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Posted in Design, Labyrinth, Typology, tagged Christmas Tree Labyrinth, christmas-labyrinth on December 24, 2018| 9 Comments »

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Posted in Design, Labyrinth, Typology, tagged circuit sequence, complementary, Concentric style, dual, level sequence, path sequence, Troy Town on October 7, 2018| 2 Comments »

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:

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:

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.

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:

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:

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:

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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Posted in Design, Labyrinth, Typology, tagged 11 circuit labyrinth, Chartres labyrinth, complementary, Concentric style, path sequence, seed pattern, self-dual on September 9, 2018| 2 Comments »

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 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.

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 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 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 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 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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Posted in Design, Labyrinth, Typology, tagged complementary, Concentric style, path sequence, Reims labyrinth, self-dual on August 12, 2018| 1 Comment »

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.

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 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.

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 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).

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 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.

**Related Posts**

- How to make a Classical (Minoan) Labyrinth from a Medieval Labyrinth, Part 1
- Un- / interesting Labyrinths
- The Six Very Interesting Labyrinths with 7 Circuits
- How to Draw Eight 7 Circuit Labyrinths
- How to make a Circular 7 Circuit Classical Labyrinth and Seven New (up to now unknown) Circular 7 Circuit Labyrinths

Posted in Design, Labyrinth, Typology, tagged Auxerre labyrinth, complementary, Concentric style, Cretan labyrinth, Cretan type, Minoan labyrinth, path sequence, self-dual on July 15, 2018| 2 Comments »

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.

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 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.

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 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).

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 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.

**Related Posts**

- The Complementaries to the Three Very Interesting Historical Labyrinths with 4 Arms and 11 Circuits
- The Dual Labyrinth
- The Complementary Labyrinth
- The Round Classical Labyrinth
- How to Draw Eight 7 Circuit Labyrinths
- How to Make a Circular Classical 7 Circuit Labyrinth and Seven New (up to now unknown) 7 Circuit Labyrinths
- The Heart of the Chartres Labyrinth is the Classical Labyrinth

Posted in Design, Labyrinth, Typology, tagged 3 circuit Chartres Labyrinth, 5 circuit Chartres labyrinth, 7 circuit Chartres Labyrinth, Knossos Labyrinth, Petit Chartres, Story Path Labyrinth on June 17, 2018| 2 Comments »

By halving a 7 circuit labyrinth in labyrinthine logic, as it was successful for the 5 circuit Chartres labyrinth.

The 4th circuit cuts the labyrinth in two parts. Then I receive an external (circuits 1 – 3) and an internal labyrinth (circuits 5 – 7). Both are identical in its path sequences. Even if the “barriers” are at different places.

The path sequence defines the type: 3-2-1-2-3-2-1-4. It is identical for both versions. This 3-2-1-4 reminds very much of the smallest possible labyrinth: the Knossos labyrinth (and of the meander).

If I leave out the barriers, I receive this labyrinth. This once again shows the quality of the Chartres labyrinth.

To make the layout more appealing, I can arrange the barriers in steady distances, in a way make a labyrinth with three arms.

This is the smallest possible version of a Chartres labyrinth. And there are just two barriers possible for it. Otherwise it does not work. Also three are not possible, but with four barriers it works aganin.

How should one name this type now? I suggest Petit Chartres because it is a sort of a basic element of the Chartres labyrinth. Still other names are conceivable.

It am speaking here about the type and not about the style. The petals in the middle and the lunations around the perimeter belong to the style.

To create this labyrinth is possible in a variety of ways, not necessarily in the manner described. For more read also the related posts below.

There is even a copyrighted labyrinth of this kind: The Story Path©. Warren Lynn and John Ridder of Paxworks have developed it and call the style “3-circuit-triune”. I do not know how they have found it.

**Related Posts**

- The Classical 3 Circuit Labyrinth Type Knossos
- A 3-Circuit 3-Axle Labyrinth
- The Heart of the Chartres Labyrinth is the Classical Labyrinth

**Further Link**

Posted in Design, Labyrinth, Typology, tagged 5 circuit Chartres labyrinth, Chartres labyrinth, Demi-Chartres on May 20, 2018| 10 Comments »

The Chartres labyrinth occurs in many variations. Here I speak of the 11 circuit Chartres labyrinth as a type. Some elements of the original labyrinth in the Cathedral at Chartres, such as the six petals in the middle and the lunations around the outermost perimeter, belong to the style Chartres.

For me the type Chartres exists above all in the layout of the paths. One goes in quickly (on the 5th circuit) and one quickly approaches the middle (6th and 11th circuit). Then follows the wandering through all quadrants. The access of the centre happens from completely outside (1st circuit) quickly about the 6th and 7th circuit into the centre.

Theoretically there are lot of possibilities to build similar types to the Chartres labyrinth. They can be found worldwide. However, the original Chartres labyrinth owns many special qualities which make it an extraordinary example among the Medieval labyrinths. Among others, it is self-dual and symmetrical.

Hence, the original can be divided in labyrinthine mathematics (11:2=5) in two equal labyrinths. I cut it into two parts, by omitting the 6th circuit. Thereby I get two new, yet identical 5 circuit labyrinths in a Chartres-like layout: I quickly reach the middle and finally enter the centre directly from the outermost circuit. The way in between shows the labyrinthine pendular movement, that Hermann Kern describes as characteristic for a labyrinth.

How should we now name this type of labyrinth? To me the name 5 circuit Chartres labyrinth seems properly to differentiate it from other 5 circuit Medieval labyrinths with another layout for the paths.

I would like to call it ** Demi-Chartres**.

Just now you may see a nice example for the practical realisation in Vienna on the Schwarzenbergplatz in the temporary plant labyrinth to the European Year of Cultural Heritage 2018:

**Related Posts**

- The Heart of the Chartres Labyrinths is the Classical Labyrinth
- A 7-Circuit Centred Medieval Labyrinth
- The Chartres Labyrinth is self-dual and symmetrical
- How to make a 5-circuit Classical Labyrinth from a 5-circuit Chartres Labyrinth

**Further Links**

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