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In dealing with the double-barrier technique in recent posts, I found this installation of Mark Wallinger’s Labyrinths on the London Underground:

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

The special feature of this is that two double barriers are located next to each other in the upper part of the central axis. In the routing chosen by him you move at the transition from the 2nd to the 3rd quadrant first away from the center.

I’ve changed that so much that you would “experience” a movement to the center in a walkable labyrinth.

This is what it looks like:

A new labyrinth in concentric style

A new labyrinth in concentric style

I have also moved the side double barriers and this makes the routing in all quadrants also different. So a new type of labyrinth is born.

Here in Knidos style:

A new centered sector labyrinth in Knidos style

A new centered sector labyrinth in Knidos style

Why not as a two-parted labyrinth?

A new two-parted 5 circuit labyrinth

A new two-parted 5 circuit labyrinth

The left part has the path sequence: 3-4-5-2-1 and the right part: 5-4-1-2-3, so there are two 5 circuit labyrinths in it.

And here again in Knidos style:

A new two-parted and centered 5 circuit labyrinth in Knidos style

A new two-parted and centered 5 circuit labyrinth in Knidos style

The remarkable thing about this type is that both the entry into the labyrinth in the 3rd lane takes place, as well as the entry into the center.

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There are eight possibilities for a one arm 5 circuit labyrinth (see Related Posts below).

The structure of the different labyrinths can be expressed through the path sequence. Here is a list:

  1.  3-2-1-4-5
  2.  5-4-1-2-3
  3.  5-2-3-4-1
  4.  1-4-3-2-5
  5.  3-4-5-2-1
  6.  1-2-5-4-3
  7.  1-2-3-4-5
  8.  5-4-3-2-1

The sector labyrinth presented in my last post (see Related Posts below) has a different path sequence in all 4 quadrants. In other words, there are 4 different labyrinths hidden in it. These were the path sequences in the 1st to the 4th line of the list above.


Today another 5 circuit sector labyrinth modeled with Gossembrot’s double barrier technique:

A new 5 circuit sector labyrinth in concentric style

A new 5 circuit sector labyrinth in concentric style

The path sequence in quadrant I is: 3-4-5-2-1, in quadrant IV: 1-2-5-4-3. These are the aforementioned courses at the 5th and 6th place. The two upper quadrants have: 1-4-3-2-5 and 5-2-3-4-1. These correspond to the upper pathways at the 4th and 3rd places. Not surprising, because the transition in these sector labyrinths takes place either on the 1st or the 5th course.

Here in a representation that we know from the Roman labyrinths:

The new sector labyrinth in square shape

The new sector labyrinth in square shape

Or here in Knidos style:

The new sector labyrinth in Knidos style

The new sector labyrinth in Knidos style

On Wikimedia Commons I found this picture of Mark Wallinger’s unique Labyrinth installation at Northwood Hills station, installed as part of a network-wide art project marking 150 years of the London Underground. It is part of the emboss family (one of the 11 labyrinth design families).

Mark Wallinger Labyrinth 10/270, Photo: credit © Jack Gordon

Mark Wallinger Labyrinth 10/270, Photo: credit © Jack Gordon

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.


Now only two path sequences are missing, then we would have the eight complete.
There is also a new sector labyrinth for this:

Another new sector labyrinth in concentric style

Another new sector labyrinth in concentric style

In the two lower quadrants we have the courses 1-2-3-4-5 and 5-4-3-2-1. These are the above mentioned pathway sequences at the the 7th and 8th places. The upper two sequences (5-2-3-4-1 and 1-4-3-2-5) are again identical to the aforementioned two labyrinths and the one in the previous post.

The quadratic representation shows that it is actually a mixture of serpentine type and meander type (see Related Posts below).

The new sector labyrinth in Roman Style

The new sector labyrinth in Roman Style

Here in Knidos style:

The new sector labyrinth in Knidos style

The new sector labyrinth in Knidos style

Simply put, in only three sector labyrinths can all theoretically possible eight 5 circuit labyrinths be proved.


But it is also possible to move the “upper” pathways down, so that again arise new display options.
Then you can swap the right and left “lower” quadrants.
Or mirror everything and create right-handed labyrinths.

Here are two examples:

Even one more new sector labyrinth in round shape

Even one more new sector labyrinth in round shape

Another new sector labyrinth in Knidos style

Another new sector labyrinth in Knidos style

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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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Further links

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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 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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Find our Typology Confirmed

In chapter 3 of his book, Herman Wind (see below: Literature 1) aims at introducing a new categorization of labyrinths. For this purpose he has used images of labyrinths primarily from Kern (Literature 2) and also from some other sources. Wind has abstracted the sequences of circuits from the ground plans of the individual labyrinths. In the labyrinth library, table 3.2.1 A-F on pages 73-78 of his book, entries of 235 labyrinths can be found. Each line represents one labyrinth with a reference to figure, location, date when recorded and sequence of circuits. Labyrinths with the same sequences of circuits were arranged subsequently. By this, Wind has attributed similar labyrinths to the same groups, divergent labyrinths to different groups and thus created a typology. However, he does not term his groups „types“ but „families“ instead. These families have not been given different names and are also not always clearly distinguished one from another. Therefore in the labyrinth library, the reader himself must draw parentheses around the lines with the same sequences of numbers in order to identify the families.

In the book, five examples of the use of the labyrinth library are presented. Let us have a look at the first example (p. 81). This shows examples of labyrinths that were attributed to the same family as the labyrinth of Ravenna.

Figure 1. Labyrinths Attributed to the Same Family as Ravenna

Examples A „Filarete“, C „Ravenna“, and F l(eft) „Watts 7 circuits“ all have the same sequence of circuits. Example B „Hill“ was equally attributed to this family, even though it is completely different. It can be seen at first sight, that this labyrinth does not belong to this family. This is a faulty drawing of a labyrinth of the Saffron Walden type. It seems, there has been some mistake in the attribution of the labyrinth in the labyrinth library. Interestingly, neither the author nor the editor have noticed this. Although they have noticed the difference in the much more resembling example F r(ight) „Watts 11 circuits“, but only stated a certain similarity with the family of Ravenna. This is just what can be seen in a direct comparison of both images F l and F r.

The way Wind uses the sequence of circuits causes two problems:

First: This sequence of circuits is unique only in alternating one-arm labyrinths. If we consider also non-alternating labyrinths, examples with different courses of the pathway may have the same sequence of circuits (fig. 2).

Figure 2. Labyrinths with the Sequence of Circuits 7 4 5 6 1 2 3 0

So, Wind attributes the two non-alternating labyrinths (a) St. Gallen and (b) Syrian Grammar to the same family. This is correct. Should he find an alternating labyrinth of the shape (c), however, he would have to attribute this to the same family, although it has a clearly different course of the pathway. This because it’s sequence of circuits is 7 4 5 6 1 2 3 0, just the same as in examples (a) and (b). (For other examples with ambiguous sequences of circuits see related posts 1, 2).

Second: Wind’s sequences of circuits for the labyrinths with multiple arms are incomplete. They only indicate which circuits are covered at all but provide no information on how long the respective pieces of the pathway are. Such sequences of circuits are not even unique in alternating labyrinths. As Jacques Hébert explains, the sequence of circuits in labyrinths with multiple arms must take into account the division into segments and the resulting variation in length of path segments (Literature 3). This can be done in different ways.

Figure 3. Sequences of Circuits of the Wayland’s House Labyrinth

Figure 3 shows one of the possibilities using a pure sequence of numbers with the example of the Wayland’s House 1 labyrinth. The sequence of circuits of this labyrinth according to Wind (lower row W:) has 21 numbers. If we consider also the length of the path segments following Hébert (upper row H:) the sequence has 30 numbers. From Wind’s sequence of circuits the labyrinth cannot be restored without an image of it or only after multiple attempts. From Hébert’s sequence of circuits it can be restored without difficulty.

That there may exist alternating labyrinths with different courses of the pathway for the same incomplete sequence of circuits is shown in fig. 4.

Figure 4. Labyrinths with Different Courses of the Path and the Same Incomplete Sequence of Circuits

The two labyrinths shown have different courses of the pathway. This is represented in the complete sequence of circuits (upper lines). In the incomplete sequence of circuits (lower lines), however, the difference has disappeared. It is the same for both labyrinths.

Conclusion

The categorization by Wind is not new. We have done this already (Literature 4). We have used about the same material, have attributed similar labyrinths to the same groups and divergent labyrinths to different groups and refer to this as a typology (related posts 3, 4, 5). We also obtain more or less the same results (further links). Thus, the categorization by Wind confirms our typology to a great extent. As the criterion for similar or divergent, we use the course of the pathway. However, we don’t describe this with the sequence of circuits but with the pattern. This allows us a unique and complete representation of the course of the pathway and an unambigous attribution of the labyrinth examples to types of labyrinths.

Literature

  1. Listening to the Labyrinths, by Herman G. Wind, editor Jeff Saward. F&N Eigen Beheer, Castricum, Netherlands, 2017.
  2. Kern H. Through the Labyrinth: Designs and Meanings over 5000 years. London: Prestel 2000.
  3. Hébert J. A Mathematical Notation for Medieval Labyrinths. Caerdroia 34 (2004), p. 37-43.
  4. Frei A. A Catalogue of Historical Labyrinth Patterns. Caerdroia 39 (2009), P. 37-47.

Related Posts

  1. Circuits and Segments
  2. The Level Sequence in One-arm Labyrinths
  3. Type or Style / 6
  4. Type or Style / 5
  5. Type or Style / 1

Further Links

Katalog der Muster historischer Labyrinthe

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