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This type of labyrinth has already been described and appreciated in detail. Nevertheless, I would like to come back to this today.
I was particularly drawn to the five-pointed star (the pentagram) in the center. This appears in many national flags, so also in the European flag. That is why this type of labyrinth would be well suited for a “European labyrinth”. The Augsburg humanist Sigismund Gossembrot the Elder would also be a good “godfather” for such a labyrinth.

The Gossembrot labyrinth in European colors

The Gossembrot labyrinth in European colors

Here with boundary and path lines of the same width. That would be e.g. well suited as a template for a finger labyrinth:

The Gossembrot fingerlabyrinth in European colors

The Gossembrot fingerlabyrinth in European colors

It would be nice if this type of labyrinth were built as a walkable and public labyrinth.
To make this easier, I present a kind of prototype in the following drawing. The axis dimension is 1 m. This makes it very easy to convert to different sizes. Since the line axes are specified, different line and path widths can be implemented. The diameter of the center is four times the axis dimension, i.e. 4 m.
How this is done with a scaling factor has already been explained in various articles in this blog, most recently in the labyrinth calculator.

The design drawing

The design drawing

Here you can see, print or download the drawing as a PDF file

The rights of use are the same as for the labyrinth calculator.

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In my last post I have shown, that there exist 64 labyrinths with 3 real and / or pseudo double-barriers, 4 arms, and 5 circuits (see: related posts, below). But, how many of these do have exclusively pseudo double-barriers?

This question could actually be answered with the material from the last post. In order to show this, I once again make use of the tree diagram (fig. 1). This shows the combinations that can be obtained based on labyrinth D. There we can see, that the uppermost combination results in a pattern made-up exclusively of real double-barriers (that is, labyrinth D). This is the only one of the eight patterns using only real double-barriers. Similarly, the lowermost combination results in the only pattern made-up exclusively of pseudo double-barriers. I will term this D’. The six combinations in between all result in patterns with mixed combinations of real and pseudo double-barriers.

Figure 1. Combinations with Real, Pseudo, and Mixed Double-barriers

Now, if we proceed the same way as in fig. 1 for all labyrinths A – H, we always will obtain a lowermost combination made-up exclusively of pseudo double-barriers. These patterns and the corresponding labyrinths are shown in fig. 2. I have termed them A’ – H’. Labyrinths with the same uppercase letter belong to the same tree diagram.

Figure 2. The 8 Labyrinths with 3 Pseudo Double-barriers, 4 Arms, and 5 Circuits

Thus, we can conclude, that among the 64 labyrinths there are

  • 8 labyrinths with only real double-barriers
  • 8 labyrinths with only pseudo double-barriers
  • 48 labyrinths with real and pseudo double-barriers

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The question of a formula or table for calculating the construction elements in the labyrinth has arisen several times. For me, I solved the problem by designing and constructing the various labyrinths using a drawing program (AutoCAD). This creates drawings that contain all elements geometrically and mathematically exactly.
However, I do not print these on a specific scale, but adjust the size of the drawing so that it always fits on a sheet in A4 format.
Only the dimensions are decisive for the implementation of the labyrinth in the location. If possible, I also try not to use “crooked” measurements, but simple units, usually the meter.
The dimensions are therefore suitable to be scaled and so labyrinths can be constructed in different sizes.
The drawings thus represent a kind of prototype. Since the axes of the lines are always given, the widths of the boundary lines and the path can be varied.

The construction elements of the lines in the labyrinth mostly consist of arcs and straight lines. In the program used (AutoCAD), these individual elements can be combined into so-called polylines and their total length is then calculated.

The length specifications for the boundary lines and the path (the Ariadne thread) in the drawing are thus created. The boundary lines consist of 2 straight lines and 22 arcs (24 elements in total). The Ariadne thread consists of 1 straight line and 25 arcs (26 elements in total). The entire labyrinth consists of 50 individual elements.

It would be possible to calculate all of this in a table with the corresponding formulas, but it would be more cumbersome and extensive.

The scaling factor makes it easier to calculate variants in different sizes. The labyrinth calculator is something like a summary and general instructions for use. Here especially for the well-known Classical 7 circuit labyrinth.
However, this method has also been described for other types of labyrinths in this blog.

The Labyrinth Calculator

The Labyrinth Calculator

Here you can view, print or download the drawing as a PDF file

Here are some comments on copyright:
All drawings and photos in this blog are either mine or Andreas Frei, unless otherwise stated, and are subject to license CC BY-NC-SA 4.0
This means: You may use or change the drawings and photos without having to ask us if you name our names as authors, if you do not use the drawings and photos for commercial purposes and if you publish or distribute them under the same license. A link to this blog would be nice and we would be happy, but it is not a requirement.

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As is known, there exist 8 labyrinths with 3 double-barriers, 4 arms and 5 circuits (see: related posts 1, below). In fig. 1, I show the patterns and labyrinths again and give them subsequent names from „A“ to „H“.

Figure 1. The 8 Labyrinths with real Double-barriers

The most important restriction when deriving these labyrinths was, that the double-barrier must look the same as in Gossembrot’s labyrinth. Erwin has commented and wants also to include courses of the pathway, in which the path may change from the outermost to the innermost circuit or vice versa. I don’t consider these as double-barriers. Furthermore, this principle for the design has already occurred in earlier historical labyrinths. I will now refer to the new double-barrier, consistently used by Gossembrot as „real double-barrier“ and term the older course of the pathway „pseudo double-barrier“. Undoubtedly, the pseudo double-barrier is also an interesting element for the design. In it’s pure form it has been realized in the labyrinth type Avenches. Similarly, the real double-barrier in it’s pure form occurs in the 8 labyrinths shown in fig. 1. It is also possible to mix both principles of design, as has been done by Erwin and Mark Wallinger (see related posts 2).

Here I am interested in the question, how many labyrinths there are if real and / or pseudo double-barriers are used. The most important fundamental concepts for this have been elaborated previously (see: related posts 1). However, what changes is, that now not only options a) or b), but also options c) or d) are allowed for the connection of the sectors (fig. 2).

Figure 2. Admissible Connections

The other constraints, however, remain still valid. Sector patterns no. 1 and no. 6 cannot be used at all. The four one-sided sector patterns no. 2, no. 4, no. 5, and no. 7, can still be placed only in quadrants I and IV. This, because we want to preserve both halves of a double-barrier in all side-arms even if the path changes the circuit when traversing a side-arm. Therefore again, also only patterns no. 3 and no. 8 can be placed in every quadrant.

From this it follows, that we can start with the eight already known labyrinths with real double-barriers. In order to illustrate the following considerations, I pick out the labyrinth D. This has the sequence of patterns 8 3 8 3.

If we now also allow for pseudo double-barriers, this results in not only one but two possibilities at each side arm for the connection of one sector with the next: one on the same circuit, that I will term „direct“ connection, and another one with a change from one extreme circuit to the other, that I will term „indirect“ connection. Since at each side-arm both options are at the disposal, this leads to a much greater number of possible combinations.

Figure 3 shows the possible combinations if we start with labyrinth D and allow also indirect connections. In the first quadrant stands pattern no. 8. At the first side-arm there are two possibilities for a connection between quadrant I and quadrant II. Pattern no. 8 from quadrant I can be directly connected with no. 3 or also indirectly connected with no. 8 in quadrant II. Pattern no. 8 is the complementary of no. 3. An indirect connection requires the pattern complementary to the one for a direct connection. This is a general rule. At the 2nd side-arm for each of the patterns from quadrant II, there are again two possibilities to connect them with quadrant III, and similarly, the same applies to the 3rd side-arm. Whereas previously there were only 1*1*1 = 1 combinations for a direct connection, there are now together 2*2*2 = 8 possible combinations for a direct and / or indirect connection of all quadrants.

Figure 3. Possible Combinations with Direct or Indirect Connections Based on Labyrinth D

Each combination results in a new 4-arm sector labyrinth. I will illustrate this in fig. 4 with the first combination. This results in the already known labyrinth D with exclusively real double-barriers and the sequence of patterns 8 3 8 3.

Figure 4. The First Combination: Exclusively Direct Connections with Real Double-barriers – Labyrinth D

As a second example I show in fig. 5 a pattern that is formed by a combination of real and pseudo double-barriers. Namely, this has pseudo double-barriers with indirect connections at the first and third side-arms, whereas there is a real double-barrier with a direct connection at the second side-arm. This combination results in a sequence of patterns of 8 8 3 3.

Figure 5. The Sixth Combination: Mix of Real and Pseudo Double-barriers

Finally, fig. 6 presents all eight patterns that can be obtained starting with labyrinth D by combining real and pseudo double-barriers.

Figure 8. All Eight Combinations Based on Labyrinth D

The same approach as for labyrinth D is also feasible for the seven other labyrinths, i.e. labyrinths A, B, C, E, F, G, and H. In sum, this then results in 8 * 8 = 64 different types of labyrinths with three real or pseudo double-barriers, four arms and five circuits.

Related Posts:

  1. The Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits
  2. New 5 Circuit Labyrinths with Double Barriers

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In one of his articles, Andreas introduced the relatives of the labyrinth type Gossembrot 51 r.

The labyrinth type Gossembrot 51 r centered and in concentric shape

The labyrinth type Gossembrot 51 r centered and in concentric shape

Today I want to take a closer look at the dual version.
I particularly like this. This is because the outermost circuit (1) lead around all sectors. In the original, the innermost (7) completely circles the center.
And because it can be centered. The entrance takes place on the 3rd ring and the center is reached from the 5th ring. This also creates a small empty space.

I also like the rhythm, it is reminiscent of that of the Chartres labyrinth. It goes quickly to the middle and then follows the long walking in all sectors. Pretty much at the end you come back very close to the entrance, circle the entire labyrinth and then quickly get to the center.

An inverted pentagram is drawn in the middle in Gossembrot’s handwriting. I tried to fully integrate that. The top of the star points down. It is oriented in the direction of the 5 axes. Gossembrot designed the labyrinth around 1480. So this arrangement has no satanic character at all. These ideas only appeared in the middle of the 19th century.

I expressly recommend the excellent article on Wikipedia (see link below) about the geometry and meaning of the pentagram.

The dual and centered labyrinth type Gossembrot 51 r in Knidos style

The dual and centered labyrinth type Gossembrot 51 r in Knidos style

The five axes also tempt you to choose a pentagonal layout. The same widths for the lines (in black) and the paths (in white) and the angular edges also give it a spatial effect.

The dual labyrinth type Gossembrot 51 r centered in pentagonal shape

The dual labyrinth type Gossembrot 51 r centered in pentagonal shape

Here is Ariadne’s thread in pentagonal shape:

Ariadne's thread in the dual labyrinth type Gossembrot 51 r centered in pentagonal shape

Ariadne’s thread in the dual labyrinth type Gossembrot 51 r centered in pentagonal shape

The effect is spatial, but it seems to go deeper.

Who dares to build such a labyrinth as a walkable labyrinth?
As far as I know there is not a single copy worldwide. It looks new and modern, but its origins go back over 500 years.
The star in the middle, on the other hand, could be left out or only slightly indicated. It is not an element in the type, but is part of the style, similar to the six “petals” in the Chartres labyrinth.
An empty center is always inviting and open to a wide variety of applications.

Further Link

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We do wish all our readers a happy and successful new year!

 

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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