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Welcome to the Labyrinth

The topic of this blog is the labyrinth. Under nearly all aspects, I would like to arouse your interest on the fascinating lines and the meaning of this old object. Being an old surveyor I put my focus on the geometrical shape.
A new post should be published about twice a month. Meanwhile I am accompanied by Andreas Frei as coauthor.

Contents

In a blog the single posts (articles) are disposed in reverse order: the latest posts first, the older ones following. The display of the content is thus different from a website where it is always permanent.

Anyone who is looking for something special about labyrinths or just wants to know what can be found on this blog, maybe would like to have an overview.

I can provide this now and offer it as an own page titled Contents.

The register with the table of Contents is on top of the blog under the header image next to About us.

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Most of the pictures and graphics were created by Andreas Frei and me (Erwin Reißmann), unless stated otherwise, and are provided under license CC BY-NC-SA 4.0.

The Labyrinths with 4 Real Double-barriers, 5 Arms and 5 Circuits

Up to now I have examined only sector labyrinths with four arms. The real double-barrier, however, originates from the five-arm labyrinth type Gossembrot 51r. As we know, this is not a sector labyrinth and it has 7 circuits. Now I want to find out, how many sector labyrinths with five arms and exclusively real double-barriers there are. Such labyrinths must have five cirucits. Therefore, we can use the same six sector patterns we have already used for the four-arm sector labyrinths.

In these four-arm labyrinths, only two sector patterns could be placed in every quadrant, i.e. sector patterns no. 3 and no. 8. Four sector patterns could only be placed in the quadrants next to the main axis (related posts 2). Now this is not different either in labyrinths with five arms. However, we then have to fill not only four quadrants, but the five sectors I til V with sector patterns. Sectors I and V lie next to the main axis. In the three sectors II, III, and IV between them, only sector patterns no. 3 or no. 8. can be placed. These can be arranged in only two different sequences, 3 8 3 or 8 3 8.

Figure 1 shows how the first sequence can be completed with patterns for sectors I and V. In sector I the two sector patterns no. 5 or no. 8 can be connected to the sequence 3 8 3. In sector V, sector patterns no. 7 or no. 8 can be attached.

Figure 1. Combinations with the Sequence 3 8 3 in the Sectors II – IV

In fig. 2 we can see, how the second sequence can be completed to a full five-arm labyrinth. Here, in sector I the sector patterns no. 3 or no. 4, in sector V the sector patterns no. 2 or no. 3 can be attached to the sequence 8 3 8 between them.

Figure 2. Combinations with the Sequence 8 3 8 in the Sectors II – IV


Figure 3 shows the four patterns and labyrinths that can be generated with the first sequence (3 8 3 from fig. 1).

Figure 3. Patterns and Labyrinths with the Sequence 3 8 3 in the Sectors II – IV


Figure 4 shows the four patterns and the corresponding labyrinths that can be generated with the second sequence (8 3 8 from fig. 2).

Figure 4. Patterns and Labyrinths with the Sequence 8 3 8 in the Sectors II – IV


Just as in the four-arm labyrinths, there exist also 8 different five-arm labyrinths made-up exclusively of real double-barriers. Even though they have only five circuits, they strongly remind us to the labyrinth type Gossembrot 51 r. We can also name them following the same rule as for the four-arm labyrinths (related posts 1). The name, thus, is composed of an uppercase letter followed by a number of marks. However, since these labyrinths have four side-arms, also four marks have to be attached to each uppercase letter. These marks indicate how the sectors are connected to each other. We have here exclusively real double-barriers with direct connections. Therefore, each name is composed of an uppercase letter followed by four horizontal marks.

Related Posts:

  1. Classifying the Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits
  2. The Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits

World Labyrinth Day 2020: Drawing a Labyrinth

Once again (for the 12th time) the Labyrinth Society invites us to celebrate World Labyrinth Day.
As every year, it is the first Saturday in May, this year May 2nd, 2020.

Flyer of the Labyrinth Society

Flyer of the Labyrinth Society

The global corona pandemic is also affecting this day. Larger group events are usually not possible.
That is why the day should be celebrated differently than usual. There are many ways to do this.

The Labyrinth Society offers a 24-hour online event entitled Walk around the World, which virtually connects people on the GoToMeeting platform across time zones.
This requires registration. More information is available on the Labyrinth Society website:


Lars Howlett offers to use a finger labyrinth virtually and online.

Design © Lars Howlett

Design © Lars Howlett

This will take place in a zoom meeting on May 2nd, 2020 from 12:45 to 14:00 PM Pacific Time (USA and Canada), at which you can register here:

Here is a converter for the different time zones:


My proposal: Drawing a labyrinth

There are also many options and methods for this. Some have already been featured in this blog.
How about a labyrinth on empty toilet paper rolls? After all the hamster purchases, should there be enough?
Ariadne’s thread, the path in the labyrinth, is drawn directly. So we do not need a pattern, as we do it usually to draw the classic labyrinth.

Toilet paper rolls labyrinths

Toilet paper rolls labyrinths

How to do this is explained here.


Or do we draw the (boundary) lines for a Wunderkreis? The path runs between the lines here. It is a walk-through labyrinth with a choice.

The lines for a Wunderkreis

The lines for a Wunderkreis

The details are explained below:


But we can also draw the path in the labyrinth, Ariadne’s  thread:

Ariadne's thread in a Wunderkreis

Ariadne’s thread in a Wunderkreis (Kaufbeuren)

We see the method here:


For many, however, it will also be possible, as usual, to walk a labyrinth.

No matter how, World Labyrinth Day can be celebrated.
The Labyrinth Society is again organizing a survey.

If you are looking for a labyrinth near you, maybe you will find one here:

Related Posts

The 3 Circuit Labyrinth Cube

I have already written about the labyrinth on Rubik’s Cube. I had presented a template for making a model. I’ve used that myself and made a labyrinth cube out of it. This is how it looks like: Ariadne’s  thread runs on the surface of the vsible miniature cubes. The 9 faces of the miniature cubes on “down” and the 7 on “up” were not used.

Ariadne's thread on Rubik's Cube

Ariadne’s thread on Rubik’s Cube

That gave me the idea of whether it would not be possible to include all the miniature cubes?
And I offer a solution for this:
The Rubik’s Cube has all in all 26 pieces. The core mechanism is fixed and holds the 6 center pieces with one coloured face. The 12 edge pieces show two coloured faces, and the 8 corner pieces have three coloured faces.
Ariadne’s thread should now run through all the little cubes. So 6 times one face is visible, 12 times two faces and 8 times three faces.
It looks like this on the template:

The unfolded cube

The unfolded cube

Ariadne’s thread is shown in color here. The labyrinth should be the type Knossos labyrinth with the path sequence 3-2-1-4. Green stands for 3, blue for 2, brown for 1. The center is white (4), and can only be reached if you pass through the invisible center (= core mechanism) from the “up” surface.
The beginning of Ariadne’s thread is the green edge cube with three coloured faces in the lower left “front”. Then you run up to the next edge cube with three coloured faces, you run around the “up” surface, following the “back” surface, then the “right” surface to the green/blue coloured face edge cube, which initiates the line of the blue cubes. The next change is in the “left” surface from blue to brown.

There is also a template that can be used to build a model. Here you can see, print or download the template as a PDF file.

I tried it myself and made a more or less successful cube.

The labyrinth cube

The labyrinth cube

In the large picture on the left you can see the “up” surface, the “left”, and the”front” surface from above. In the three smaller pictures the cube is each rotated by 90 degrees.

Here in a tilted representation:

The tilted cube

The tilted cube

In the large picture on the left, you can see the “front”, “left” and “down” surface from above. In the three smaller pictures the cube is rotated by 90 degrees again.

I cannot judge how difficult that would be to solve. Especially if you do not say beforehand what that should be and you should find the beginning and the end of the thread yourself.
Perhaps one could make the specification: connect the green cube with the white cube and go over all miniature cubes.

Related Post

Further Links

Classifying the Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits

There exist 64 patterns of labyrinths with 3 real or pseudo double-barriers, 4 arms, and 5 circuits (see: related posts 1, below). Up to now, only a few of them have been published in any form at all, and among those probably most were published in this blog. The earliest one, however, is a Roman mosaic labyrinth. Now I am tempted to attribute all examples that have been shown last year in this blog to one of the 64 patterns.

Until now, I have used the following names for the patterns:

A – H for the 8 labyrinths with exclusively real double-barriers
A’ – H’ for the 8 labyrinths with exclusively pseudo double-barriers

Therefore, so far, only 16 of the 64 patterns have been given a name. These names have evolved during the elaboration of the previous posts. In order to attribute a name to each of the 64 patterns a more systematic approach is needed. For this, the naming has to be refined. This can be done by drawing upon the tree-diagram (related posts 1). I have given the name D to the uppermost pattern with exlusively real double-barriers, and D’ to the lowermost that uses exclusively pseudo double-barriers. Now we need a differentiation that provides unique names to all eight patterns, i.e also to the six remaining patterns.

Fig. 1 shows (still with the example of the basic labyrinth D) the kind and the sequence of connections between the sectors. Direct connections (real double-barriers) are represented with a horizontal (–), indirect (pseudo double-barriers) with a vertical (|) mark. This order of the combinations is not arbitrary, but systematically ordered. The uppermost combination consists of only direct connections and is represented by the sequence – – –. In the second combination, the last direct connection is replaced by an indirect connection, resulting in the sequence – – |. The third combination replaces the central direct by an indirect connection and results in the combination – | –. The fourth combination uses indirect instead of direct connections in the central and last connection (– | |). And so forth.

Figure 1. Sequence of Connections

If we substitute „–„ with „0“, and „|“ with „1“ , we can see, that the order of the combinations simply corresponds with the binary numbers from 000 to 111. Namely, these are the first eight numbers from Zero to Seven written in the binary system.

Figure 2. Order of the Combinations

With this we can give a unique denomination to each of the 8 patterns that have been derived in fig. 1 (starting from labyrinth D). And not only that. This denomination also provides insight into how the sectors are connected one with another. In this new denomination, I name the first pattern D – – –. It’s previous name was D. The second pattern previously had no name yet and it’s new name is D – – | and so on til the seventh pattern, all without previous names. The lowermost, eighth pattern prevously was named D’ and it’s new name is D | | |. This systematics is independent of the basic labyrinth. We can apply it to all labyrinths A – H. By this, we can give an unique denomination to each of the 64 patterns made-up of a uppercase letter followed by three horizontal or vertical marks.

Now I will attribute some real examples.

Three labyrinth examples can be attributed to one of the patterns from tree diagram D. The earliest one is Roman mosaic labyrinth of the Avenches type (related posts 5). This has the lowermost pattern D | | |.

The second example was introduced by Erwin in his post from August.2019 (related posts 4) and has the uppermost pattern D – – –.

The third example is the one by Mark Wallinger 233/270 shown in the post from October 2019. This has the third pattern D – | – (related posts 2).

Figure 3. Labyrinths of Group D

The labyrinths from fig. 4 cannot be attributed to any variant of the basic labyrinth D. These are all labyrinths with only real double-barriers. That means, they are the basic labyrinths and have the uppermost patterns of other tree diagrams.

The first labyrinth was shown in Erwin’s post from August 2019 and has the pattern G – – – (related posts 4).

The second labyrinth from the post of Erwin from September 2019 has the pattern F – – – (related posts 3).

Third is the labyrinth 10/270 by Mark Wallinger from the same post, and this has the pattern A – – –.

Figure 4. Classification of Other Labyrinths with Real Double-Barriers

The new sector labyrinth in fig. 5 from Erwin’s post from October 2019 has the pattern G – | – (related posts 2). Thus, it is one of the 48 patterns with mixed real and pseudo double-barriers.

Figure 5. Labyrinth from Group G with Real and Pseudo Double Barriers

The labyrinth shown in fig. 6, however, cannot be attributed to any one of the 64 patterns, because it does not have double-barriers in all side-arms. This labyrinth originates also from the post of Erwin from September 2019 (related posts 3). This example shows very well, that it is not possible to generate double-barriers using sector patterns no 1 and no 6.

Figure 6. Not a Labyrinth with Only Double Barriers

However, this was not the intention either. Erwin just wanted to once make use of all eight sector patterns in a four arm sector labyrinth.

Related Posts:

  1. The Labyrinths with Real or Pseudo Double-barriers, 4 Arms, and 5 Circuits
  2. New 5 Circuit Labyrinths with Double Barriers
  3. A new Generation of Sector Labyrinths
  4. A new Type of Sector Labyrinth inspired by Gossembrot
  5. How to Draw a Man-in-the-Maze Labyrinth / 15

How to Make a Labyrinth Type Gossembrot (51 r, dual in Knidos Style)

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.

Related Posts

The Labyrinths with 3 Pseudo Double-barriers, 4 Arms, and 5 Circuits

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

Related Posts:

How to Calculate the Classical 7 Circuit Labyrinth

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.

Related Posts