correspond the dashed lines in the pattern. This pattern appears for the first time in the labyrinth by Gossembrot. Therefore it is a type of it’s own. I refer to it as type Gossembrot 53 r.

Figure 1. The Labyrinth on Folio 53 r

The labyrinth on fol. 54 v has 11 circuits and is designed in the concentric style (fig. 2). This type of labyrinth is also referred to as the scaled-up basic type or scaled-up classical / Cretan type of labyrinth. This, because the seed pattern in the classical style consists of a central cross with two nested angles and a coaxial bullet point between each two arms of the cross. The seed pattern of the basic type is made-up of a central cross with one angle and bullet point between each two arms of the cross.

Figure 2. The Labyrinth on Folio 54 v

There exist several historical examples of this type of labyrinth. The two earliest examples (fig. 3) are frescos in the church of Hesselager, Fünen, Denmark and in the church of Sibbo, Finnland (see literature, below).

Figure 3. Earliest Historical Examples (15 th Century)

Both were dated from the 15 th century without any further precision. Also, Gossembrot 54 v dates from the 15 th century (1480). Therefore, based on the dating, it is not possible to certainly identify the earliest preserved example of this type of labyrinth. So it is even conceivable, that the drawing by Gossembrot is earliest and thus Gossembrot was also the originator of this type of labyrinth.

Literature
Kern H. Through the Labyrinth – Designs and Meanings over 5000 Years. München, London, New York: Prestel 2000. P. 280, fig. 593; p. 281, fig. 601.

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I was particularly fascinated by the technique of double barriers in Gossembrot’s 7 circuit labyrinths presented in recent posts. This makes possible completely new types of labyrinths. He probably did not “invent” the double barriers, but he was the first to consistently and systematically use them.

How does this technique affect 5 circuit labyrinths?
I tried that and came across a whole new kind of sector labyrinths.
As you know, one sector after another is traversed in these before the center is reached.

The historical Roman labyrinths are divided into three different variants: the meander type, the spiral type and the serpentine type (see the Related Posts below).
The entry into the labyrinth is usually up to the innermost lane. And in all four sectors the structures are the same.
The change to the next sector either always takes place outside or even once inside (or alternately).

Now the new type:

The new sector labyrinth in concentric style

The new sector labyrinth in concentric style

What is so special about that?
Already the entrance: It takes place on the 3rd lane. This does not occur in any historical sector labyrinth. And the entrance into the center is also from the 3rd lane.

Then the structure expressed by the path sequence is different in each quadrant.

Quadrant I:   3-2-1-4-5
Quadrant II:  5-2-3-4-1
Quadrant III: 1-4-3-2-5
Quadrant IV: 5-4-1-2-3

The transitions to the next sector are always alternately.

Nevertheless, the new labyrinth is very balanced and mirror-symmetrical.

Here in a square shape:

The new sector labyrinth in square shape

The new sector labyrinth in square shape

This makes it easier to compare with the previously known Roman labyrinths (see below), which are mostly square.

The difference to these becomes clear especially in the presentation as a diagram. Because this shows the inner structure, the pattern.

The diagram for the new sector labyrinth

The diagram for the new sector labyrinth

Very nice to see are the nested meanders.

But even in Knidos style, this type is doing well:

The new sector labyrinth in Knidos style

The new sector labyrinth in Knidos style

How should one call this type? And who builds one as a walkable labyrinth?

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The Four Labyrinths with 4 Arms und 8 Circuits

Four drawings by Gossembrot show labyrinths with 4 arms and 8 circuits. Among these, two each are on a circular and rectangular layout. Figure 1 shows these four figures compared. Figures a (circular) and c (rectangular) have the same course of the pathway (=). This is also true in figures b (circular) and d (rectangular). The two circular figures (a, b) as well as the two rectangular (c, d) have different courses of the path (≠).

Figure 1. The Four Designs Compared

All four figures bear inscriptions in their centers.

Figure a (fol. 51 v): „Laborintus inducens et educens“ – labyrinth leading in and leading out


Figure b (fol. 52 r): „Laborintus tamen educens nunquam intus perveniens fines“ – labyrinth leading out but nowhere arriving at the center

Figure c (fol. 52 v below): „Ibi introis et exis“ – here you enter and exit.

Figure d (fol. 52 v above): „Der Irrgang clausus est et numquam introibis“ the maze is closed and nowhere you enter.

From this we can see, that Gossembrot was engaged with the difference between labyrinth and maze. Figure 2 shows, using the lower, rectangular images, that the design of the side-arms in all four images is the same (areas within blue frames). The figures on the right side only differ with respect to the design of the main axis from those on the left side (areas within red frames). This becomes also clear from the patterns shown at the bottom of fig. 2. The left figures are labyrinths, the right figures are a special form of a simple maze. The pathway enters on the 6th circuit and there it branches. One branch continues to the first side-arm. There it turns to the 7th circuit, makes a full circuit and thereby traverses the main axis. It again turns at the first side-arm, leads back through the outer circuits 6 – 1 and arrives back in the other branch of the bifurcation. The innermost 8th circuit is completely isolated from the rest of the course of the pathway. It begins in a dead-end, does one round and ends in the center.

Figure 2. Labyrinth and Maze

So it seems, Gossembrot had derived a maze from the labyrinth. As a matter of fact, there exists a second historical labyrinth with the same pattern. This is sourced in a autograph (1456/63) of the Nuremberg physician and humanist Hartmann Schedel (see literature, below). The labyrinth drawn freehand was affixed to one of the last blank pages of the autograph. This autograph is accessible online in the same digital library as the manuscript by Gossembrot (further links, below). The original drawing of the labyrinth is oriented with the entrance to the left side. In fig. 3, for a better comparability, I have rotated it with the entrance to the bottom.

Figure 3. Type Schedel

Based on the earlier date (1456/63) of the publication by Schedel, I have named this type of labyrinth with „type Schedel“. Gossembrot was friends with Hermann Schedel, the uncle of Hartmann. The manuscript by Gossembrot dates from 1480. Having stated this, it has also to be considered that the labyrinth drawing of the Schedel autograph was affixed. Therefore it could also have been added later. Thus, it is well concievable that the drawings by Gossembrot were earlier and thus Gossembrot could have been the originator of this type of labyrinth.

Literature

  • Kern H. Through the Labyrinth: Designs and Meanings over 5000 years. London: Prestel 2000, p. 126, fig. 216.

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My co-author Andreas Frei reported in his last article about the labyrinth drawing rejected by Sigmund Gossembrot on folio 53 v. And thereby made the amazing discovery that in it principles of design have been applied to which so far not one known historical labyrinth was developed.
Not for the sector labyrinths of the Roman labyrinths or the various Medieval ones. Even among the contemporary labyrinths (for example, the London Underground’s 266 new types by Mark Wallinger), this new type does not show up.

However, the labyrinth derived by Andreas Frei has some extraordinary features that I would like to describe here in more detail.
First of all see a representation of the new type in concentric style:

The 7 circuit labyrinth of folio 53 v in concentric style

The 7 circuit labyrinth of folio 53 v in concentric style

Contained is the classic 7 circuit labyrinth, as it can be developed from the basic pattern. In the upper area and in the two side parts 3 barriers are inserted, which run over 4 courses and again create 6 new turning points. These barriers are arranged very evenly, they form an isosceles cross. This significantly changes the layout.

The entrance to the labyrinth is on lane 3, then in the 1st quadrant on the lower left side you immediately go to the lanes 6, 5, 4 and 7. Thereby the center is completely encircled (in all 4 quadrants).
In the 4th quadrant on the bottom right, you go back over the lanes 6, 3, 2 through the remaining quadrants to the 1st quadrant.
From here, you go around the whole labyrinth, in the 4th quadrant, you quickly reach the center via the lanes 4 and 5.
Twice the entrance is touched very closely: at the transition from lane 2 to 1 in the 1st quadrant and at the transition from lane 1 to 4 in the 4th quadrant.

Fascinating are also the two whole “orbits” in lanes 7 and 1. The two semicircles in lane 2 are remarkable too. Lanes 3, 4 and 5 are only circled in quarter circles.

All this results in a unique rhythm in the route, which appears very dynamic and yet balanced.

Of course, this is hard to understand on screen or in the drawing alone. Therefore, it would be very desirable to be able to walk such a labyrinth in real life.

So far there is no such labyrinth. Who makes the beginning?

The centered labyrinth of folio 53 v

The centered labyrinth of folio 53 v

This type can also be centered very well. This means that the input axis and the entrance axis can be centrally placed on a common central axis. This results in a small open area, which is also referred to as the heart space.

Also in Knidos style, this type can be implemented nicely. This makes it even more compact. However, the input axis is slightly shifted to the left, as it is also the case in the original.
Here the way, Ariadne’s thread has the same width everywhere.

The labyrinth of folio 53 v in Knidos style

The labyrinth of folio 53 v in Knidos style

And here, as a suggestion to build such a labyrinth, the design drawing for a prototype with 1 m axle jumps. The smallest radius is 0.5 m, the next one is 1 m larger.
With a total of 11 centers, the different sectors with different radii can be constructed.

The design drawing

The design drawing

The total diameter is depending on the width of the path at about 18 m, the path length would be 225 m.

As the axes of the path are dimensioned, Ariadne’s thread is constructed.
All dimensions are scalable. This means that the labyrinth easily can be enlarged or reduced.

And here you may download or print the drawing as a PDF file.

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The Labyrinth on Folio 53 v

Originally I had intended to show the design on folio 53 v already in my previous post (see related posts, below). It can be seen as a mistaken attempt to the labyrinth on fol. 51 r. But then I took a closer look at it. And the result has prompted me to dedicate a separate post to this design. Fig. 1 shows the design on fol. 53 v.

Figure 1. Labyrinth on folio 53 v

The design on fol. 53 v was rejected, crossed out and overwritten with text. It is clearly recognizable a five-arm labyrinth with 7 circuits. Also the design of the side-arms is very similar as in the labyrinth on fol. 51 r.
As the labyrinth on fol. 51 r, also this labyrinth rotates anti-clockwise. In fig. 2 I have mirrored it, inscribed the Ariadne’s Thread and in parallel presented the pattern. The Ariadne’s Thread traverses the lines of the labyrinth in two places. These are marked with blue circles. I have assumed that these were still provisional auxiliary lines that would have been removed if the final version of this labyrinth had been completed.

Figure 2. Ariadne’s Thread and Pattern

The result is surprising. Segment 4 is not filled out by the pathway. The path on the innermost and the two outer circuits passes this segment and marks only the left side of the third and the right side of the fourth side-arm. In addition the main axis includes one superfluous axial piece of the path. The pathway leads into the center, and a second piece of the path in the center of the main axis leads from the center into a dead-end.

This design can be easily corrected such that there results a four-arm unicursal labyrinth as shown in fig. 3.

Figure 3. Corrections

In order to achieve this, each of the two pairs of walls delimiting the pathway drawn in blue must be shifted against another until they come to lie one above the other. This results in the extinction of the fourth segment and of the central piece of the pathway with the dead-end on the main axis.

Figure 4 shows the new pattern and the four-arm labyrinth derived from it.

Figure 4. The Labyrinth Hidden in the Draft on Fol. 53 v

So, in the rejected five-arm design, a four-arm labyrinth is contained or hidden. The course of the pathway of this follows about the same principle as in the labyrinth on fol. 51 r. I am not aware of any existing labyrinth of this type.

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In the context of the theme Labyrinth and Flower of Life, the similarity to a cube has been mentioned more often. The hexagonal shape of the labyrinth was just too reminiscent of a cube. And that got me looking for the labyrinth on the cube.

I have a magic cube and as a small brain training I solve it once a day. This is now memorized and routinely.

In Further Link below you can find out what a magic cube is.

First, I tried to put Ariadne’s thread on the small squares. This is relatively easy.

For better representation, the 6 sides of a cube are “flattened”:

The layout

The layout

You can draw in there Ariadne’s thread for a 3 circuit labyrinth type Knossos. Generally known, this has the path sequence: 3-2-1-4.
The beginning is on the frontside below at left. Then we go to the third line, to the second and the first line and finally to the center in 4 up in the middle square.

Ariadne's thread

Ariadne’s thread

And here in an isometric view:

Three views

Three views

I hope you can imagine that on the drawings?
We see the lines on 5 sides of the cube, the bottom remains empty. The middle is slightly larger, but we do not touch all the small squares.

Ariadne’s thread for the template with slightly thicker lines:

Ariadne's thread

Ariadne’s thread

If you want, you can download, print or copy the template as a PDF file.

Such a cube would certainly be quite easy to solve as a magic cube. Especially if you have a template of it in mind.

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The Labyrinth on Folio 51 r

In the previous post I have presented the nine labyrinth designs by Gossembrot and gave references to the sources (see below: related posts 1). The first labyrinth on folio 51 r undoubtedly is the most important of all. It is the earliest preserved example of a five-arm labyrinth at all. Furthermore, it’s course of the pathway is unprecedented and deviates from every previous type of labyrinth. Here I will show the course of the pathway and it’s special features stage by stage. For this, I use the Ariadne’s Thread inscribed into the labyrinth and in parallel the pattern. This is the same approach I had applied with the labyrinth by Al Qazvini (related posts 2). As a baseline I always use a labyrinth with the entrance on bottom and in clockwise rotational direction. Gossembrot labyrinth fol. 51 r, however, rotates anti-clockwise. Therfore, in figure 1, I first mirror the labyrinth horizontally.

Figure 1. Labyrinth on Folio 51 r (left), horizontally mirrored (right)

The image on left shows the original labyrinth of fol. 51 r, the right image shows the same labyrinth mirrored. Mirroring does not affect the course of the pathway with the exception of the pathway traversing in the opposite direction.

Fig. 2 shows the first stage of the course when it enters the labyrinth. This is nothing special. The path fills the space left over by the pattern and continues to the innermost circuit as directly as possible.

Figure 2. Way into the Labyrinth

This circuit is then traversed in a forward direction through all five segments, as can be seen in fig. 3. This is also nothing special either.

Figure 3. Forward Direction on the 7th Circuit Through all Segments

The special characteristic of the course of the path starts after it has turned at the end of the fifth segment. Then it proceeds to a movement in backward direction, following a line that alternates between forming a curve wrapping and being wrapped and also marking the axes. This process continues to the first side-arm (fig 4).

Figure 4. Backward Direction Onset of Special Course

At this point the former course is interrupted. Again the path marks the axis (first side-arm), but then continues as a meander through segment 2, as shown in fig. 5.

Figure 5. Backward Direction, Interruption, Insertion of Meander

From there the original course is resumed. Still in a backward direction, the pathway fills the rest of segment 2 and segment 1 and finally turns from the 2nd to the 1st circuit (fig. 6).

Figure 6. Backward Direction, Resumption of Special Course

From here now it continues again in forward direction and takes it’s course through all segments until it reaches the opposite side of the main axis. In passing, it fills the inner space it had left over on its course in backward direction in segments 3 and 4 (fig. 7).

Figure 7. Forward Direction Through all Segments

From there it reaches the center after having filled the space left over in segment 5 (fig. 8).

Figure 8. Completion, Reaching the Center

This course of the pathway, like in some sector labyrinths, results in symmetric pairs of nested turns of the pathway at each side-arm. Unlike in sector labyrinths, however, the pathway does not complete one sector after another, but traverses through all sectors in each direction. First in forward direction on the innermost circuit, then in backward direction modulating through circuits 6 to 2, and finally again in forward direction on circuits 1, 4, and 5.

Related Posts:

  1. Sigmund Gossembrot / 1
  2. The Labyrinth by Al Qazvini

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