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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

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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 the last post I have introduced the eleven-circuit Cakra Vyuh Labyrinth. Even though the seed pattern has a central cross and also can be easily drawn freehand, it is not a labyrinth in the Classical style. In fig. 1 I show the seed pattern in different variants.

CaVy_SP_var

Figure 1. Variants of the Seed Pattern

Image a shows the original seed pattern, image b the seed pattern in the Classical style, image c in the Concentric style, and image d in the Man-in-the-Maze style.

This figure clearly shows that the original seed pattern deviates from the Classical style. It is true that this seed pattern has a central cross as for instance the Cretan labyrinth also. However in the Cakra Vyuh seed pattern, from this cross further junctions branch off.

This is different in the Classical style. The Classical style consists of verticals, horizontals, ankles and dots. For this, no central cross is required. This page illustrates well, what I mean (left figure of each pair). If a seed pattern includes ankles these lie between the cross arms and do not branch off from them.

The four images in fig. 1 in part look quite different one from each other. So how do I come to the assertion that they are four variants of the same seed pattern? Let us remember that these figures show seed patterns for the walls delimiting the pathway. Now let us inscribe the seed patterns for the Ariadne’s Thread into these figures (fig. 2).

CaVy_SPab

Figure 2. With the Seed Pattern for the Ariadne’s Thread Inscribed

At first glance this looks even more complex. However, if we focus on the red figures, we will soon see what they have in common.

CaVy_SPa

Figure 3. Seed Pattern for the Ariadne’s Thread

The seed pattern represents a section of the entire labyrinth. More exactly, it is the section along the axis of the labyrinth. The turning points of the pathway align to the axis. This can be better seen on the seed pattern for the Ariadne’s Thread compared with the seed pattern for the walls delimiting the pathway.

In all four seed patterns, turns of the pathway with single arcs interchange with turns made-up of two nested arcs. This constitutes the manner and sequence of the turns and is the basic information contained in the seed pattern. In the four seed patterns shown, the alignment of the turns may vary from circular (image a, image d) to longisch, vertical, slim (image b, image c). The shape of the arcs is adapted to the shape of the walls delimiting the pathway. However in all images it is a single turn in alternation with two nested turns.

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