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GF IW 7A7U7S7E1

Seven times Seven. Labyrinth with 7 Arms, 7 Circuits, and Double-barriers in all Side-arms. No Sectorlabyrinth – Rather the Opposite. Self-dual. My own design.

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We wish all visitors of this blog a Merry Christmas and a Happy New Year!

Christmastree labyrinth

Christmastree labyrinth

The basis for this labyrinth is the labyrinth shown often in this blog, originally rejected by Sigmund Gossembrot on folio 53 v.
Andreas Frei discovered and described the pattern for this new type in his blog posts.
I have taken up this structure and shown it in different ways of representation (= style).
Finally, in this year’s Christmas tree labyrinth in a triangular and centered shape.

One could say in a somewhat sophisticated way: Andreas is the master of the type and I the master of the style.

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Sigmund Gossembrot has used the double-barrier as a new element for the design of labyrinths. His five-arm labyrinth on fol. 51 r (see: related posts 5) and the four-arm labyrinth hidden in the design on fol. 53 v (related posts 4) are made up exclusively of double-barriers in all side-arms. They have 7 circuits and are no sector labyrinths.

In a series of posts, Erwin has introduced new sector labyrinths with four arms, five circuits and double-barriers (related posts 1, 2, 3). He has based these designs on the 8 possible courses the pathway can take in a one-arm labyrinth with five circuits. Sector labyrinths can be obtained by stringing together such courses of the pathway. Theoretically there exist 4096 variations for an arbitrary stringing together of four out of 8 courses of the pathway. Erwin has shown some of them. However, not all did consistently employ the principle of the double-barriers.

Here I will address the question how many sector labyrinths there are with four arms, five circuits and consistently employing double-barriers. I also start with the 8 possible courses of the pathway. These are based on Arnol’ds’ meanders in fig. 1 (related posts 6).

Figure 1. The Meanders by Arnol’d

In fig. 2 I show the patterns that correspond with the meanders. The patterns bear the same numbers as the meanders they were derived from. The left side of the figure shows the patterns of all alternating one-arm labyrinths with five circuits. Each of hese patterns also contains a connection from the outside into the labyrinth (from top left) and a connection to the center (to bottom right). These connections are represented in grey. In order to be used as segments (sectors) in sector labyrinths, these patterns at first have to be considered without the grey connection lines. The question here is the course of the path inside the sector. In a sector labyrinth, multiple such patterns are stringed together. Only the first pattern contains a connection to the outside and only the last a connection to the center. The patterns representing the 8 possible courses in a sector are displayed in the box on the right side.

Figure 2. The Corresponding Patterns – Left Half: Patterns of the One-arm Labyrinths; Right Half: Patterns of the Sectors

Next we intend to string together such patterns of sectors and to generate four-arm labyrinths using exclusively double-barriers. Let us first have a look at such a double-barrier in the labyrinth type Gossembrot 51 r. Figure 3 shows the labyrinth with the Ariadne’s Thread drawn in (red). Except for the one-arm labyrinths, an axis always lies between two segments, is formed by two different segments. Let us choose the double-barrier at the third side-arm. This connects segments III and IV and lies on the outermost four circuits. In the scaled-up section the seed pattern for the walls delimiting the path is drawn-in in blue color. It can be seen, that two nested turns of the Ariadne’s Thread are mirrored symmetrically against the central piece of the wall delimiting the pathway. Four circuits are needed for the double-barrier. In a labyrinth with five circuits, only one circuit remains free for the passage from one sector to the next. From this it becomes clear, that labyrinths with five circuits and using exclusively double-barriers must be sector labyrinths. There is only one circuit for the path to traverse the arms. This implies that the pathway must have completed the previous sector entirely before changing to the next.

Figure 3. The Double-barrier as by Gossembrot

Figure 4 shows the admissible connections between the sectors. (Pro memoria: the lines represent the pattern, i.e. the Ariadne’s Thread in rectangular form). The double-barriers occupy four circuits side by side. So they can lie in two places on circuits 2 – 5 or on circuits 1 – 4. Only connections on the same circuits are allowed, that is, the two options on the outermost (a) or on the innermost (b) circuit. If we would consider to change to another circuit when connecting the segments, as shown in options c or d, this would result in the insertion of an additional axial piece of the path between the halves of the double-barrier and these halves would be shifted by one circuit one against the other. But this is not a double-barrier any more.

Figure 4. Admissible Connections between Sectors

This circumstance strongly limits the range of the possibilities for stringing together the patterns. Figure 5 shows, how the different patterns can be used. The red numbers, characters at the free ends of each pattern indicate with which patterns it can be connected there (number of pattern, E for entrance, Z for center). A four-arm labyrinth has four segments. These are therefore also referred to as „quadrants“.

Figure 5. Possibilities for the Use of the Patterns

  • Two patterns, no. 1 and no. 6 cannot be used at all. With these it is not possible to generate a double-barrier.
  • Four „one-sided“ patterns, that is no. 2, no. 4, no. 5, and no. 7 have only on one side a half of a double-barrier (circled in red). On this side they can be connected with other patterns to double-barriers. It is true that it is also still possible to connect pattern no. 2 with no 5, and pattern no. 4 with no. 7 (not indicated). However, such a connection results in a two-arm labyirinth with one double-barrier only. On the second side of these one-sided patterns the free end lies on the third circuit. There, no double-barrier can be generated. Therefore, on this side only a connection to the entrance or the center is possible. Thus, these one-sided patterns can only be placed next to the main axis. Pattern no. 2 and no. 7 can only be placed in quadrant IV, where they are connected with the center. Pattern no. 2. can further only be connected with no. 8, and pattern no. 7 can be connected with no. 3.
  • Only two patterns, no. 3 and no. 8 can be completed to double-barriers at both sides. And only these can be placed in quadrants II or III. Furthermore, they may also be placed in quadrants I or IV, and thus be connected with the entrance or with the center (not indicated). Patterns no. 3 and no. 8 can be stringed together alternately or they can be connected with other one-sided patterns (pattern no. 3 with no. 5 and no. 7; pattern no. 8 with no. 4 and no. 2).

This provides us with the basis for the generation of the patterns for the sector labyrinths with the double-barriers. We start with the patterns for the quadrants II and III. For these, there are only two arrangements. Pattern no. 8 can be attached to no. 3 (above) or pattern no. 3 can be attached to no. 8 (below). The upper combination can be supplemented with patterns no. 5 or no. 8 towards quadrant I, and with patterns no. 2 or no. 3 towards quadrant IV. The lower combination can be supplemented with patterns no. 3 or no. 4 towards quadrant I, and with patterns no. 7 or no. 8 towards quadrant IV.

With the upper combination of the patterns no. 3 and no. 8 in quadrants II and III, thus, four patterns of labyrinths with four arms, five circuits and a consistent use of double-barriers can be generated. These patterns are shown in fig. 6.

Figure 6. The Patterns with the Combination no. 3 in Quadrant II – no. 8 in Quadrant III

Also with the lower combination of the patterns no. 8 and no. 3 in quadrants II and III, four patterns of labyrinths with four arms, five circuits and a consistent use of double-barriers can be generated. These patterns are shown in fig. 7.

Figure 7. The Patterns with the Combination no. 8 in Quadrant II – no. 3 in Quadrant III

Figure 8 now shows the labyrinths corresponding to the patterns of fig. 6.

Figure 8. The Labyrinths Corresponding with the Patterns of Fig. 6

Finally, figure 9 shows the labyrinths corresponding to the patterns of fig. 7.

Figure 9. The Labyrinths Corresponding with the Patterns of Fig. 7

The question concerning the number of possible labyrinths can be clearly answered:

  • There exist 8 labyrinths with 3 double-barriers, 4 arms and 5 circuits.

Moreover, in addition to this question, we obtain the following findings:

  • Labyrinths with 5 circuits and fully employing double-barriers must be sector labyrinths.
  • Such labyrinths cannot have double-barriers at the main axis. Double-barriers are only possible at the side-arms.

Related Posts:

  1. New 5 Circuit Labyrinths with Double Barriers
  2. A new Generation of Sector Labyrinths
  3. A new type of Sector Labyrinth inspired by Gossembrot
  4. Sigmund Gossembrot / 3
  5. Sigmund Gossembrot / 2
  6. Considering Meanders and Labyrinths

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I have already seen many pictures of finger labyrinths on the internet. Mostly they are made of wood or ceramics. They show the way in the labyrinth, Ariadne’s thread.
I do not like many of them. Especially the last part of the way, the entrance to the center, is often not so satisfactory. This is not very clear in some finger labyrinths; the path often runs from the side rather than from the bottom and perpendicular to the center.

That’s why I would like to present some own ideas on that.
In the classical 7 circuit labyrinth, the center is usually only as wide as the path itself and therefore less accentuated. That’s why I prefer to choose a slightly larger center, as we have it in the Knidos style. But not four times the axle width, but only the double.

That’s how it looks:

Ariadne's thread in Knidos style

Ariadne’s thread in Knidos style

The turning points are slightly shifted, the center is slightly enlarged. As a result, the last piece of the path runs perpendicular to the center.


In addition, the classical 7 circuit labyrinth can be centered very well. Since the first and the last part of the path are on the 3rd and 5th ambulatory.

That’s how it looks:

Ariadne's thread centered for a finger labyrinth

Ariadne’s thread centered for a finger labyrinth

The four turning points are shifted a bit more.
The empty space in the interior also is a bit more distorted.


Below is a kind of preview drawing for a round labyrinth of 33 cm diameter with all construction elements.

However, the dimensions are very well scalable. That is, for a smaller labyrinth, I use a corresponding scaling factor, which multiplies all measures.

For example, if I want it to be half the size, I multiply all measurements by 0.5.

If I want to reach a certain size, I determine the required scaling factor by dividing this size by 33 cm.

For a desired diameter of 21 cm, e.g. I calculate the scaling factor as follows: 21 cm : 33 cm = 0.636. I then multiply all the measurements with 0.636.

To convert the cm to inches, I divide by 2.54: The diameter becomes 33 : 2.54 = 13 inches.

Note for my inch-using visitors: Simply replace the measure unit “cm” by “inch” in the drawing and then calculate the desired size of your labyrinth as described above.

Design drawing

Design drawing

Here you might see, print or download it as a PDF file

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The labyrinth on folio 51 r is Gossembrot’s most important one. It is the earliest preserved example of a five-arm labyrinth at all. It’s course of the pathway using double barriers in all side-arms is unprecedented (see: related posts 1, below). However, it is not self-dual. Therefore, it can be expected that there exist three relatives of it (related posts 4).

I term as relatives of an (original) labyrinth the dual, complementary, and dual-complementary labyrinths of it (related posts 2 and 3). In fig. 1 the patterns of the Gossembrot 51 r-type labyrinth (a, original), the dual (b), the complementary (c), and the dual-complementary (d) of it are presented.

Figure 1. Patterns of the Relatives of Type Gossembrot 51 r

Figure 2 shows the labyrinths corresponding to the patterns in their basic form with the walls delimiting the pathway on concentric layout and in clockwise rotation.

Figure 2. The Relatives of Type Gossembrot 51 r in the Basic Form

These four related labyrinths all look quite similar. To me it seems, the dual (b) and the complementary (c) look somewhat less balanced than the original (a) and the dual-complementary (d). Presently, I am not aware of any existing examples of a relative to the Gossembrot 51 r-type labyrinth.

Related Posts:

  1. Sigmund Gossembrot / 2
  2. The Relatives of the Wayland’s House Type Labyrinth
  3. The Relatives of the Ravenna Type Labyrinth
  4. The Complementary versus the Dual Labyrinth

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It seemed obvious to apply the principles for two-parted 5 circuit labyrinths in the previous posts also for 7 circuit labyrinths.

For this purpose, the double barrier is extended to 6 passages, thus creating a triple barrier.

This is what it looks like:

A two-parted 7 circuit concentric labyrinth (entrance 3rd track)

A two-parted 7 circuit concentric labyrinth (entrance 3rd track)

It is noteworthy that both the entry into the labyrinth as well as the entry into the center is possible in and from the same circuit, here from the third circuit.
The path sequence is: 3-6-5-4-7-2-1-2-7-4-5-6-3-8

But it is also possible to design that from the 5th circuit. This creates a new type again.
Here is the example:

A two-parted 7 circuit concentric labyrinth (entrance 5th track)

A two-parted 7 circuit concentric labyrinth (entrance 5th track)

The path sequence is then: 5-4-3-6-7-2-1-2-7-6-3-4-5-8


And here are the two variants in Knidos style:

A two-parted 7 circuit labyrinth in Knidos style (entrance 3rd track)

A two-parted 7 circuit labyrinth in Knidos style (entrance 3rd track)

 

And here from the 5th circuit:

A two-parted 7 circuit labyrinth in Knidos style (entrance 5th track)

A two-parted 7 circuit labyrinth in Knidos style (entrance 5th track)

Now you could move the barrier and its related elements upwards. Then it would not be the first circuit who is completely gone through, but the innermost, the 7th circuit.

So the whole structure (pattern), expressed in the path sequence, would be changed. That would also create a new type of labyrinth again.

Here in a simplified representation:

A two-parted concentric 7 circuit labyrinth (entrance 3rd track)

A two-parted concentric 7 circuit labyrinth (entrance 3rd track)

Here with entrance in the 5th circuit:

A two-parted concentric 7 circuit labyrinth (entrance 5th track)

A two-parted concentric 7 circuit labyrinth (entrance 5th track)

But someone else had this idea. On the Harmony Labyrinths website, Yvonne R. Jacobs has introduced hundreds of new labyrinth designs and copyrighted them.

She calls these types Luna V (Desert Moon Labyrinth) and Luna VI (Summer Moon Labyrinth). On her website you can look at the corresponding drawings and even order finger labyrinths (only in the USA).

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Further Link

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Summary

At the end I want to summarize some findings of the previous six posts on Gossembrot. In my opinion, two main aspects seem important.

New types of labyrinths

Gossembrot has created two labyrinths with unique courses of the pathway, and thus designed two new types of labyrinths. The five-arm labyrinth on fol. 51 r is an outstanding type of labyrinth. The one-arm labyrinth with nine circuits on fol. 53 r is one of the rarer non-alternating types of labyrinths. Furthermore, a third new type of a four-arm labyrinth is hidden in the drawing on fol. 53 v.

Gossembrot could also have been first in drawing the Schedel type labyrinth (fol. 51 v) or the scaled-up basic type (fol. 54 v). It is true, that the manuscript containing the Schedel type is dated somewhat earlier than the one by Gossembrot. However, the drawing in Schedel manuscript could also have been added later. The two earliest examples of the scaled-up basic type are dated from the 15 th century without further precision. Thus, they could also have been generated later than 1480. However, I think this is unlikely. Both examples (Hesselager and Sibbo) were desinged in the classical style – which is the style that best matches with the natural way of designing this type of labyrinth.

Approaches to mazes

Gossembrot was strongly involved with the difference between labyrinth and maze. This is well attested by the mazes he had derived from the labyrinths of the Schedel type (fol. 52 r and fol. 52 above) and, following an other approach, from the Chartres type (fol. 54 r). And also by the fact that Gossembrot took this complex labyrinth for his best labyrinth.

I think also that his rejected design on fol. 53 v is not a mistaken attempt to the five-arm labyrinth on fol. 51 r. But instead, it seems to me that this is a failed attempt to derive a maze from the five-arm labyrinth. This is particularly supported by the design of the main axis. This was amended in a similar way as the ones of the mazes (fol. 52 r and fol. 52 v above) Gossembrot had derived from the Schedel type labyrinth.

It was not until the 15 th century that the creation of mazes began. The first drawing of a maze by Giovanni Fontana dates from year 1420 (see literature below: p. 138, fig. 239). Gossembrot was one of the first to draw mazes. His mazes, however, are, even compared with some other ones by Fontana (literature, p. 238, fig. 240), still rudementary and are fully based on unicursal labyrinths.

Conclusion

Gossembrot undoubtedly has his great importance in the design of unicursal labyrinths. Even if he must have been very fascinated by the maze, such that he himself took a maze for his best labyrinth, his drawings still represent tentative approaches and attempts to mazes. Contrastingly, he has created awesome original designs with fundamental innovations in unicursal labyrinths.

Literature
Kern H. Through the Labyrinth – Designs and Meanings over 5,000 Years. Munich, London, New York: Prestel 2000.

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