Feeds:
Posts
Comments

Archive for the ‘Typology’ Category

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?

Related Posts

Advertisements

Read Full Post »

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.

Related Posts

Further links

Read Full Post »

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.

Related Posts

Read Full Post »

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

Read Full Post »

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

Read Full Post »

Type and Style

In my post Type or Style / 4 from August 2015, I have discussed the typology of the website Begehbare Labyrinthe (related posts 4). In the meantime, this typology has been completely revised (additional links). The new typology adopts our principles relating to type (related posts 3) and style (related posts 2). Furthermore it combines type and style. All walkable labyrinths were now attributed to types according to their course of the pathway. Labyrinths with the same course of the pathway (pattern, sequence of circuit) are of the same type. These types were then further divided into groups according to the style. Also the naming of the types has been reworked.

This whole thing can be well explained using the basic type. „Basic type“ is the new name of the type that formerly or elsewhere has been termed „classical“ or „Cretan“ type of labyrinth. This type has one axis, seven circuits and the pattern shown in fig. 1.

Figure 1. Pattern Basic Type

Figure 1 shows the pattern in pure form on left, and on right with an aid how to read it. It is read from top left to bottom right (related posts 5). To this corresponds the course of the path in it’s sequence of circuits 3 2 1 4 7 6 5 (related posts 1). By this, the type is accurately described. It is the most frequent type of labyrinth worldwide. And also in the typology of Begehbare Labyrinthe it is by far the most frequent type. Means that of the currently included 305 walkable labyrinths, 133 are of the basic type. These are designed in various styles:

  • „triangle“ (1 example)
  • „rectangle“ (1 example)
  • „classic“ (97 examples)
  • „Knidos“ (15 examples)
  • „concentric“ (15 examples)
  • „Man-in-the-Maze“ (1 example)
  • „other“ (3 examples).

Each type of a labyrinth in each of its styles is depicted with a figure of one corresponding labyrinth example. Figure 2 shows as an example the section representing the basic type in the classical style with its 97 examples.

Figure 2. Section Basic Type in Classical Style

If you move the cursor over the image of the labyrinth, the pattern is overlaid. At the side of the image all attributed walkable labyrinths are listed. Moving the cursor over a name makes an image of the corresponding labyrinth fade in. A click into a link brings you to the page with the full entry of the corresponding labyrinth. This often includes a comprehensive image of the whole labyrinth, and an extensive description of it including type, style, number of circuits, number of axes, size and measurements, materials and other information.

At present, the typology includes about 60 different types and some 10 styles. However, not every type is represented in each style. Despite this, the typology at the moment contains 92 groups composed of types and styles, what is more than the 60 pure types, that are based exclusively on the course of the pathway. These groups cover all walkable labyrinths listed in the website. However, from time to time, new labyrinth examples are added and therefore also the number of types and styles may increase further.

The full list of the types of labyrinths is ordered in increasing order first by the number of axes, then by the number of circuits. So, first all one-arm types of labyrinths are listed, and these in ascending order by the number of circuits from the smallest with 3 circuits to the largest with 11 circuits. Next follow the types with 2, 3, 4, 5, 6 und 8 arms, each again in ascending order by the number of circuits.

This new typology is now systematic, consistent, clearly reproducable, and completely covers the listed walkable labrinths. Furthermore it can be easily extended if labyrinth examples in new types or styles are added to the list.

Related Posts:

  1. The Level Sequence in One-arm Labyrinths
  2. Type or Style / 7
  3. Type or Style / 6
  4. Type or Style / 4
  5. How to Read the Pattern

Additional Links:

  1. Typology Begehbare Labyrinthe

Read Full Post »

Wishing all visitors of this Blog a Merry Christmas and a Happy New Year!

An 11 circuit Christmas tree labyrinth

An 11 circuit Christmas tree labyrinth

Related Posts

Read Full Post »

Older Posts »

%d bloggers like this: