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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.

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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).

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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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Quite simply: By leaving out the barriers in the minor axes. I have already tried this with the Chartres labyrinth (see related posts below). But is that also possible with any other Medieval labyrinth?

In part 1 I had made it for the type Auxerre. Now I take the type Reims which is also self-dual like Chartres and Auxerre. And again I take the complementary version. All examples are presented in the concentric style.

The Reims labyrinth

The Reims labyrinth

 

Here the original with all lines and the path in the labyrinth, Ariadne’s thread. The barriers in the upper minor axis are identical with those in the type Chartres, the barriers in the horizontal axis are different from Chartres, as well as the arrangement of the turning points in the main axis below the center.

The Reims labyrinth without the barriers

The Reims labyrinth without the barriers

The barriers are left out. When drawing the path I had to discover that four lanes cannot be included. These are the both outermost and the both innermost tracks (1, 2, 10, 11). Hence, I have anew numbered the circuits and there remain only 7 circuits instead of the original 11. However, this also means that by changing the Reims  Medieval labyrinth into a concentric Classical labyrinth through this method not an 11 circuit labyrinth is generated, but a 7 circuit.

The circular 7 circuit labyrinth

The circular 7 circuit labyrinth

This is an up to now hardly known and not so interesting labyrinth. Since one enters the labyrinth on the first circuit and arrives at the center from the last. The path sequence is very simple: 1-2-3-4-5-6-7-8, a simple serpentine pattern.


Now we turn to the complementary labyrinth:

The complementary Reims labyrinth

The complementary Reims labyrinth

The complementary labyrinth is generated by mirroring the original one. The upper barriers remain, right and left they run differently and in the main axis, the turning points shift. The entrance into the labyrinth changes to the middle (lane 9) and the entrance into the center is from further out (lane 3).

The complementary Reims labyrinth without the barriers

The complementary Reims labyrinth without the barriers

As with the original four lanes can not be inserted (1, 2, 10, 11). Hence, a 7 circuit labyrinth arises again. I have anew renumbered the lanes and have drawn the labyrinth anew.

Then thus it looks:

The circular 7 circuit labyrinth

The circular 7 circuit labyrinth

The labyrinth is entered on the 7th lane, the center is reached from the first lane. The path sequence is: 7-6-5-4-3-2-1-8. This labyrinth does not belong to the historically known labyrinths. However, it has already appeared in this blog (see related posts below).

The surprising fact is that again no 11 circuit Classical labyrinth could be generated through the transformation. Rather two 7 circuit labyrinths.

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Quite simply: By leaving off the barriers in the minor axes. I have already tried this with the Chartres labyrinth (see related posts below). But is that also possible with every other Medieval labyrinth?

As an example I have chosen the type Auxerre that Andreas showed here recently. This labyrinth is self dual as are Chartres and Reims, therefore of special quality. And they all have a complementary version.

The Auxerre labyrinth

The Auxerre labyrinth

Here the original with all the lines and the path in the labyrinth, Ariadne’s thread. The barriers in the minor axes are identical with those of the Chartres type. There is only another arrangement of the turning points (the lanes 4, 5, 7, 8) in the middle of the main axis.

The original Auxerre labyrinth without the barriers

The original Auxerre labyrinth without the barriers

The barriers are omitted. When drawing Ariadne’s thread, I found that four tracks could not be inserted. Hence, I have anew numbered the circuits and there remain now 7 circuits instead of the original 11. However, this also means that by changing this Medieval labyrinth into a concentric Classical labyrinth through this method no 11 circuit labyrinth is generated, but a 7 circuit.

The 7 circuit circular Cretan labyrinth

The 7 circuit circular Cretan labyrinth

If one looks more exactly at it, one recognises the well-known path sequence: 3-2-1-4-7-6-5-8. We got a Cretan labyrinth in concentric style.


Now we turn to the complementary labyrinth:

The complementary Auxerre labyrinth

The complementary Auxerre labyrinth

The complementary labyrinth is generated by mirroring the original one. The upper barriers remain, right and left they run differently and in the main axis, the turning points shift. The entrance into the labyrinth changes to the middle (lane 9) and the entrance into the center is from further out (lane 3).

The complementary Auxerre labyrinth without the barriers

The complementary Auxerre labyrinth without the barriers

As with the original, four lanes can not be inserted (4, 5, 7, 8). Therefore, the result is again a 7 circuit labyrinth. I renumbered the lanes and have redrawn the labyrinth.

This is how it now looks like:

The complementary 7 circuit circular Cretan labyrinth

The complementary 7 circuit circular Cretan labyrinth

The labyrinth is entered on the 5th lane, the center is reached from the 3rd lane. The path sequence is: 5-6-7-4-1-2-3-8. This labyrinth is not one of the historically known labyrinths. But it showed up in this blog several times (see related posts below). Because it belongs to the interesting labyrinths among the mathematically possible 7 circuit labyrinths.

The surprising fact is that no 11 circuit Classical labyrinth could be generated through the transformation. But for that  the 7 circuit Cretan labyrinth. Therefore we can say that the heart of the Medieval Auxerre labyrinth is the Cretan (Minoan) labyrinth as it is in the Chartres labyrinth.

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In addition to the universally known labyrinth of Chartres and the less popular labyrinth of Reims a third, much less known, very interesting (interesting and self-dual) medieval labyrinth with 4 arms and 11 circuits has been preserved. This is sourced from a manuscript that is stored in the municipal library of Auxerre. Therefore I have named it as Type Auxerre.

At the end of this series I want to show these three labyrinths and their complementaries.

In the three following figures I start with the original labyrinth (image on top left).

From this I obtain the pattern by unrolling the Ariadne’s Thread of it (image on top right).

Then I mirror the pattern vertically without interrupting the connections to the exterior and to the center. This results in the pattern of the complementary labyrinth (image on bottom right).

Then I curl in this pattern to obtain the complementary labyrinth (image on bottom left).

Fig. 1 shows this procedure with the example of the labyrinth of Auxerre. This labyrinth is not recorded in Kern [1]. The image of the original labyrinth was taken from Saward [2] who sourced it from Wright [3].

Figure 1. Labyrinth of Auxerre and Complementary

Fig. 2 shows the labyrinth of Reims and the complementary of it. The image of the original labyrinth was sourced from Kern [1].

Figure 2. Labyrinth of Reims and Complementary

Finally, the labyrinth of Chartres and it’s complementary are presented in fig. 3. The image of the original labyrinth was sourced from Kern [1].

Figure 3. Labyrinth of Chartres and Complementary

With these considerations I wanted to point out that three historical labyrinths exist with a similar degree of perfection as Chartres. Together with their complementaries we now have present six very interesting labyrinths with 4 arms, 11 circuits and a similar degree of perfection.

[1] Kern, H. Through the Labyrinth. Prestel, Munich 2000.
[2] Saward J. Labyrinths and Mazes. Gaia, London 2003.
[3] Wright C. The Maze and the Warrior. Harvard University Press, Cambridge (Massachusetts) 2001.

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Just like the labyrinth from Ravenna, the Wayland’s House labyrinth is also a historical type of labyrinth with 4 arms and 7 circuits. There exist even two different Wayland’s House labyrinths (figure 1).

Figure 1. The Two Wayland’s House Labyrinths

 

I have named them as Wayland’s House 1 and Wayland’s House 2. Wayland’s House 1 first appeared in a manuscript of the 14th century, Wayland’s House 2 in a manuscript of the 15th century, both from Iceland. This can be easily looked up in Kern. In the following I refer to Wayland’s House 1.

In this type of labyrinth, the pathway does not enter on the first circuit and does not reach the center from the last circuit either. Therefore it is an interesting labyrinth. And also the complementary of it is an interesting labyrinth. This, however, is not the most important reason for why I present this type of labyrinth and its’ relatives here. Whereas no existing examples of any relative of the Ravenna labyrinth are known, there exists a contemporary type of labyrinth for each, the dual, complementary and complementary-dual of the Wayland’s House labyrinth.

Figure 2 shows the patterns of the original Wayland’s House labyrinth (a), the dual (b), complementary (c) and complementary-dual (d) labyrinths.

Figure 2. The Relatives of the Wayland’s House Type – Patterns

The original (a) and dual (b) both are interesting labyrinths. The complementaries of them, (c) and (d), are likewise interesting labyrinths.

Figure 3 shows the labyrinths corresponding to the patterns in their basic form with the walls delimiting the pathway shown on concentric layout and in clockwise rotation.

Figure 3. The Relatives of the Wayland’s House Type – Basic Forms

The relatives of the Wayland’s House type (a) are three of the so-called neo-medieval labyirnth types (there are other neo-medieval types of labyrinths too). These relatives are: dual (b) = „Petit Chartres“, complementary (c) = „Santa Rosa“, and complementary-dual (d) = „World Peace“ labyrinth.

So these contemporary types of labyrinths can be easily generated simply by rotating or mirroring of the pattern of Wayland’s House. This having stated I do not mean to pretend, these types of labyrinths have intentionally or knowingly been derived in such a way from the Wayland’s House type. Rather, available information suggests that they were created in a naive way, i.e without their designers having known about this relationship with the Wayland’s House type labyrinth. Nevertheless, actually, these modern neo-medieval labyrinths are the relatives of Wayland’s House.

The Wayland’s House labyrinth at first glance has some similarities with the Chartres type labyrinth. However it is not self dual and its course of the pathway is guided by an other principle yet. And this applies to its relatives too. Therefore the choice of the name „Petit Chartres“ to me seems inconvenient. It seems like this name was chosen because this type of labyrinth originally was designed in the Chartres-style. So this type seems to have been named after its style.

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