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On my own behalf

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 he could find 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 above the header image next to About us.

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For a better view

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In the previous articles on this topic, I have already explained the method of the stamp-folding calculation brought into play by Tony Phillips.

Now it should go on here. Namely, it is possible to generate further variants of labyrinths by simply rotating the polygon used.

I take again the net with the polygon from the last post on this topic (part 2).

The net with the polygon

This diagram can be used to create four different labyrinths. Two directly (line 2 and 3), the other two by a simple calculation.

Other constellations can be gained by rotating the network 12 times by 30 degrees. Or in other words, it’s just like changing the clock for the summer or winter time.
Since only interesting labyrinths are of interest here, I omit all positions where the lines would point to the first and / or last circuit. So from the 12 you should not reach the 1 or the 11. Only the “times” are interesting, which point farther away, that is, run more sharply.
That would be in the above net the 1, 5 and 6. So I turn only to these times. In other words, I bring the 1, 5, and 6 into alignment with the 12. I turn the net by 30, 150, and 180 degrees. To rotate is the net with the polygon, the numbers stay in place.

Here’s the first turn:

rotated by 30 degrees

rotated by 30 degrees

I get four completely different path sequences than in the original above.

The second rotation:

rotated by 150 degrees

rotated by 150 degrees

I get four new variants again.

The last rotation:

rotated by 180 degrees

rotated by 180 degrees

Here I just get a different order of the sequences than in the original polygon. So there are no new variants, just another arrangement. This is because the rotation of 180 degrees corresponds to a symmetrical reflection.

It is not always possible to find new variants. With the help of this net I have generated a total of 12 different path sequences for 12 new labyrinths.

The path sequences can be directly converted into a labyrinth drawing.
Here only one (again in concentric style) is to be shown (the 2nd path sequence from the first polygon above):

A new 11 circuit labyrinth

A new 11 circuit labyrinth

Related Posts

Fine Tuning

A careful view of the Chartres type labyrinth in the Man-in-the-Maze style (see related posts 1, below) revealed that the design can still be improved. In addition to the not accessible middle, this figure has another 16 smaller segments that are not covered by the pathway. These are highlighted as crossed-through areas in fig. 1. Among others one requirement is that the pathway in a labyrinth should cover as much of the space of the figure as possible and should not let remain any „dead spots“. Such uncovered segments are therefore not liked to be seen in labyrinths.

Figure 1. Segments not Covered by the Pathway

These segments have evolved as a result of the construction of the seed pattern (see related posts 2). However, upon a closer view it turned out that they can be resolved. For this, as shown in fig. 2, the outer delimiting line has to be removed and the wall delimiting the pathway in between must be prolonged further to the inside.

Figure 2. Corrections to the Seed Pattern

The necessary corrections are shown for the segment on low left in orange color. Such corrections must be carried out for all 16 segments in a similar way.

The result can be seen in fig. 3. After these corrections, the labyrinth looks even somewhat more dynamic. Finally, only the one non accessible middle is left over. This is just the same as in the alternating one-arm labyrinths in the MiM-style.

Figure 3. The Final Labyrinth

However, now it cannot anymore be seen easily where the pathway traverses the side-arms. Therefore, the gain in dynamics and unity is associated with a loss in transparency.

Related Posts:

  1. How to Draw a Man-in-the-Maze Labyrinth / 12
  2. How to Draw a Man-in-the-Maze Labyrinth / 11

Again we deal with the simple, alternating, transit mazes, defined by the New York Professor of Mathematics Tony Phillips. In his calculations he ascertains a number of 1014 theoretically possible variants of interesting 11 circuit labyrinths (12-level mazes).

He also describes a simplified method for calculating these variants, which John E. Koehler developed in 1968 to solve a related problem of stamp-folding.

The following pictures should explain this method. To this I first use the already known path sequence for the 11 circuit labyrinth which can be generated from the basic pattern, namely: 5-2-3-4-1-6-11-8-9-10-7-12.
The path sequence must begin with an odd number and then the row must be composed of even and odd numbers alternately. The center is named with “12”, as it is the outside.

I draw a circle and divide it into 12 parts, as for a dial. Now I have to connect all points with lines, but same-colored lines must not cross.

5-2-3-4-1-6-11-8-9-10-7-12

5-2-3-4-1-6-11-8-9-10-7-12

I start with blue in 12 and go to 5, 2, 3, 4 (Fig. 1). Then from 4 to 1, thereby I change the colour (Fig. 2). I continue with 6, 11, 8, 9, 10 (Fig. 3). I again change the colour and complete the lines from 10 to  7 and 12 (Fig. 4).

But you can do it differently. For example, draw all the lines first in one color and then the intersecting ones in the other. Here again, the same-colored  lines should not cross each other. But more than once, as long as they are different (see 4 – 7).

The web

The web

But since we are looking for new labyrinths, we now go the opposite way: We draw a network of 12 lines, which connects all 12 points according to the above specifications and derive from this the path sequence.

Here is an example:

The web with the polygon

The web with the polygon

I write the first path sequence in line 2 (here in blue), starting at 12 and reading the lower digit, here 5. This is the beginning of the path. Then I follow the polygon until I land at 12 again and get: 5-2-3-4-1-6-11-10-9-8-7-12. That’s the original.
Now I go backwards and write the path sequence in line 3. So from 12 to 7, etc. That gives: 7-8-9-10-11-6-1-4-3-2-5-12. This is the complementary to the original.

I receive the lines 1 and 4 by arithmetic. I add the corresponding numbers of each row to “12”. In line 4, I get the dual to the original. In line 1, I get the complementary to the dual.

I verify this by comparing the numerical columns thus obtained with the others in “reverse”. This applies to the lines 1 and 4, as well as 2 and 3.
This is reminiscent of what has been described before when dealing with the dual and complementary labyrinths (see Related Posts below).

But there are alternatives. I turn the dial around, write the numbers for the 12 dots to the left, counterclockwise.
This is how it looks like:

The web with the two dials

The web with the two dials

The left side shows the dial as before. I start at 5, count to 12 and get the original. Then I start at 7 and count again to 12 and get the complementary to the original.
Now the right dial. I also start at 5 and count to 12 and so get the dual to the original. Then again from 7 to 12 and I get the complementary to the dual.

What should the blue written path sequences mean? They point out that the entry into the labyrinth can be placed on the same axis as the entry into the center. Here on circuit 5 and 7. Walter Pullen calls this that a labyrinth layout is mergeable. This allows you to construct a small recessed spot in the labyrinth, which some name the heart space. Especially in the concentric style, this can be implemented well.

From these two newly obtained path sequences, I now construct two new 11 circuit labyrinths in concentric style:

They have a different pattern of movement than the upt to now known labyrinths. In addition, we see 6 turning points for the circuits.

This is the dual to the previous labyrinth. Again, there is another “feeling”.

Who makes the beginning and builds such a labyrinth?

The other two paths sequences also result in new labyrinths, which I don’t show here. These belong to the remaining 1000 variants that are theoretically possible for 11 circuit labyrinths.

Related Posts

Finalizing the Labyrinth

Now we have all elements we need (see related posts 2, below) and are able to finalize the Chartres type labyrinth in the Man-in-the-Maze style. These elements are the auxiliary figure and the seed pattern (fig. 1).

Figure 1. Auxiliary Figure and Seed Pattern

 

The auxiliary figure has 90 spokes and 22 rings. In it’s interior there are two rings that are not used for the labyrinth. The reason for this is, that the distance between two spokes would be too narrow and no room would be left for the pathway (related posts 1).

In order to complete the labyrinth we proceed exactly as shown in the first post of this series (related posts 4). First the situation of the center has to be determined (fig. 2).

Figure 2. Situation of the Center

The center lies at the top end of the right half of the seed pattern of the main axis. This is exactly the same as in the one-arm labyrinths in the MiM-style (related posts 4). However, as here are arranged four seed patterns with two halves each on the central auxiliary circle, the center is shifted to the upper end of the eighth part at bottom right.

Figure 3. Step 1, Quadrant IV

In fig. 3 then we draw a wall delimiting the pathway around the center along the lines of the auxiliary figure.

From there we add one wall delimiting the pathway after each other and by this complete the IV th quadrant (fig. 4).

Figure 4. Completion of Quadrant IV

The walls delimiting the path of the IV th quadrant envelop the center and connect the right half of the seed pattern of the main axis with the left half of the seed pattern of the 3rd side-arm.

Next, the walls delimiting the pathway of the other quadrants have to be completed. Where do we have to begin with? This is shown in figure 5.

Figure 5. Step 1, Quadrants I – III

The places where the seed patterns of two different axes connect to each other each are on the innermost, 11th circuit. These places are already well known from the 9th post of this series (related posts 3). They are highlighted with dashed lines. On these spokes lie the inner walls delimiting the pieces of the pathway that have to be connected. So first, we prolong these walls delimiting the pathway until they reach the radius that is highlighted with a dashed circle. On this auxiliary circle lie the inner walls delimiting the 11th circuit.

Figure 6. Step 2, Quadrants I-III

These are then connected with chains of lines along the spokes and rings of the auxiliary figure (fig. 6). These lines represent the outer walls delimiting the path of the 11th circuit and the inner walls delimiting the path of the 10th circuit (fig. 6).

Figures 7 to 9 show, how the other quadrants are completed sequentially

Figure 7. Completion of Quadrant III

 

Figure 8. Completion of Quadrant II

 

Figure 9. Completion of Quadrant I

If we then remove the auxiliary figure, we can easily view the final result (fig. 10).

Figure 10. Labyrinth of the Chartres Type in the Man-in-the-Maze Style

Related Posts

  1. How to Draw a Man-in-the-Maze Labyrinth / 11
  2. How to Draw a Man-in-the-Maze Labyrinth / 10
  3. How to Draw a Man-in-the-Maze Labyrinth / 9
  4. How to Draw a Man-in-the-Maze Labyrinth

In my last posts I had shown the method of transforming the Medieval labyrinth by leaving out the barriers.

The first possibility to generate a labyrinth is of course the use of the seed pattern. Thus most of the Scandinavian Troy Towns with 7, 11, or 15 circuits were created.

Some years ago I wrote about the meander technique. Thereby many new, up to now unknown labyrinths have already originated.

Andreas still has demonstrated another possibility in his posts to the dual and complementary labyrinths. New versions of already known types therein can be generated by rotating and mirroring.

Now I want to use this technology to introduce some new variations.

I refer to simple, alternating transit mazes (labyrinths). Tony Phillips as a Mathematician uses this designation to explore the labyrinth. He also states the number of the theoretically possible variations of 11 circuit interesting labyrinths: 1014 examples.

The theoretically possible interesting variations of the 3 up to 7 circuit labyrinths once already appeared in this blog.

I construct the examples shown here in the concentric style. One can relatively simply effect this on the basis of the path sequence (= circuit sequence or level sequence). There is no pattern necessary.  The path sequence is also the distinguishing mark of the different variations.

I begin with the well known 11circuit classical labyrinth which can be generated from the seed pattern:

The 11 circuit labyrinth from the seed pattern

The 11 circuit labyrinth from the seed pattern

To create the dual version of it, I number the different circuits from the inside to the outside, then I walk from the inside to the outside and write down the number of the circuits in the order in which I walk one after the other. This is the new path sequence. The result is: 5-2-3-4-1-6-11-8-9-10-7- (12).
In this case it is identical to the original, so there no new labyrinth arises. Therefore, this labyrinth is self-dual. This in turn testifies to a special quality of this type.

Now I generate the complementary version. For that to happen I complement the single digits of the path sequence to the digit of the centre, here “12”.
5-2-3-4-1-6-11-8-9-10-7
7-10-9-8-11-6-1-4-3-2-5
If I add the single values of the row on top to the values of the row below, I will get “12” for every addition.

Or, I read the path sequence in reverse order. This amounts to the same new path sequence. But this is only possible with self-dual labyrinths.

I now draw a labyrinth to this path sequence 7-10-9-8-11-6-1-4-3-2-5-12.
Thus it looks:

The complementary 11 circuit labyrinth from the seed pattern

The complementary 11 circuit labyrinth from the seed pattern

This new labyrinth is hardly known up to now.


Now I take another labyrinth already shown in the blog which was generated with meander technique, however, a not self-dual one.

The original 11 circuit labyrinth from meander technique

The original 11 circuit labyrinth from meander technique

First, I determine the path sequence for the dual labyrinth by going inside out. And will get: 7-2-5-4-3-6-1-8-11-10-9- (12).

Then I construct the dual labyrinth after this path sequence.
This is how it looks like:

The dual 11 circuit labyrinth

The dual 11 circuit labyrinth

Now I can generate the complementary specimens for each of the two aforementioned labyrinths.

Upper row the original. Bottom row the complementary one.
3-2-1-4-11-6-9-8-7-10-5
9-10-11-8-1-6-3-4-5-2-7
The bottom row is created by adding the upper row to “12”.

The complementary labyrinth looks like this:

The complementary labyrinth of the original

The complementary labyrinth of the original

Now the path sequence of the dual in the upper row. The complementary in the lower one.
7-2-5-4-3-6-1-8-11-10-9
5-10-7-8-9-6-11-4-1-2-3
Again calculated by addition to “12”.

This looks thus:

The complementary labyrinth of the dual

The complementary labyrinth of the dual

I have gained three new labyrinths to the already known one. For a self-dual labyrinth I will only receive one new.

Now I can continue playing the game. For the newly created complementary labyrinths I could generate dual labyrinths by numbering from the inside to the outside.

The dual of the complementary to the original results in the complementary of the dual labyrinth. And the dual of the complementary to the dual one results in the complementary one of the original.

The path sequences written side by side makes it clear. In the upper row the original is on the left, the dual on the right.
In the row below are the complementary path sequences. On the left the complementary to the original. And on the right the  complementary to the dual one.

3-2-1-4-11-6-9-8-7-10-5  *  7-2-5-4-3-6-1-8-11-10-9
9-10-11-8-1-6-3-4-5-2-7  *  5-10-7-8-9-6-11-4-1-2-3

The upper and lower individual digits added together, gives “12”.

It can also be seen that the sequences of paths read crosswise are backwards to each other.

I can also use these properties if I want to create new labyrinths. By interpreting the path sequences of the original and the dual backwards, I create for the original the complementary of the dual, and for the dual the complementary of the original. And vice versa.

If I have a single path sequence, I can calculate the remaining three others purely mathematically.

Sounds confusing, it is too, because we are talking about labyrinths.

For a better understanding you should try it yourself or study carefully the post from Andreas on this topic (Sequences … see below).

Related Posts

Completion of the Seed Pattern

Two more steps are still needed in order to bring the Chartres-type labyrinth into the Man-in-the-Maze style. First, the seed pattern has to be completed.

We already have the seed pattern for the walls delimiting the pathway, but still without the pieces of the pathway that traverse the axes. These are still represented as pieces of the Ariadne’s Thread (fig. 1).

Figure 1. Seed Pattern and Pieces of Path Traversing the Axes

 

The labyrinth should be represented entirely by the walls delimiting the pathway. For this, the walls around the pieces of the path traversing the axes have to be completed (fig. 2).

Figure 2. Completion of the Walls Delimiting the Pathway – 1

We begin from the outside to the inside and first draw the walls around the outermost of these pieces of the pathway.

As a next step we add the walls delimiting the next inner pieces of the pathway (fig. 3).

Figure 3. Completion of the Walls Delimiting the Pathway – 2

As one can see, in each step, for each piece of the path, 2 or 4 for spokes have to be prolonged inwards, which are then connected with an arc of a circle.

And so we continue until all pieces of the path traversing the axes are enveloped by walls delimiting them (fig. 4).

Figure 4. The Final Seed Pattern for the Walls Delimiting the Pathway

This results in the complete seed pattern for the walls delimiting the pathway. In the center of the seed pattern and where the path traverses the axes there exist areas that are not accessible. This is quite analogue with the seed patterns in alternating labyrinths in the MiM-style, in which the center is not accessible either.

Related Posts:

Quite simply: By leaving out the barriers in the minor axes. I have already tried this with the Chartres labyrinth some years ago. And in the last both posts on this subject with the types Auxerre and Reims. You can read about that in the related posts below.

Today I repeat this for the Chartres labyrinth. Here the original in essential form, in a concentric style.

The Chartres labyrinth

The Chartres labyrinth

The original with all lines and the path in the labyrinth, Ariadne’s thread. The lunations and the six petals in the middle belong to the style Chartres and are left out here.

Now without the barriers in the minor axes.

The Chartres labyrinth without the barriers

The Chartres labyrinth without the barriers

All circuits can be included in the labyrinth originating now, differently from the types Auxerre and Reims. The path sequence is: 5-4-3-2-1-6-11-10-9-8-7-12. We have eight turning points with stacked circuits. It is self-dual. That means that the way out has the same rhythm as the way in.

But this 11 circuit labyrinth is quite different from the more known 11 circuit labyrinth, that can be generated from the enlarged seed  pattern.
Since this looks thus:

The 11 circuit labyrinth made from the seed pattern

The 11 circuit labyrinth made from the seed pattern

The path sequence here is: 5-2-3-4-1-6-11-8-9-10-7-12. We have got four turning points with embedded circuits. It is developed from quite another construction principle than the Chartres labyrinth. However, it is self-dual.


Now we turn to the complementary labyrinth.

The complementary labyrinth is generated by mirroring the original. Then thus it looks:

The complementary Chartres labyrinth

The complementary Chartres labyrinth

The entry into the labyrinth happens on the 7th circuit, the center is reached from the 5th circuit. The barriers are differently arranged in the right and left axes, the upper ones remain. It is self-dual.

Without the barriers it looks thus:

The complementary Chartres labyrinth without the barriers

The complementary Chartres labyrinth without the barriers

The transformation again works, as it does for the original. The path sequence is: 7-8-9-10-11-6-1-2-3-4-5-12. Also this labyrinth is self-dual.

We confront it with the complementary labyrinth, generated from the seed pattern.

The complementary 11 circuit labyrinth made from the seed pattern

The complementary 11 circuit labyrinth made from the seed pattern

The path sequence on this is: 7-10-9-8-11-6-1-4-3-2-5-12.
Contrarily to the original this type did not show up historically.

So we have created two completely new 11 circuit labyrinths from the Chartres labyrinth, which look different than the 11 circuit labyrinths that can be developed from the seed pattern.

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